Many-query simulation tasks like computational design and uncertainty quantification require repeated evaluations of high-fidelity numerical models. However, the computational cost and runtime of a single high-fidelity simulation can make many-query tasks computationally infeasible. In these scenarios, it is desirable to substitute the high-fidelity model with a computationally efficient surrogate to drastically decrease the computational costs of many-query tasks with minimal accuracy loss. This presentation details recent advancements in non-intrusive reduced-order modeling, a data-driven surrogate modeling paradigm, leveraging tools from kernel methods. Kernel methods provide a means for data-driven function approximation that requires minimal training cost, enables the enforcement of desired structural properties (e.g., smoothness, polynomial structure, closure terms), has robust mathematical foundations, and admits rigorous error estimates. We apply kernel methods in two key components of a reduced-order model (ROM): 1) nonlinear dimension reduction and 2) identification of ROM dynamics in the latent space. In the first part of the talk, we present kernel-based nonlinear manifolds, which generalizes recent work on quadratic manifolds (QMs). This class of approaches augments POD with a nonlinear correction term in the reconstruction map, which we learn via kernel methods. We compare our approach to POD and several recent QM approaches on data from several example problems, including a 2D boundary layer problem and a 3D solid mechanics problem. In the second part of the talk, we present kernel ROMs, a non-intrusive ROM approach also relying on kernel methods. While existing approaches approximate ROM dynamics by learning low-dimensional matrix operators, our approach approximates ROM dynamics using kernel methods. We show that our approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel and admits a computable a posteriori error estimate. The approach is demonstrated in several numerical experiments that include comparisons to operator inference.
Contributions – Spring Semester 2026
January 23, 2026
(Friday)
(Friday)
Speaker: Alejandro Diaz
Sandia National Labs
Title: Applications of kernel methods in data-driven reduced-order models
February 2, 2026
Speaker: Zoran Tomljanovic
University of Osijek
Title: Efficient Solution of Sequences of Parametrized Lyapunov Equations With Applications
Sequences of parametrized Lyapunov equations arise in many application settings. Moreover, the solutions of these equations are often intermediate steps in an overall procedure whose main goal is to compute the trace of a matrix that solves a structured Lyapunov equation. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a seed, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.
February 9, 2026
Speaker: Måns Andersson
Virginia Tech
Title: Anderson Acceleration for Matrix Splittings
We will talk about the design and development of an Anderson Accelerated Preconditioned Modified Hermitian and Skew-Hermitian Splitting (AA-PMHSS) method for solving complex-symmetric linear systems with application to electromagnetics problems, such as wave scattering and eddy currents. While it has been shown that the Anderson acceleration of real linear systems is essentially equivalent to GMRES, we show here that the formulation using Anderson acceleration leads to a more performant method. We show that the method can be generalised to many matrix splittings enabling a method that can be seen as a inexactly left-preconditioned GMRES.
February 23, 2026
Speaker: Mike Ackermann
Virginia Tech
Title: Enhancing the adaptive Antoulas-Anderson algorithm with iteratively reweighted least squares
Given a set of data on an interval, the adaptive Antoulas-Anderson (AAA) algorithm is a widely used method to efficiently construct highly accurate rational approximations to the data. Broadly, AAA fits the given data adaptively by interpolating a subset of the data and fitting the remaining data via a relaxed (linearized) rational least-squares fit. While AAA can often produce rational approximations accurate to any prescribed tolerance, these approximations may have degrees larger than what is actually required to meet the given tolerance. In this work, we consider the adaptive construction of interpolating rational approximations while aiming for the smallest feasible degree to satisfy a given error tolerance. To this end, we introduce refinement approaches to the linear least-squares step of the classical AAA algorithm that aim to minimize the true rational least-squares error with respect to the given data. Furthermore, we theoretically analyze the derived approaches in terms of the corresponding gradients from the resulting minimization problems and use these insights to propose a new greedy framework that ensures monotonic error convergence. Numerical examples from function approximation and model order reduction verify the effectiveness of the proposed algorithm to construct accurate rational approximations of small degrees.
Speaker: Jorge Reyes
Virginia Tech
Title: Parameter Recovery via Continuous Data Assimilation for the steady Navier Stokes Equations
Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover unknown physical parameters via continuous data assimilation (CDA). We study the steady incompressible Navier–Stokes equations with "unknown" viscosity and assume access only to partial velocity observations. By combining CDA with nonlinear solvers (e.g. Picard, Picard–Newton iteration), we show that the assimilation error scales linearly with the viscosity mismatch. This linear relationship leads to a simple and efficient parameter recovery algorithm based on a small number of CDA solves. Our analysis establishes well-posedness and convergence of the CDA system with inconsistent viscosity, as well as quadratic convergence of the CDA–Picard–Newton iteration. Numerical experiments for 3D driven cavity flows at moderate to high Reynolds numbers demonstrate rapid and high-accuracy recovery of the viscosity. The results illustrate how data assimilation can serve not only for state reconstruction, but also as a practical and mathematically grounded tool for parameter identification in nonlinear fluid models.
March 2, 2026
Speaker: Hengyi Shi
Virginia Tech
Title: An Efficient Low-Rank Solver for Nonlinear Convection–Diffusion Equations
Convection-diffusion equations arise in a wide range of scientific and engineering applications, including fluid transport, heat conduction, and the evolution of passive scalars. In many practical settings, especially in diffusion-dominated regimes, the solution often exhibits a low-rank structure. The structure has motivated the development of low-rank methods, which have shown success to solving linear convection-diffusion equations but fewer work has been done to the nonlinear diffusion case. In this work, We present a novel low rank method which can handle these nonlinear diffusion efficiently. The approach is based on the explicit-implicit-null (EIN) method, where a linear and easy invertible stiff term is added and then subtracted. The stability restriction related to explicit schemes is then relaxed and nonlinear solver required by fully implicit schemes is avoided simultaneously. The low-rank solver is based on the cross approximation, which uses CrossDEIM algorithms for nonlinear term evaluation. Finally, we show the effectiveness of the solver for a few test cases.
Speaker: Serkan Gugercin
Virginia Tech
Title: What to Interpolate for L2 Optimal Approximation
In this talk, we revisit the L2 optimal approximation problem through various formulations and applications, exploring its rich mathematical structure and diverse implications. We begin with the classical case where the optimal approximant is a rational function, highlighting how Hermite interpolation at specific reflected points emerges as the necessary condition for optimality. Building on this foundation, we consider extensions that introduce additional structure to rational approximations and relax certain restrictions, revealing new dimensions of the problem. Throughout, we demonstrate how Hermite interpolation at reflected points serves as a unifying theme across different domains and applications.
March 16, 2026
Speaker: Qifan Chen
Ohio State University
Title: Generalized Method of Lines: New Runge-Kutta Methods for Hyperbolic Conservation Laws
The conventional method-of-lines framework for time-dependent partial differential equations (PDEs) relies on method of lines: when using a multi-stage Runge-Kutta (RK) method for time integration, the exact same spatial operator is evaluated at every stage. In this talk, we challenge this default assumption and introduce a generalized method of lines for hyperbolic conservation laws. By allowing different spatial operators to be used at different RK stages, we open up a new design space for numerical schemes. We present novel RK methods that hybridize distinct Discontinuous Galerkin (DG) spatial discretizations within a single time step. We will discuss the theoretical formulation of these multi-operator schemes, explore various hybridization strategies, and highlight their specific advantages in terms of stability, accuracy, and efficiency.
Speaker: Edward Huynh
University of Texas at Austin
Title: On Minimax Optimal Transport and Differential Games
This talk is on a modification of the optimal transport (OT) problem. We introduce a novel minimax formulation of the quadratic (OT) problem, generalizing the classical Monge and Kantorovich frameworks. We define a Nash Equilibrium Transport Plan (NETP) as a solution to a zero-sum game between two teams, where the cost function depends on pairs of paths. The teams’ strategies are transport plans over path spaces connecting given source and target measures. We establish conditions for the existence and uniqueness of NETPs, leveraging compactness and semi-continuity arguments. This path-space formulation allows us to incorporate interaction terms, such as pursuit-evasion dynamics, connecting our static problem to differential games. Furthermore, we explore duality and characterizations of the Nash-Monge-Kantorovich transport map. We conclude this talk with a discussion on strategies and associated challenges for numerically solving these problems. This is a work in progress which may provide a foundation for analyzing competitive transport phenomena with applications in multi-agent systems and game theory.
March 23, 2026
Speaker: Peter DelMastro
Virginia Tech
Title: Completely-Positive and Trace Preserving Schemes for the Lindblad Equation using Tensor Train Compression
We propose a family of low-rank, completely positive and trace preserving (CPTP) schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further compressed using the tensor train (TT) format, also know as matrix product states (MPS). This representation fits naturally into our recent Kraus is King scheme for the Lindblad equation, whose underlying operations are arithmetic on the columns of the low-rank factor matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on the primary bottleneck: density matrix rank-truncation. We perform extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating open quantum systems with millions of degrees of freedom using only modest computational resources.
Speaker: Achintya Sunil
Virginia Tech
Title: Preconditioner updating for high-order DG FEM systems using Sparse Approximate Mappings from L-1 constrained sparsity pattern generator.
Preconditioning of high-order discrete Galerkin (DG) systems presents a challenge in applications such as topology optimization and PDE-based inverse problems. Computing an accurate preconditioner for the linear system in each step can make the solve process a computational bottleneck. The sequence of linear systems to solve can be similar enough to warrant a cheap preconditioner updating scheme. The sparse approximate mapping (SAM) is one such method that was introduced in Preconditioning Parametrized Linear Systems by Carr et al. Each column of the SAM is solved from a least-squares problem. The method is cheap as long as the underlying sparsity pattern is sufficiently sparse. However, as revealed in prior research, there is no intuitive choice for such a pattern. In this talk, we present a modification of the existing SAM computation algorithm that solves the lasso problem instead. We use the alternating direction method of multipliers (ADMM) algorithm to solve the lasso problem for each column. We demonstrate that the parameters of ADMM and the regularization parameter can be heuristically selected to compute patterns for SAMs that are significantly sparse.
March 30, 2026
Speaker: Stephen Becker
University of Colorado, Boulder
Title: Randomization methods for big-data
In this era of big-data, we must adapt our algorithms to handle large datasets. One obvious issue is that the number of floating-point operations (flops) increases as the input size increases, but there are many less obvious issues as well, such as the increased communication cost of moving data between different levels of computer memory. Randomization is increasingly being used to alleviate some of these issues, as those familiar with random mini-batch sampling in machine learning are well aware of. This talk goes into some specific examples of using randomization to improve algorithms. We focus on special classes of structured random dimensionality reduction, including the count sketch, tensorSketch, Kronecker fast Johnson-Lindenstrauss sketch, and pre-conditioned sampling. These randomized techniques can then be applied to speeding up the classical Lloyd's algorithm for K-means and for computing tensor decompositions, for example. If time permits, we will also show extensions to optimization, including a gradient-free method that uses random finite differences.
April 6, 2026
Speaker: Priyanka Sinha
Virginia Tech
Title: Multi-Agent Online Learning under Heterogeneity and Temporal Variations
Multi-agent online learning has emerged as a powerful framework for addressing large-scale problems with temporal variation and system heterogeneity. This talk presents two research efforts that apply these ideas in distinct settings. The first work studies clustered personalized federated learning under heterogeneous data distributions through a repeated multi-agent learning framework. Instead of enforcing static cluster assignments, we model personalization as a dynamic decision process and analyze its stability using no-regret learning dynamics. This perspective leads to stable grouping behavior and improved learning performance under evolving system conditions. In the second work, we study distributed reconstruction of a collection of images using an online ADMM framework in which consensus penalties are adapted over time rather than fixed, addressing limitations of static methods in handling spatial heterogeneity and partial observability. The ADMM iterations are interpreted as a dynamical system, with penalty weights acting as control variables updated via partial feedback to enable data-aware, pixel-level consensus. This online control formulation achieves near-optimal performance via dynamic regret guarantees while preserving convergence of weighted inexact ADMM.
Speaker: Jordan Jackson
Virginia Tech
Title: Convergence Analysis for Infinite GMRES
Many applications require the solution of large-scale linear systems that have nonlinear parameter dependence. The Infinite GMRES algorithm, developed by Jarlebring and Correnty in 2022, converts these finite-dimensional systems into infinite-dimensional systems with linear parameter dependence. This transformation involves a linearization process that results in a system with a special companion-like structure. Due to the shift-invariance property of Krylov subspaces, all choices of parameter give the same Krylov subspaces. Thus performing the Arnoldi process for one parameter enables the efficient construction of the GMRES iterates for all parameter values. The special structure of the right-hand-side vector of the infinite-dimensional system allows this Arnoldi process to be executed with finite-dimensional vectors, where the dimension grows with every iteration. As such, Infinite GMRES uses finite-length Krylov basis vectors to solve an infinite-dimensional system. The special structure of the infinite-dimensional system invites further investigation into the convergence behavior of the Infinite GMRES residual norm. We aim to understand how various components of this system, including the size and nature of the block matrices, the number of blocks, and the choice of parameter value influence convergence behavior. In this presentation, we will discuss preliminary convergence results based on the numerical range.
April 13, 2026
Speaker: Eric de Sturler
Virginia Tech
Title: Dynamically Updating Preconditioners with Solver Information
Preconditioners are essential for fast convergence of iterative linear solvers, but the computation of a good preconditioner can be quite expensive (for example, algebraic multigrid (AMG) preconditioners). In some cases, such as matrix-free solvers, computing a preconditioner is even more complicated, and some cheap and effective preconditioners cannot be used as the matrix is not available. This cost is even worse in simulations where the system matrix changes every timestep, nonlinear step, or optimization step (in PDE-constrained optimization). Even if we can compute a good preconditioner, we often still see some convergence issues due to various complicating factors, PDE coefficients that vary drastically in space or due to strong nonlinearity, highly deformed meshes, and poor meshes due to geometric complexity. On the other hand, if we solve the same system many times (time-stepping with a fixed time step) or the system changes slowly, we have substantial opportunity to learn properties of the system from these solves and we can dynamically update the preconditioner or learn a better preconditioner. I will discuss simple strategies to update the preconditioner dynamically using information that is derived during the solve and provide methods for analysis.
Speaker: Kun Huang
Virginia Tech
Title: A fast scheme for the Boltzmann equation based on low-rank approximation
The Boltzmann equation can be lifted to a higher-dimensional linear master equation, whose solution can be projected back into a lower-dimensional probability distribution. We will show that the consecutive lifting-projection flow is indeed a nice approximation of the original Boltzmann flow. To address the issue of high-dimensionality, we adopt the tensor train format and compute low-rank solutions with a greedy cross approximation algorithm. Our scheme is consequently faster than the spectral method for the variable hard sphere (VHS) model. Numerical tests are performed to illustrate the efficiency and accuracy.
April 20, 2026
Speaker: Li Zhu
University of Minnesota, Duluth
Title: Finite Element Methods for Surface Plasmon Polaritons on Complex Graphene Geometries
Graphene’s unique electromagnetic properties offer unprecedented control over light at the nanoscale. However, simulating wave propagation on complex graphene sheets requires robust numerical frameworks and rigorous mathematical analysis. Starting from a foundation in Maxwell’s equations, this talk details the development of advanced numerical algorithms tailored for graphene-based surface plasmon polaritons. We will examine the stability and convergence of these methods when applied to complex 2D manifolds. The presentation will conclude with several numerical experiments showcasing the propagation of waves on graphene sheets.
April 27, 2026
Speaker: Hayden Ringer
Virginia Tech
Title: Towards a Black-Box Laplacian MPS Eigensolver
The eigenproblem for the Laplace operator on planar shapes is well-known, and the theory is well understood. However, numerical methods for accurately computing the eigenvalues (and eigenfunctions) of a given planar shape have proven difficult to develop. The Method of Particular Solutions (MPS), which was "revived" by Trefethen and Betcke in 2005, provides the highest accuracy among existing approaches. The drawback is that MPS requires significant shape-specific tuning and basis selection, which limits its direct applicability to problems which require computing the eigenvalues of many different regions (such as in shape optimization or shape inference problems). Here, we present our work on developing a "black-box" MPS eigensolver, which is designed to automatically manage the tuning and basis-selection considerations, enabling its use as the solver within an "inner loop" of a larger problem.
Speaker: Dan Folescu
Virginia Tech
Title: Data-Driven Modal Truncation
Modal truncation has long been a fundamental approach of model order reduction: to systematically eliminate eigenmodes of a dynamical system that contribute little to the modeling behavior over a given time/frequency range. Typically, this procedure requires the ability to access and utilize intrusive state-space information about the underlying full-order system, which can be infeasible for larger state dimensions or impossible if the full-order system is not known in closed form. Data-driven approaches like proper orthogonal decomposition address these concerns by identifying empirical modes of the system derived from multiple time-domain trajectories of the full-order model, whereafter model reduction can proceed as in the state-space formulation. In this work, we show how one can truncate empirical eigenmodes with only frequency-domain samples of a system's transfer function by utilizing contour integral methods. After reviewing the necessary theoretical foundations, we will discuss how contour integration and the pole-residue decomposition of an underlying system can be combined to identify all eigenvalues within an arbitrary domain in the complex plane. This tool allows us to perform modal truncation, or identify a stable-unstable decomposition of the underlying system. Through the use of a key result of Keldysh, contour integral methods can also identify eigenvalues of analytic nonlinear matrix-valued functions. We will show numerical experiments illustrating how our data-driven modal truncation method performs on application problems and compare with other state-of-the-art methods.
May 4, 2026
Speaker: Andreas Granath
Umeå University
Title: CutIGA: a high-order spline-based method for the wave equation
When solving wave problems, it is desirable to use numerical schemes that offer high-order accuracy while maintaining geometric flexibility. One such approach is the cut finite element method (CutFEM), an unfitted method that has primarily been applied to time-independent problems in mechanics, but has also been extended to the wave equation in a series of works by Sticko and Kreiss. However, their approach utilizes a Lagrange basis, which leads to suboptimal conditioning of the mass matrix and a strongly order-dependent CFL condition. In this talk, we provide a brief introduction to CutFEM and present ongoing work on a spline-based discretization of the wave equation in second-order form. We demonstrate a clear improvement in the conditioning of the mass matrix, along with a nearly order-independent CFL condition. We conclude by discussing open questions regarding the existence of optimal stabilization strategies with respect to conditioning.
Speaker: Mirjeta Pasha
Virginia Tech
Title: Sketching the Generalized Krylov Subspace Method for Large-Scale Regularized Inverse Problems
The generalized Krylov subspace (GKS) method is an effective projection technique for solving large-scale Tikhonov regularization problems with a general regularization matrix $\Phi$. However, as the subspace expands, the method faces computational bottlenecks from QR factorizations of the tall projected matrices AV k and $\Phi$V k, and from the full reorthogonalization of the basis. In this talk, we present sGKS, a sketched variant that applies randomized dimensionality reduction to each of these bottlenecks. The thin QR factorizations are performed on sketched counterparts of much smaller dimension and maintained incrementally via rank-one updates. Reorthogonalization is carried out in the sketched inner-product space, reducing per-iteration costs while preserving approximation quality as guaranteed by the oblivious subspace embedding property. We provide an analysis showing near-optimality of the sketched projected Tikhonov solution and demonstrate the method on image deblurring, X-ray computerized tomography, and seismic travel-time tomography, where sGKS matches the reconstruction quality of standard GKS at reduced computational cost. This is joint work with Davide Palitta (university of Bologna).
May 11, 2026
(4pm, DDS 366)
(4pm, DDS 366)
Speaker: Malena Sabaté Landman
University of Bath
Title: Inexactness and Krylov Subspace Regularization
Linear discrete inverse problems arise in many areas of science and engineering, from medical imaging and geophysics to atmospheric modelling. Their numerical solution often relies on iterative algorithms, particularly Krylov subspace methods, that can efficiently handle large-scale, ill-posed systems. In many practical settings, however, exact computations of matrix–vector products, preconditioners, or right-hand sides are either infeasible or unnecessary, leading to inexact iterations. This talk explores the interplay between inexactness and the regularizing behaviour of Krylov subspace methods for inverse problems.
Contributions – Fall Semester 2025
September 8, 2025
Speaker: Mirjeta Pasha
Virginia Tech
Title: Priorconditioned Sparsity-Promoting Projection Methods for Deterministic and Bayesian Linear Inverse Problems
High-quality reconstructions of signals and images with sharp edges are needed in a wide range of applications. To overcome the large dimensionality of the parameter space and the complexity of the regularization functional, sparisty-promoting techniques for both deterministic and hierarchical Bayesian regularization rely on solving a sequence of high-dimensional iteratively reweighted least squares (IRLS) problems on a lower-dimensional subspace. Generalized Krylov subspace (GKS) methods are a particularly potent class of hybrid Krylov schemes that efficiently solve sequences of IRLS problems by projecting large-scale problems into a relatively small subspace and successively enlarging it. We refer to methods that promote sparsity and use GKS as S-GKS. A disadvantage of S-GKS methods is their slow convergence. In this talk, we discuss techniques that improve the convergence of S-GKS methods by combining them with priorconditioning, which we refer to as PS-GKS. Specifically, integrating the PS-GKS method into the IAS algorithm allows us to automatically select the shape/rate parameter of the involved generalized gamma hyper-prior, which is often fine-tuned otherwise. Furthermore, we investigate variations of the proposed PS-GKS method, including restarting and recycling. These respectively leverage restarted and recycled subspaces to overcome situations when memory limitations of storing the basis vectors are a concern. We provide a thorough theoretical analysis showing the benefits of priorconditioning for sparsity-promoting inverse problems. Numerical experiment are used to illustrate that the proposed PS-GKS method and its variants are competitive with or outperform other existing hybrid Krylov methods. This is joint work with Jonathan Lindbloom, Jan Glaubitz, and Youssef Marzouk.
Speaker: Steffen Werner
Virginia Tech
Title: Data-driven Balanced Truncation for Mechanical Systems
Learning dynamical systems from data has emerged as a pivotal area of research, bridging the realms of mathematics, engineering, and data science. Dynamical systems, which describe how states evolve over time based on underlying mathematical relations, are fundamental for understanding a wide range of time-dependent phenomena. For the use of these systems in practical applications like predictive simulations and control, high modeling accuracy as well as interpretability and explainability are essential. All of these desired properties are strongly tied to the internal mathematical structure of the dynamical systems. For example, in the modeling of mechanical or electro-mechanical processes, typically systems with second-order time derivatives. In this case, data are usually given in the frequency domain as samples of the corresponding transfer function. To learn such structured systems from frequency domain data, we develop in this work a data-driven second-order balanced truncation approach. This method allows the construction of low-dimensional second-order systems with generalized proportional damping structure by constructing appropriate Loewner-like matrices. The damping coefficients can be cheaply inferred from given data using gradient-based optimization. Several numerical examples demonstrate the effectiveness of the proposed method.
September 15, 2025
Speaker: Jorge Reyes
Virginia Tech
Title: The EMAC formulation for fluid models and longer time accuracy
This talk will focus on the energy, linear momentum, and angular momentum conserving (EMAC) scheme. We show how EMAC provides improved long-time accuracy for fluid models by leveraging its built-in conservation properties, which mitigate error growth in extended simulations. In addition to standard flow problems, we will discuss its application in the context of Large Eddy Simulation (LES), with particular attention to the Ladyzhenskaya model, and highlight how EMAC can enhance the reliability of turbulence models.
Speaker: Gregory Janes
Virginia Tech
Title: Computation and analysis of unequal counter-rotating vortex pair interactions
The influence of upstream inflow boundary perturbations on the evolution of counter-rotating vortex pairs with unequal strength and size is investigated using implicit large-eddy simulations. Lamb-Oseen vortex pairs are generated at a time-varying inflow boundary condition with parameters based on published experimental work. Previous research efforts have traditionally focused on the stronger (primary) vortex, as it is associated with the lowest average minimum pressure. However, experimental evidence indicate rapid, localized pressure drops in the weaker (secondary) vortex first, motivating the present investigation into the dynamics and mechanisms leading to this phenomenon. Three sinusoidal perturbations are applied to both the primary and secondary vortices at the inflow boundary to determine their effect on long-wavelength instability growth and their subsequent dynamics. The results demonstrate a direct correlation between the initial perturbation amplitude and the streamwise location of the precipitous pressure drop in the secondary vortex. The larger inflow perturbations are shown to accelerate the growth of the Crow instability, leading to accelerated pressure reduction in the secondary vortex closer to the inflow boundary. We seek to quantify the stretching kinematics and evolution along the entire length of the vortex filament, using an adaptive centerline interpolation method informed by the Liutex vector. Furthermore, current investigations look to employ data-driven reduced models to extract the instabilities present in the individual vortex filaments and for simplified and reduced computation.
September 22, 2025
Speaker: JC Hartman
Virginia Tech
Title: Parameter Estimation for Delay Differential Equations: A New Galerkin Approximation Approach
Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. We will discuss a method for determining the unknown model parameters, discrete delays, and integral kernels for these equations from data through a Galerkin approximation approach based on a type of orthogonal polynomials due to T. H. Koornwinder. Our approach reduces the problem into the estimation of some scalar parameters in the resulting Galerkin ordinary differential equations. Thanks to the analytical nature of the approximation, this latter estimation problem can be handled efficiently even in the challenging cases of distributed delays. We will provide an overview of the proposed method along with its application to some examples that exhibit chaotic dynamics.
Speaker: Amit Rotem
Virginia Tech
Title: An Adaptive Low-Rank Method for the Navier Stokes Equations
For many time-dependent problems in fluid dynamics, snapshots of the solution are often well approximated by low rank representations. Such approximations are natural when the dynamics are characterized by coherent structures, but such approximations are also (eventually) valid for chaotic flows owing to the effects of diffusion. The low rank property of these dynamics motivates the use of reduced order models and compression techniques. We present a novel low rank method for approximately solving the Navier Stokes equations that dynamically updates the rank of the solution and never evaluates the solution on the entire grid. The method is based on a pseudo-spectral discretization in space and Runge-Kutta time-stepping. The incompressibility condition is preserved by solving an elliptic problem for which we present a direct low rank solver based on the fast Fourier transform and the CrossDEIM algorithm. The nonlinear terms in the Navier Stokes equations are also approximated with CrossDEIM. Finally, we show the effectiveness of the solver for a few classical test cases.
September 29, 2025
Speaker: Massimiliano Lupo Pasini
Oak Ridge National Laboratory
Title: Two-Level Sketching Alternating Anderson acceleration for Complex Physics Applications
We develop a progression of methods for accelerating fixed-point iterations by leveraging approximate and reduced formulations of Anderson acceleration (AA). Firstly, we establish rigorous theoretical error bounds that enable approximate calculations within AA when solving linear problems. We prove that, as long as the approximate computations satisfy these bounds, the convergence guarantees of AA are preserved while computational cost can be reduced. Guided by these results, we introduce computable heuristic quantities that automate accuracy tuning in practice, ensuring monotonic residual reduction and convergence within prescribed tolerances. Motivated by this analysis, we propose a reduced variant of AA in which the least-squares system underlying Anderson mixing is projected onto an adaptively chosen subspace. Secondly, we extend the paradigm to a novel two-level sketching Alternating Anderson–Picard (AAP) method for challenging single- and multi-physics PDE simulations. Our extension combines a static, physics-informed projection (e.g., via Schur-complement reduction to the most informative field) with a dynamic, algebraic sketching stage derived from backward stability analysis under Lipschitz continuity. To balance stability and efficiency, we design inexpensive error estimators and cache-aware randomized sketching strategies that minimize memory footprint and computational overhead. Implemented in Julia, our two-level sketching AAP achieves up to 50% reductions in time-to-solution over standard AA, while maintaining convergence robustness, on benchmark problems including Stokes, p-Laplacian, Bidomain, and Navier–Stokes systems. Together, these contributions provide a rigorous theoretical basis and scalable algorithmic extensions that make Anderson-type accelerations more efficient, adaptive, and applicable across a wide range of scientific computing problems.
October 6, 2025
Speaker: Tom Werner
TU Braunschweig
Title: Preconditioning techniques for almost symmetric operator equations in Newton-NEPv
This talk aims at presenting different techniques to accelerate the convergence of Krylov subspace solvers for the correction equation in Newton’s method for nonlinear eigenvector problems (NEPv). While this operator equation is generally unsymmetric, its structure allows a splitting into symmetric and nonsymmetric parts which can be used to derive efficient direct and iterative preconditioners. We will show how to obtain that splitting of the update equation, discuss a selection of preconditioning techniques present in the literature and adapt them to the particular problem. Some numerical comparisons on a discrete Kohn-Sham example complete the talk.
Speaker: Paul Cazeaux
Virginia Tech
Title: Atomistic-to-Continuum modeling and Dirac operators in topological materials
We present a theoretical framework with numerical validation for upscaling atomic-level discrete tight-binding models to effective Dirac partial differential equations governing macroscopic electron propagation in locally periodic topological materials. Applications include the Haldane model, commensurate as well as twisted multi-layer 2D materials such as graphene. We briefly discuss the spectral theory for obtaining multi-band macroscopic models through Taylor expansion of the pseudo-differential operator symbol near band structure degeneracy points. We then propose efficient numerical discretization strategies for the nonstandard effective Dirac equations and confirm theoretical predictions for both bulk states and topological edge states.
October 13, 2025
Speaker: Ionut Farcas
Virginia Tech
Title: Fast prediction of plasma instabilities with sparse-grid-accelerated optimized dynamic mode decomposition
Parametric data-driven reduced-order models (ROMs) that embed dependencies in a large number of input parameters are crucial for enabling many-query tasks in large-scale problems. These tasks, including design optimization, control, and uncertainty quantification, are essential for developing digital twins in real-world applications. However, standard training data generation methods are computationally prohibitive due to the curse of dimensionality, as their cost scales exponentially with the number of inputs. This presentation focuses on efficient training of parametric data-driven ROMs using sparse grid interpolation with Leja points, specifically targeting scenarios with higher-dimensional input parameter spaces. Leja points are nested and exhibit slow growth, resulting in sparse grids with low cardinality in low-to-medium dimensional settings, making them ideal for large-scale, computationally expensive problems. Focusing on gyrokinetic simulations of plasma micro-instabilities in fusion experiments as a representative real-world application, we construct parametric ROMs for the full 5D gyrokinetic distribution function via optimized dynamic mode decomposition (optDMD) and sparse grids based on Leja points. For a real-world electron temperature gradient driven micro-instability simulation featuring six input parameters, we demonstrate that an accurate parametric optDMD ROM can be constructed at a cost of only 28 high-fidelity gyrokinetic simulations thanks to sparse grids.
October 20, 2025
Speaker: Mike Ackermann
Virginia Tech
Title: Towards a reduced order model for hyperbolic partial differential equations
In many scientific tasks such as partial differential equation (PDE) constrained optimization, repeated simulation of complex, parameter dependent PDEs is required. Simulation of the PDE for many different parameters is often the dominant cost in these procedures. The goal of model order reduction (MOR) is to produce a reduced order model (ROM) which approximates the PDE solution at a much lower computational cost. MOR is typically accomplished by projecting the PDE dynamics onto a low dimensional linear subspace which well approximates the PDE solution manifold. Hyperbolic PDEs are an important subclass of PDEs which typically arise from modeling conservation laws (i.e., conservation of mass or momentum). Standard MOR techniques are often not effective for producing ROMs of hyperbolic PDEs as their solution manifolds cannot be well approximated by linear subspaces of small dimension. In this talk, we introduce a MOR technique specifically designed for parameter dependent hyperbolic PDEs. Our method first computes an intermediate ROM solution via a finite volume approximation, then constrains the ROM solution to a spatially shifted dictionary of precomputed PDE solutions. The ROM approximation is taken to be the best approximation from the shifted dictionary to the intermediate ROM solution in the L1 norm. Throughout the talk, we use Burgers' equation to motivate and demonstrate our method.
Speaker: Rodrigo Figueroa Justiniano
Virginia Tech
Title: Tensor-Structured Sketching for Very High-Dimensional Problems
Very high-dimensional data with multilinear structure – such as tensors arising from discretized PDEs, quantum chemistry, and multiway feature maps – render standard sketching tools impractical due to infeasible storage and computational requirements. In this talk, we review structured random projections and tensor-aware sketches and introduce a unified framework that fuses separable (Khatri-Rao) projections with tensor train-compatible compression. The resulting sketches are memory-sparse, fast to apply, and come with numerical guarantees of approximate subspace-geometry preservation, along with theoretical results that circumvent exponential scaling in the dimension while incurring minimal and controllable additional error.
October 27, 2025
Speaker: Ning Wan
Virginia Tech
Title: nlKrylov: A Unified Framework for Nonlinear GCR-type Krylov Subspace Methods
In this talk, we introduce a unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations. Building on classical GCR-type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.
Speaker: Rudi Smith
Virginia Tech
Title: A Tangential ADI Method for the Efficient Solution of Large-scale Indefinite Lyapunov Equations
The Low-Rank Alternating Direction Implicit (LR-ADI) method is an effective approach to solve large-scale Lyapunov equations by iteratively assembling the solution factors using the right-hand side terms. In the case of medium and high-rank right-hand sides, the ADI naturally suffers from increasing computational costs due to the block-wise growth of the constructed solution factors. We present a tangential ADI framework for indefinite Lyapunov equations that overcomes this limitation by replacing full block updates with a sequence of memory-efficient rank-1 updates. At each step, an adaptive shift and a corresponding tangential direction are selected to column-wise construct the solution's low-rank factorization. The implicit residual formulation of the iteration avoids forming the dense full-dimensional solution matrix making the approach usable for Lyapunov equations with large-scale sparse coefficient matrices. By mitigating the rapid growth of the solution rank, our tangential approach offers a more scalable and computationally efficient algorithm for practical applications.
November 3, 2025
Speaker: Timo Heister
Clemson University
Title: Adaptive Finite Element Simulations of Phase-field Fracture Problems
We present an efficient and scalable Finite Element scheme for quasi-static fracture problems. The crack is discretized using a phase-field approach, which allows merging and joining of cracks. An active active set strategy is used to solve the constrained minimization problem. The developed method supports large scale parallel computations and a novel adaptive mesh refinement strategy. Various benchmark problems are shown that demonstrate method.
November 10, 2025
Speaker: Aryeh Keating
Virginia Tech
Title: A Scalable Sequential Framework for Dynamic X-Ray CT via Expectation-Maximization with Optical Flow
Large-scale dynamic inverse problems are typically ill-posed and suffer from complexity of the model constraints and large dimensionality of the parameters. A common approach to overcome ill-posedness is through regularization that aims to add constraints to parameters for robust solves. In this talk, we describe a memory efficient method for reconstructing a dynamic sequence of undersampled images from computerized tomography data that requires minimal hyperparameter tuning. We begin by introducing a prior-based dimension reduction Kalman filter with smoothing. While this method is suitable for image reconstruction, the challenge lies in applications when initializing covariance and state transition model hyperparameters without prior knowledge an under the constraint that we may only do one pass of the method. To address this problem under such constraints, we utilize a regularized Optical Flow and Expectation Maximization methods alongside the Kalman filter for state transition model and error covariance parameter estimations, respectively. We also illustrate the effectiveness of the prescribed approach through numerical experiments comparing accuracy, memory, and time, from a wide range of limited angle and single shot computerized tomography.
Speaker: Cankat Tilki
Virginia Tech
Title: H2 Model Reduction for Infinite-Dimensional Systems on Banach Spaces
We extend the H2 model reduction to time-invariant systems on Banach spaces governed by an abstract Cauchy problem with output. Specifically, we assume that the abstract Cauchy problem produces an exponentially stable, strongly continuous semigroup, and the control and observation operators are bounded. In this setting, we define an operator-valued transfer function on the right half-plane and an H2 norm suitable for these systems. Our main result shows that the classical first-order H2 optimality conditions persist in this Banach-space framework: an H2-optimal reduced model satisfies the usual tangential (Hermite) interpolation conditions at the reflections of its reduced poles.
November 17, 2025
Speaker: Eric de Sturler
Virginia Tech
Title: An Efficient Streaming Type Algorithm for the General Gauss-Markov Linear Model
We discuss new methods for solving very large, generalized regression problems, including those were data comes in continually, as in the case of satellite data, so that the 'full' problem is never available. Our methods must also be able to handle very large general observation and prior covariance matrices. This requires iterative methods to compute solution search spaces and approximate solutions using variants of the CS decomposition that are easily updated without having to manipulate large, generally dense, orthogonal factors. For problems where data comes in continually, we need methods to efficiently and effectively combine current solution data and search spaces with incoming data to continually update solutions. This is joint work with Chris Beattie.
Speaker: Achintya Sunil
Virginia Tech
Title: Solving Alternating Current Optimal Power Flow (ACOPF) systems on Heterogeneous Hardware: Background & Heuristics
Solving power flow systems on heterogeneous hardware present various implementation challenges, owing to the irregular and sparse structure of these systems. In order to exploit the performance gains of hardware accelerators, it is necessary to devise algorithmic level changes to existing sparse-direct libraries. One such change is to maximize the reuse of the pivoting sequence across systems, a method called refactorization. Unfortunately, this arrives at the cost of a decline in the solution quality for subsequent systems. While iterative refinement with the LU factors can be used to compute an update to the solutions and improve their accuracy, the more serious problem of division by near zero pivot is difficult to predict. In this talk, we present pre-processing steps that produce a pivoting sequence with greater reusability and methods to avoid the costly symbolic factorization altogether. We begin with an introduction to the formulation of these ACOPF systems, discuss the development of the current method for solving these systems and how we improved on it. We conclude with the potential for sparse approximate mappings to recycle the LU factors across systems, opening the door to iterative methods.
December 1, 2025
Speaker: Deven Mhadgut
Virginia Tech
Title: Estimating Deployment Loads for Bistable Spacecraft Booms using System Identification Techniques
Bistable tape springs are used in space for their ability to self-deploy using stored strain energy. However, their uncontrolled deployment can induce mechanical shocks that may damage sensitive satellite components and disrupt attitude control. In the past, researchers have focused on mitigating these shocks through damping materials and active control techniques. Accurate prediction of deployment-induced loads from dynamic response measurements, however, remains a challenge. This can be done by solving the inverse problem for estimating these loads, which would enable the characterization of forces and moments that are otherwise difficult to measure directly. By quantifying these loads, engineers can optimize damping and isolation at the sensitive locations. This work presents a data-driven framework where a Frequency Response Function (FRF) model of a composite tape spring is developed using force and laser vibrometer velocity measurements from an electrodynamic shaker excited by a reference signal. Input loads are inversely calculated from measured responses across various signal types (sine, triangle, square and random), and the transfer function is processed using Vector Fitting. A parametric load model is then generated using the P-AAA algorithm and cross-validated for accuracy. Finally, the entire process is validated: the force inversely calculated from an actual deployment test's tip velocity is applied back to the boom via the shaker, and the resulting tip velocity is measured and compared to the original deployment velocity, thereby enabling reliable and non-intrusive dynamic load characterization.
Speaker: Lucas Onisk
Emory University
Title: Mixed-to-Low Precision Iterative Methods for Linear Inverse Problems
Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. The matrices that define these problems are typically severely ill-conditioned and may be rank deficient. Because of this, regularization is often needed to stem the effect of perturbations caused by error in the available data. In this talk we consider the solution of the regularized least-squares problem using both mixed precision and Krylov subspace projection techniques. We utilize a filter factor analysis to investigate the regularizing behavior of the proposed iterative methods.
December 8, 2025
Speaker: Elizaveta Rebrova
Princeton University
Title: On implicit preconditioning and other benefits of randomized iterative linear solvers
Sketch-and-project is a unifying framework for many well-known projective iterative methods for solving linear systems, as well as for extensions to certain nonlinear problems. It encompasses popular algorithms such as Randomized Kaczmarz and Gauss-Seidel, including their block variants. In this talk, I will present our recent work on sharper ways to quantify the convergence of sketch-and-project methods. We show that, when properly designed, these methods quickly capture the dominant (and potentially outlying) singular directions of the linear system, thereby acting as an implicit preconditioner. This insight leads to particularly efficient solvers for approximately low-rank systems, which commonly arise in machine learning (e.g., kernel matrices and signal-plus-noise models). I will also discuss how to adapt these solvers to exploit problem structure for computational efficiency and to improve robustness to structured sparse noise.
Contributions – Spring Semester 2025
February 3, 2025
Speaker: Fan Tian
Tufts University
Title: Tensor BM-Decomposition for Compression and Analysis of Video Data
In the era of big data, high dimensional data has become more prevalent. Tensor methods, as essential tools for multidimensional data analysis, have gained significant attention for addressing challenges with analyzing, storing, and computing these complex data arrays. In this talk, we will discuss a recent tensor decomposition technique known as the Bhattacharya-Mesner decomposition (BMD). For tensors 𝓐, 𝓑, and 𝓒 of dimensions m×1×n, m×p×1, and 1×p×n respectively, their BM-product yields a third-order tensor of size m×p×n with a BM-rank of 1 (Mesner and Bhattacharya, 1990). Consequently, representing an arbitrary m×p×n third-order tensor as a sum of a few BM-rank 1 terms provides a compressed form of the tensor. We will discuss the development of an efficient algorithm to compute the tensor BMD and its applications in image and video processing.
February 17, 2025
Speaker: Tongtong Li
UMBC
Title: A structurally informed data assimilation approach for discontinuous state variables
Data assimilation is a scientific process that combines available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. In this talk, we focus on the design of a new structurally informed prior that exploits statistical information from the simulated state variables. In particular, based on the second moment information of the state variable gradient, we construct a new weighting matrix for the numerical simulation contribution in the data assimilation objective function. This replaces the typical prior covariance matrix used for this purpose. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using standard ensemble-based Kalman filtering on shallow water equations.
February 24, 2025
Speaker: Clarence Rowley
Princeton University
Title: Covariance balancing model reduction
Data-driven reduced-order models often struggle with high-dimensional nonlinear systems sensitive to low-variance coordinates, which are typically truncated. To address this, we use ideas from balanced truncation and active subspaces to identify low-dimensional coordinate systems that balance adjoint-based sensitivity information with state variance along trajectories. Our method, analogous to balanced truncation, replaces system Gramians with state and adjoint-based gradient covariance matrices, maintaining key transformation laws. We also present a further refinement whereby the resulting oblique projection is iteratively optimized to minimize forecasting error. We demonstrate and compare these techniques with other methods on a challenging toy problem and a nonlinear axisymmetric jet flow simulation with 100,000 state variables.
Speaker: Alberto Padovan
UIUC
Title: Non-intrusive model reduction for systems with transient growth
In the previous presentation, we saw that accurate reduced-order models can be computed with knowledge of the sensitivity information (or adjoint information) of the original nonlinear system. However, this information is available only via intrusive access to the governing equations of the original system, and it may therefore be unavailable when using "black-box" solvers (e.g., commercial or legacy codes) whose source code cannot be accessed by a user. In order to address this problem, we develop a non-intrusive formulation where the components of the reduced-order model (i.e., the encoder, the decoder and the latent-space dynamics) simultaneously optimized against high-fidelity trajectories. In particular, we shall see that optimizing the encoder encourages the method to extract some of the sensitivity information from the available data, and this ultimately leads to accurate (yet purely data-driven) models. We demonstrate and compare this method with other existing formulations on the complex Ginzburg-Landau equation and on the incompressible flow inside a lid-driven cavity.
March 3, 2025
Speaker: Hayden Ringer
Virginia Tech
Title: Listening Carefully to Drums of Unknown Shape
The inverse Dirichlet Laplace eigenvalue problem, put simply, is: "Can a planar shape be determined by its Dirichlet eigenvalues?" This problem is known to be ill-posed. In this talk, we present advances in our ongoing project to develop computational and Bayesian-statistical techniques for recovering unknown shapes, based on their "spectral data." In particular, we will demonstrate the effectiveness of these approaches on small benchmark problems. We will discuss: (1) improvements we have made to our forward eigensolver, (2) a parameterization scheme which "quotients-out" the continuous symmetries of the eigenproblem, (3) an engineered loss/likelihood function for evaluating "spectral closeness", and (4) the construction of a regularization term/prior distribution which enforces non-degeneracy of the parameterized family of shapes.
Speaker: Mark Embree
Virginia Tech
Title: Finding Favorable Inner Products for GMRES Analysis
The GMRES algorithm is a popular method for solving large scale linear systems, Ax = b, when A is nonsymmetric. Despite the method's popularity, its convergence behavior can be quite subtle. Suppose the eigenvalues of A fall in the right half-plane. When the field of values of A, denoted W(A), does not contain the origin, the Crouzeix-Palencia theorem leads to simple bounds on the norm of the residual. However, this bound does not apply when W(A) contains the origin. We propose a Lyapunov equation framework for constructing inner products in which W(A) does not contain the origin, allowing for convergent GMRES bounds at the cost of a multiplicative constant (that accounts for the change of inner product). This framework, inspired by Sergei Godunov's bounds on the matrix exponential, modulates the role the departure from normality plays on the convergence bound.
March 17, 2025
Speaker: Paul Schwerdtner
New York University
Title: Dimensionality Reduction with Quadratic Manifolds for Large-Scale, Streaming, and Sparse Data Settings
Dimensionality reduction determines low-order representations of complex trajectories in high-dimensional spaces, which are crucial in model order reduction for accelerating simulations and reconstructing vector fields from sparse measurements. While linear subspace approximations are efficient to compute, they often fail to capture the complexity of transport-dominated phenomena. To address this, we are building on quadratic manifolds, which enhance linear subspaces with quadratic correction terms, significantly improving approximation accuracy. Our approach constructs these manifolds by systematically combining leading and later principal components of training data to form a basis that maximizes the benefit of quadratic corrections. We first demonstrate the scalability and effectiveness of our manifold construction on large-scale fluid dynamics datasets. Then, we present a novel empirical sparse regression method that exploits quadratic manifolds to reconstruct field variables and data vectors from sparse samples. This approach reduces approximation errors by several orders of magnitude compared to widely used methods such as gappy proper orthogonal decomposition and discrete empirical interpolation, which rely on linear approximations. Finally, we demonstrate the effectiveness of quadratic manifolds for sparse reconstruction through several numerical examples, including applications in Vlasov systems and a rotating detonation rocket engine.
March 24, 2025
Speaker: Mattia Manucci
University of Stuttgart
Title: Numerical linear algebra for model order reduction of dynamical systems
During the past three decades, model order reduction (MOR) has become an established tool to reduce computational cost by obtaining high-fidelity solutions of (partial) differential-algebraic equations required in parameter studies, controller design, and optimization. The use of MOR allows for real-time simulation on simple devices that can be used for optimal control, shape design, and medical diagnosis. In this talk we show how tailor-made numerical linear algebra techniques can effectively tackle the challenge of reducing the complexity of parametric, time-variant partial differential equations, thus allowing for accurate and efficient MOR. We start introducing contour integration methods (CIMs), that are, time integrators based on the numerical approximation of the inverse Laplace transform that directly evaluate the solution of a linear dynamical system in a prescribed time T or in a continuous time window. We highlight the advantages and critical aspects of those methods and show how they are naturally well-suited for constructing nonlinear reduced-order models, even in the parametric setting. We conclude presenting an extension of the CIM framework to systems of differential-algebraic equations.
March 31, 2025
Speaker: Sam Otto
Cornell University
Title: Operator Learning Without the Adjoint
Operator learning is a rapidly growing field of machine learning relevant to modeling continuum mechanics. It uses data from experiments or numerical simulations to learn operators between spaces of functions, often consisting of input and output spatial fields such as force and displacement on a structure. This talk will discuss how much data is needed to construct accurate approximations of operators, focusing primarily on learning finite-rank approximations of solution operators for non-self-adjoint uniformly elliptic PDEs. Low-rank approximation of large matrices using matrix-vector product queries generally requires querying the matrix transpose (adjoint) or otherwise querying the matrix as many times as columns in the matrix. In contrast, we show that finite-rank approximations of solution operators for uniformly elliptic PDEs can be learned without querying the adjoint solution operator, and we provide sample complexity estimates for learning using only forward queries. This is fortunate because the adjoint is usually inaccessible in experimental settings. The key ingredient is the additional structure imposed on the operator by elliptic regularity. These results illuminate some of the fundamental challenges in operator learning and provide a path towards truly data-efficient learning for linear and nonlinear operators by exploiting known structure.
April 7, 2025
Speaker: Hamza Adjerid
Virginia Tech
Title: Approximation of Control Energy Functions for Nonlinear DAEs of Stokes-type
We study optimal feedback control for a class of differential algebraic equations (DAEs) with quadratic nonlinearity. Systems of this specific form would arise in finite element approximations to control problems associated with the Navier-Stokes equations, but here we focus our attention on finite dimensional systems that share this form. Our approach is to combine the strangeness framework for DAEs with polynomial approximations to control energy functions described by Hamilton-Jacobi-Bellman equations. We develop quadratic feedback control laws by exploiting the sparseness in the original DAEs, avoiding the dense matrices that would appear in an index reduction step or transformation to a strangeness-free (s-free) system of DAEs.
Speaker: Mirjeta Pasha
Virginia Tech
Title: A transport map approach for Bayesian inference of dynamic inverse problems with heavy-tailed priors
Tackling the immediate challenges that arise from growing model complexities (spatiotemporal measurements) and data-intensive studies (large-scale and high-dimensional measurements collected as time-series), state-of-the-art methods can easily exceed their limits of applicability. Conventional methods based on Gaussian processes (GP) often fall short in providing satisfactory solutions since they tend to offer over-smooth priors. Recently, the Besov process (BP), defined by wavelet expansions with random coefficients, has emerged as a more suitable prior for Bayesian inverse problems of this nature. While BP excels in handling spatial inhomogeneity, it does not automatically incorporate temporal correlation inherited in the dynamically changing objects. In this talk we describe how to generalize BP to a novel spatiotemporal Besov process (STBP) by replacing the random coefficients in the series expansion with stochastic time functions as Q-exponential process which governs the temporal correlation structure. We thoroughly investigate the mathematical and statistical properties of STBP. A transport map representation of STBP is also proposed to facilitate the inference. We demonstrate the utility of our approach on several image reconstruction problems.
April 14, 2025
Speaker: Ning Wan
Virginia Tech
Title: Inexact Newton's methods at singular point
In the 1980s, the convergence of Newton’s method was extensively studied in the context of singular derivatives at the root. Researchers identified key challenges related to the nullity of the Jacobian and focused on analyzing Newton's method within regions where the Jacobian remains invertible. One well-known result demonstrated the existence of a starlike domain in which Newton’s method converges linearly to the root. In contrast, the theoretical understanding of inexact Newton methods near singular points remains limited. Most existing results assume that these methods are either sufficiently or even arbitrarily accurate. However, for nonsingular problems, the convergence of inexact Newton methods can be achieved by simply ensuring that the relative residuals remain bounded by a constant less than one. In this talk, we investigate a class of inexact Newton methods known to converge for nonsingular problems, and examine their behavior in the presence of singularities, both theoretically and experimentally.
Speaker: Paul Cazeaux
Virginia Tech
Title: Ground State Preparation with Open Quantum System-inspired Quantum Algorithms
An important subroutine in any quantum computing algorithm is the preparation of the initial useful and relevant quantum state of the machine, preferably efficiently and using a small number of ancilla qubits. In this talk, we will discuss a novel algorithm for the preparation of the ground state of a given Hamiltonian and its analysis, motivated by the novel approach [Z. Ding, C-F. Chen and L. Lin, Single-ancilla ground state preparation via Lindbladians, Phys. Rev. Research 6, 033147, 2024] based on the approximation of Lindblad dynamics often used to model open quantum systems. Under certain assumptions on the eigenvalue distribution of the Hamiltonian and ability to apply certain random unitary operations, explicit mixing time estimates may be obtained showing polynomial scaling in the number of qubits.
April 21, 2025
Speaker: Dan Folescu
Virginia Tech
Title: Contour Integral Methods for the Masses
Linear and nonlinear eigenvalue problems alike can be formulated (and solved) via contour integral methods. These methods compile structured quadrature data and problem dependent hyper-parameters into data matrices, whereby local spectral information can be extracted through the careful solution of a generalized eigenvalue problem. As the construction of these data matrices generally result in ill-conditioned Hankel/Loewner matrix pencils, various approaches can be taken to improve the stability of the system realization step. In this talk, we will compare and contrast some of these approaches utilizing a new MATLAB package. It empowers the construction of aforementioned data matrices from both exact data and inexact contour integration via quadrature rule. Self-consistent and visually reactive code structures allow for real-time exploration and refinement of method parameters, either programmatically or through the user-friendly graphical user interface. Various numerical examples and demonstrations will showcase how to get the most out of both contour integral methods and this new tool.
Speaker: Yichen Guo
Virginia Tech
Title: An Adaptive Mixed-Precision and Dynamically Scaled Preconditioned Conjugate Gradient Algorithm
Solving large-scale linear systems is often the most computationally demanding component of numerical simulations. With modern hardware accelerators offering significantly higher arithmetic throughput at reduced precision, and lower-precision storage cutting both memory usage and data transfer costs, there is growing interest in algorithms that can use these advantages to accelerate sparse linear solvers. In this talk, we present an Adaptive Mixed-Precision and Dynamically Scaled Preconditioned Conjugate Gradient algorithm (AMP-PCG) for solving symmetric positive definite systems. AMP-PCG dynamically adjusts the precision used for storing vectors and performing operations throughout the iteration process, exploiting low-precision arithmetic when appropriate, while maintaining a convergence rate comparable to that of double-precision PCG. We demonstrate the effectiveness of this strategy through numerical experiments that highlight both the robustness and performance gains of AMP-PCG compared to fully double-precision PCG.
April 28, 2025
Speaker: Linus Balicki
Virginia Tech
Title: Contour Integral Methods for Parametric Nonlinear Eigenvalue Problems
In many applications where large-scale nonlinear eigenvalue problems arise, we are interested in computing a subset of the spectrum that lies within a specific domain of interest. An effective class of algorithms for computing such subsets is given by contour integral methods. At their core, these methods leverage numerical quadrature to compute sample data from an underlying eigenvalue problem that encodes the spectral information for a domain of interest. Extracting eigenvalues and eigenvectors from this data is then equivalent to identifying the poles and residues of a matrix-valued rational function from data --- a classic system identification problem. In this talk, we explore how these methods can be applied in a setting where the nonlinear eigenvalue problem (and hence the eigenvalues and eigenvectors) depend on a varying parameter. In particular, we investigate conditions under which contour integration can properly capture the parametric spectral information. Through various numerical examples, we demonstrate how multivariate rational approximation methods lead to an effective procedure for solving parametric nonlinear eigenvalue problems.
Speaker: Cankat Tilki
Virginia Tech
Title: Koopman Theory and Wavelets: A Framework for Wavelet-based Dynamic Mode Decomposition
In this work, we present an in-depth analysis of the Koopman operator using wavelet theory and introduce the wavelet-based observables. We establish that for the Morlet wavelet, the action of the Koopman operator on these observables takes a simple closed form within a weighted L2 space. To approximate the Koopman operator numerically, we employ the Extended Dynamic Mode Decomposition method combined with the proposed wavelet-based observables. The observables are computed using the Synchrosqueezing Transform. We demonstrate the effectiveness of this approach on the Lorenz and FitzHugh-Nagumo systems, showing that it performs robustly even in the presence of moderate noise.
May 5, 2025
Speaker: Ionut Farcas
Virginia Tech
Title: Towards data-driven reduced modeling of exascale simulations
High-performance computing (HPC) has revolutionized our ability to perform detailed simulations of real-world problems. Prominent contemporary examples are from aerospace propulsion, where HPC is used for rotating detonation rocket engine (RDRE) simulations in support of the design of next-generation rocket engines, and plasma fusion, where HPC is used for assessing turbulent transport in fusion devices. However, these simulations can take millions of core hours, which makes them impractical for tasks like design exploration, risk assessment, and uncertainty quantification. Reduced-order models (ROMs) aim to address this limitation by constructing computationally cheap yet sufficiently accurate approximations that serve as surrogates for the high-fidelity model. However, scalable methods to construct ROMs have not kept pace with the increased scale and resolution made possible by HPC. In this presentation, we discuss a distributed memory algorithm that can achieve fast and scalable construction of predictive physics-based, data-driven ROMs trained from datasets of large state dimension. We demonstrate this algorithm's scalability using up to 2,048 cores on the Frontera supercomputer at the Texas Advanced Computing Center. We focus on a real-world RDRE for which one millisecond of simulated physical time requires one million core hours on a supercomputer. Using a training dataset of 2,536 snapshots each of state dimension 76 million, our distributed algorithm enables the construction of a predictive reduced model in just 13 seconds on 2,048 cores on Frontera. Finally, we discuss a streaming extension of this algorithm, which can enable in situ ROM construction for large-scale problems by eliminating the need for extensive data storage.
Distribution Statement A: Approved for Public Release; Distribution is Unlimited. AFRL-2024-1411
Distribution Statement A: Approved for Public Release; Distribution is Unlimited. AFRL-2024-1411
Contributions – Fall Semester 2024
September 16, 2024
Speaker: Reetish Padhi
Virginia Tech
Title: Realization-free balanced truncation for quadratic-bilinear systems
Quadrature-based balanced truncation (Quad-BT) is a recently developed reduced-order modelling technique for linear (and bilinear) time invariant systems. Quad-BT is non-intrusive, i.e., does not require access to the original system’s realization (matrices) or state-space observations. It computes reduced models purely from data-based quantities, i.e., samples of the time-domain kernels and their derivatives. In this talk, I will present the extension of Quad-BT to quadratic bilinear systems.
Speaker: Petar Mlinarić
Virginia Tech
Title: Numerical Methods for Cultivated Meat Production
Cultivating meat from animal cells is a relatively recent technology that promises to reduce the impact of conventional meat production. In this talk, I will present challenges in bringing cultivated meat production to the commercial scale and suggest how numerical methods could address some of them. In particular, we will discuss optimizing the culture media composition, bioreactor design, and optimal control for cell proliferation.
September 23, 2024
Speaker: Mirjeta Pasha
Virginia Tech
Title: Efficient Dynamic Image Reconstruction with Motion Estimation
Large-scale dynamic inverse problems are typically ill-posed and suffer from complexity of the model constraints and large dimensionality of the parameters. A common approach to overcome ill-posedness is through regularization that aims to add constraints on the desired parameters in both space and temporal dimensions. In this work, we propose an efficient method that incorporates a model for the temporal dimension by estimating the motion of the objects alongside solving the regularized problems. In particular, we consider the optical flow model as part of the regularization that simultaneously estimates the motion and provides an approximation for the desired image. To overcome high computational cost when processing massive scale problems, we combine our approach with a generalized Krylov subspace method that efficiently solves the problem on relatively small subspaces. The effectiveness of the prescribed approach is illustrated through numerical experiments arising in dynamic computerized tomography and image deblurring applications.
September 30, 2024
Speaker: Rudi Smith
Virginia Tech
Title: Exploiting sparsity in trace estimation
An important problem in numerical linear algebra is to calculate the trace of an unknown matrix with access to a matrix-vector multiplication oracle, both in randomized and deterministic settings. In this talk, we will review some important stochastic trace estimators, discuss improvements using low rank and statistical methods, and discuss some deterministic methods from compressive sensing and banded matrix recovery.
Speaker: Steffen Werner
Virginia Tech
Title: Adaptive data sampling for low-dimensional controller inference
Stabilizing dynamical systems in science and engineering poses a significant challenge, especially in scenarios where only limited data are available. The quality and quantity of information within the data related to the task at hand are crucial factors to consider. This is particularly true when dealing with unstable dynamics, as the presence of instabilities typically leads to redundant or destructive system behavior, making it difficult to collect large amounts of informative data. In our new approach, we address the issue of generating small yet informative data sets for the stabilization of dynamical systems. The key lies in the adaptive construction of suitable input signals for the data generation via intermediate low-dimensional controllers that stabilize the system dynamics over limited subspaces. Numerical experiments with chemical reactors and fluid dynamics behind obstacles demonstrate that the proposed approach stabilizes systems reliably after observing about ten data samples even though the dimension of states is orders of magnitude higher.
October 7, 2024
Speaker: Graduate Students
Virginia Tech
Title: Graduate Student Internship Exchange
Yichen Guo (Argonne National Laboratory)
Amit Rotem (Lawrence Livermore National Laboratory)
Linus Balicki (ASML, San Diego)
Hayden Ringer
Amit Rotem (Lawrence Livermore National Laboratory)
Linus Balicki (ASML, San Diego)
Hayden Ringer
October 14, 2024
Speaker: Aimee Maurais
MIT
Title: Dynamic Transport for Sampling: Kernel Fisher-Rao Flow and Beyond
In modern day data science and machine learning, the prevailing paradigm for describing information and data is that of probability distributions. For instance, in Bayesian inference we use a prior probability distribution to encode our knowledge about an object (e.g., the weather) and transform that prior using data (e.g., noisy weather measurements) to obtain a posterior distribution reflecting our updated knowledge. In generative modeling, we view a collection of data — for example, images, audio, or molecular configurations — as samples drawn from an unknown underlying probability distribution. In both cases, sampling from the probability distribution at hand is a fundamental task, relied upon for uncertainty quantification in the Bayesian setting and for creation of new data in the generative setting. In this talk we will discuss the dynamic transport approach to sampling, in which samples from a tractable reference distribution are transformed into samples from a desired target distribution by application of appropriate dynamics, such as an SDE or ODE. We will explore the flexibility of the dynamic transport framework and particularly focus on one dynamic transport algorithm, Kernel Fisher Rao Flow (KFRFlow). This algorithm, realized as an interacting particle system, corresponds to a mean-field ODE that transports samples along the geometric mixture of the two densities and has connections to Fisher-Rao gradient flows and optimal transport. On a practical level, KFRFlow only requires access to an unnormalized target-to-reference density ratio, is gradient-free, and empirically outperforms comparable gradient-free dynamic transport algorithms. We will conclude with an outlook on the challenges faced by KFRFlow and similar algorithms and discuss design and implementation choices that show promise in addressing them.
October 21, 2024
Speaker: Johann Rudi
Virginia Tech
Title: Extreme-Scale High-Resolution Mantle Convection: Inference through Large-Scale Optimization and Uncertainty Quantification for Earth's Nonlinear Viscosity
We target large-scale computational methods and parallel algorithms centered around a challenging application: global-scale and high-resolution mantle convection. We estimate parameters in the viscosity of models for mantle convection and plate tectonics from surface observations, which results in an optimization problem (i.e., outer loop). The forward problem is governed by highly nonlinear, heterogeneous, and incompressible Stokes equations (i.e., inner loop). Solving the forward problem is a major challenge at extreme computing scales. An outer loop for inference and uncertainty quantification adds another dimension of solver challenges. The computational methods for the forward problem heavily rely on advanced numerical methods, parallel algorithms, and high-performance implementations tailored to the nonlinear and highly variable viscosity. The methods we developed include adaptive mesh refinement, inexact Newton-Krylov, block preconditioners for saddle-point linear systems, and multigrid. We achieve scalable solutions of the outer loop optimization by deriving adjoint problems to efficiently compute derivatives, and we employ these in a BFGS Newton's method. We further quantify uncertainties of parameters by approximating the Bayesian posterior at the maximum a posteriori point using a Gauss-Newton Hessian approximation. This approach allows for the first time the inference of viscosity parameters of global Earth mantle models with 1-km resolution (reference: https://doi.org/10.1073/pnas.2318706121).
Speaker: Chris Beattie
Virginia Tech
Title: Solving Ill-conditioned Weighted Regression Problems
As observation systems acquire greater capacity and become more sophisticated, regression problems making use of this data must take into account the possibility of significant coupling and correlations among observations leading frequently to ill-conditioned weighted regression problems for which standard orthogonalization approaches become problematic and possibly may fail. I will describe an application setting coming from next-generation wide-swath satellite altimetry where such problems are central and then discuss more broadly extensions of an approach introduced by Paige that allow computationally effective resolution of these problems even in extreme cases where the observation covariance (and possibly its inverse) are not only ill-conditioned but possibly singular (!).
October 28, 2024
Speaker: Max Heldman
Virginia Tech
Title: A monotone finite element method for reaction-diffusion equations with discontinuous reaction terms
We present a dual-mesh based quadrature rule for diagonalizing reaction operators arising from finite element discretizations of reaction-diffusion equations with discontinuous reaction terms. The method can be viewed as a generalization of mass lumping, in particular preserving quasi-optimal convergence rates for the discrete solution.
Speaker: Jorge Reyes
Virginia Tech
Title: Bridging Large Eddy Simulation and Reduced Order Modeling of Convection-Dominated Flows
For convection-dominated flows, standard Reduced Order Models (ROMs) typically require numerical stabilization or closure models to avoid spurious oscillations and account for the effects of discarded modes. This talk focuses on a specific type of ROM closure and stabilization inspired by Large Eddy Simulations (LES). A central component of these ROMs, known as LES-ROMs, is spatial filtering, which follows the same principle used in classical full-order LES models. This approach ensures consistency between the LES-ROMs and the methods that generated the data used for training. We prove key results, such as stability and convergence, supported by several numerical simulations that demonstrate agreement with theoretical predictions. Finally, we perform parameter scaling for the tunable stabilization parameter, aiming to either a priori calculate an acceptable value or develop more efficient algorithms for this purpose.
October 31, 2024
(9am, NEC 2420)
(9am, NEC 2420)
Speaker: Samuli Siltanen
University of Helsinki
Title: Electrical impedance tomography and virtual X-rays
Electrical Impedance Tomography (EIT) is a nonlinear PDE-based imaging modality where a patient is probed with harmless electric currents, and the resulting surface voltages are measured. EIT image reconstruction is an ill-posed inverse problem, meaning very sensitive to noise in the data and modelling errors. However, one can use complex geometric optics (CGO) solutions, and a nonlinear Fourier transform to do robust medical imaging; this is the so-called regularized D-bar method. A connection between EIT and X-ray tomography was found in [Greenleaf et al. 2018] using microlocal analysis. Fourier transform applied to the spectral parameter of CGO solutions produces virtual X-ray projections, enabling a novel filtered back-projection type nonlinear reconstruction algorithm for EIT. It is remarkable how this new approach decomposes the EIT image reconstruction process in several steps, where all ill-posedness is confined in two linear steps. Therefore, we can separate the nonlinearity and ill-posedness of the fundamental EIT problem. Furthermore, the new decomposition enables targeted machine learning approaches as only one or two (mathematically well-structured) steps in the imaging chain are solved using neural networks.
November 4, 2024
Speaker: Till Peters
TU Braunschweig
Title: Bilinear port-Hamiltonian systems and related model reduction techniques
Increasingly large and complex dynamical systems play a major role in mathematical modeling today. In mechanics, electrical circuits or flow dynamics, we can often represent these passive systems as port-Hamiltonian (pH) systems. The pH formulation is a very general structure which is based on a Hamiltonian function associated with the total stored energy. Furthermore, the formulation allows us to include dissipation, interconnection and control terms into the possibly implicit differential equations of the system. For high-dimensional pH systems the task of structure preserving model order reduction arises. In the linear case there are many methods to do this. A first extension to the nonlinear case can be so-called bilinear pH systems. These systems and their model order reduction will be discussed in this talk.
Speaker: Eric de Sturler
Virginia Tech
Title: A variant of majorization-minimization generalized Krylov subspace solvers for large-scale and streaming inverse problems
The high-quality reconstruction of images with sharp edges requires inversion with edge-preserving regularization operators. A common approach is to use a general lq-norm on the gradient of the image. Such an approach is often implemented using an iteration with weighted l2-norms of the gradient terms, with the weights determined by the current solution estimate. While hybrid Krylov methods, which can pick regularization parameters dynamically, are suitable for this problem, they would need to build a new Krylov space for every update of the regularization term. This problem can be addressed with so-called generalized Krylov subspace methods, which can be combined with a majorization-minimization approach to enforce convergence to the original minimization functional, based on a (smoothed) lq-norm regularization. However, for large-scale and/or streaming applications this leads to several complications. If the solution requires many iterations, data comes in continually, or the data set is very large, the approach leads to overwhelming memory requirements and computational costs. We can mitigate this problem by periodically compressing the search space to a lower dimensional subspace. Unfortunately, minimizing only over low-dimensional subspaces makes previous convergence proofs for these methods not applicable. However, this can be fixed by a slight modification of the algorithm. We will demonstrate the resulting algorithm on a few model problems. This is joint work with Mirjeta Pasha and Misha Kilmer.
November 5, 2024
(9:30am, DDS 366)
(9:30am, DDS 366)
Speaker: Hitay Ozbay
Bilkent University
Title: Computation and Implementation of Robust Controllers for a Class of Distributed Parameter Systems
The solution to the mixed sensitivity minimization problem is reviewed for distributed parameter systems (DPS) with rational weights. The H-infinity-optimal controller’s internal structure is examined and its finite dimensional and infinite dimensional parts are identified. Typically, one approximates the infinite-dimensional parts of the optimal controller to obtain a finite-dimensional controller. We discuss the effect of such approximations on robust stability and the performance level. We also review the strong stabilization problem for time delay systems: interpolation methods and a small gain approach are presented. A new family of robust controllers is given for unstable DPS and related open problems are discussed.
November 11, 2024
(DDS 366)
(DDS 366)
Speaker: Jiuhua Hu
Virginia Tech
Title: Space-Time Non-Local Multi-Continua Multiscale Method for Channelized-Media Parabolic Equations
Many applied problems feature coupled spatial and temporal heterogeneities that complicate homogenization or upscaling. Non-local multi-continua are proposed to address complex spatial heterogeneities in various papers. We extend this approach to a parabolic problem with time-dependent heterogeneous coefficients, developing the space-time Non-local multi-continua. This method efficiently constructs multiscale basis functions through local energy minimization within oversampled space-time regions. These functions demonstrate exponential decay outside their domain and offer computational advantages over classical methods, providing a systematic and flexible way to achieve accurate solutions while reducing computational cost.
Speaker: Mike Ackermann
Virginia Tech
Title: Learning mechanical systems from data via a structured AAA algorithm
Data-driven reduced order modeling is a tool to construct models of physical phenomena when access to governing equations or a state-space model is not possible, but input-output data is abundant. One popular method for constructing data-driven models is the Adaptive Anderson Antoulas (AAA) algorithm, which blends interpolation and least-squares techniques. While AAA can produce models that fit the given data to high accuracy, the model produced by AAA does not preserve underlying differential structures which are important for interpretation of the models. A common structure to appear in mechanical systems is an explicit dependence on the second time derivative. Based on our previous work on structured barycentric forms, we present a structure preserving second-order AAA algorithm that is capable of learning highly accurate second-order surrogate models from frequency response measurements.
November 18, 2024
Speaker: Hayden Ringer
Virginia Tech
Title: Listening Carefully to Drums of Unknown Shape
The inverse Dirichlet Laplace eigenvalue problem, put simply, is: "Can a planar shape be determined by its Dirichlet eigenvalues?" This problem is known to be ill-posed. In this talk, we present advances in our ongoing project to develop computational and Bayesian-statistical techniques for recovering unknown shapes, based on their "spectral data." In particular, we will demonstrate the effectiveness of these approaches on small benchmark problems. We will discuss: (1) improvements we have made to our forward eigensolver, (2) a parameterization scheme which "quotients-out" the continuous symmetries of the eigenproblem, (3) an engineered loss/likelihood function for evaluating "spectral closeness", and (4) the construction of a regularization term/prior distribution which enforces non-degeneracy of the parameterized family of shapes.
Speaker: Yingda Cheng
Virginia Tech
Title: Structure-preserving Low Rank Scheme for the Lindblad Master Equation
The Lindblad master equation is the fundamental model for open quantum systems. A defining feature of the Lindblad equation is the completely positivity and trace preserving (CPTP) property of the solution. We present a high order accurate low rank numerical method with CPTP property. This is joint work with Daniel Appelo (VT).
December 2, 2024
Speaker: Andrew Christlieb
MSU
Title: Mathematical and modeling challenges of fusion energy
In this talk we review the challenges and multi scale modeling associated with inertial confinement fusion energy. I will give a brief overview of the Center for Hierarchical and Robust Modeling of Non-Equilibrium Transport (CHaRMNET), a DoE MMICC center founded on this class of problems. The fundamental issue is that in space the system covers 8 orders of magnitude and in time the system covers 12 orders of magnitude. Further complicating things is that it is a driven system in a non equilibrium state for the majority of its evolution. As such simplifying assumptions that would enable using reduced models such as rad hydro, Navier Stokes equations coupled with simplified non-linear transport models, do not capture known effects coming from kinetic phenomenon. After this overview, I will talk about blended computing, a process of developing structure preserving self-consistent machine learning surrogates to pair with traditional numerical algorithms as a way of capturing scales. I will review work in the field. Following this, I will talk about this in the context of a simplified problem, capturing kinetic phenomenon in fluid models of the BGK equations in 1D. I will present the models that motivate our “fluid model” closure, which leads to our work on developing structure preserving Neural Networks, touch on training, then talk about results and next steps. I end the talk by touching on other projects my group is involved with.
Dec 5 (10:00am, MCB 455)
Speaker: Youngsoo Choi
Lawrence Livermore National Lab
Title: Latent space dynamics identification
Many computational models for physical systems have been developed to expedite scientific discovery, which would otherwise be impeded by the lengthy nature of traditional, non-computational experimentation (e.g., observation, problem identification, hypothesis formulation, experimentation, data collection, analysis, conclusion, and theory development). However, as these physical systems grow more complex, the computational models themselves can become prohibitively time-consuming. To address this challenge, we introduce a framework called Latent Space Dynamics Identification (LaSDI), which transforms complex, high-dimensional computational domains into reduced, succinct coordinate systems—a sophisticated change of variables that preserves essential dynamics. LaSDI offers significant potential for extension to other innovative, data-driven algorithms. It is an interpretable, data-driven framework composed of three core steps: compression, dynamics identification, and prediction. In the compression phase, high-dimensional data is reduced to a more manageable form, facilitating the construction of an interpretable model. The dynamics identification phase then derives a model, typically expressed as parameterized differential equations, that accurately captures the behavior of the reduced latent space. Finally, in the prediction phase, these differential equations are solved within the reduced space for new parameter sets, with the resulting solutions projected back into the full space. One of the key advantages of LaSDI is its computational efficiency, as the prediction phase operates entirely in the reduced space, bypassing the need for the full-order model. The LaSDI framework supports various identification methods, including fixed forms like dynamic mode decomposition and thermodynamics-based LaSDI, regression methods such as sparse identification of nonlinear dynamics (SINDy) and weak SINDy, as well as physics-driven approaches like projection-based reduced order models. The LaSDI family has demonstrated substantial success in accelerating various physics problems, achieving up to 1000x speed-ups in fields such as kinetic plasma simulations, pore collapse phenomena, and computational fluid dynamics.
December 9, 2024
Speaker: Jenifer De Jager
Virginia Tech
Title: Lebesgue Boundedness of the Koopman Operator
Koopman theoretic methods, which convert an arbitrary nonlinear system on a finite dimensional state space into a linear system on an infinite dimensional observable space, are showing empirical promise in data-driven dynamical model order reduction, although the theory is not as well understood as for system theoretic methods. In this talk, we will discuss the boundedness of the Koopman operator and go over some of the implications thereof.
Speaker: Ian Moore
Virginia Tech
Title: Approximate Deconvolution Leray Reduced Order Modeling
In the field of reduced order modeling, filtering methods have received significant recent attention for treating poorly performing models as a method of regularization. However, filtering by itself can be susceptible to over-smoothing of generated solutions. We propose new reduced order models that integrate the methods of approximate deconvolution with a goal of balancing accuracy and smoothing properties. The models are tested with particular application to fluid flows simulated by the Navier-Stokes equations.
December 10, 2024
(2pm, DDS 366)
(2pm, DDS 366)
Speaker: Jonathan Lindbloom
Dartmouth College
Title: Priorconditioned generalized Krylov subspaces for sparsity-promoting inversion
Reconstructing high-quality signals and images with sharp edges is crucial in many applications. Hybrid projection methods offer a competitive class of algorithms for this task. This talk focuses on such methods based on generalized Krylov subspaces (GKS), known for their efficiency in solving large-scale linear inverse problems with 2-norm ("smoothing") regularization and requiring only a low-dimensional subspace for the solution. However, adapting GKS methods to handle sparsity-promoting regularization penalties incurs high computational costs due to the high-dimensional subspaces needed to capture sparse solutions. In this talk, we present a new preconditioned GKS method that addresses this issue and apply the method to both deterministic and Bayesian inverse problems. We also make some comparisons with other hybrid projection methods. This is joint work with Mirjeta Pasha (Virginia Tech), Jan Glaubitz (Linköping University), and Youssef Marzouk (MIT).
Contributions – Spring Semester 2024
February 2, 2024
Speaker: Tom Werner
TU Braunschweig
Title: The inexact matrix-Newton framework for solving NEPv
The eigenvector-dependent nonlinear eigenvalue problem (NEPv), also known as nonlinear eigenvector problem, is a special type of eigenvalue problem where we seek to find \(k\) eigenpairs of a Hermitian matrix function \(H : \mathbb{C}^{n,k} \to \mathbb{C}^{n,n}\) that depends nonlinearly on the eigenvectors itself. That is, we have to find \(V \in \mathbb{C}^{n,k}\) with orthonormal columns and \(\Lambda \in \mathbb{C}^{k,k}\) such that \(H(V) V = V \Lambda\). NEPv arise in a variety of applications, most notably in Kohn-Sham density theory and data science applications such as robust linear discriminant analysis. This talk is concerned with solving NEPv by viewing it as a set of nonlinear matrix equations and using an inexact Newton method on a matrix level. In this setting, Newton's method is applied using the Fréchet derivative and exploiting the structure of the problem by using a global GMRES-approach to solve the Newton-update equation efficiently.
Speaker: Mark Embree
Virginia Tech
Title: Finding the Nearest Correlation Matrix: A Nick Higham Tribute
Recalling contributions of Nick Higham, including his most cited paper.
February 9, 2024
Speaker: Shixu Meng
Virginia Tech
Title: Learning theory and algorithm for inverse scattering
Inverse problem plays an important role in various applications associated with target identification, non-destructive testing, and parameter estimation. Of particular concern is the inverse scattering problem by inhomogeneous media, where the goal is to estimate the unknown inhomogeneity from the knowledge of measurement data. Due to its ill-posed nature, our aim is to alleviate the ill-posedness through data processing and computing the unknown in a suitable low-dimensional space. In the Born or linearized case, this space consists of the disk prolate spheroidal wave functions, computed efficiently via a Sturm-Liouville problem. We show how to image the unknown from noisy and potentially large scale measurement data. This technique further motivates some new understanding of sampling-type imaging methods in connection with nonlinear regression learning. In conclusion, we discuss several future research directions.
February 16, 2024
Speaker: Agnieszka Międlar
Virginia Tech
Title: Towards Resilient and Robust Asynchronous Linear Systems Solvers for Edge Computing
Newly emerging technology of edge computing, where networked, autonomous devices work collaboratively in applications such as sensing platforms for radioactive source detection, fully-decentralized smart grids, or unmanned autonomous vehicle (UAV) systems to achieve a common goal through the synergy of humans and machines, requires asynchronous, flexible and resilient algorithms for decentralized processing, communication and predictions in heterogeneous environments. While the benefits of developing autonomous capabilities are multifold, they are subject to device failures, partial data losses, or security attacks, and hence require multidimensional resilience solutions. In this talk, we will present our recently proposed resilience and robustness improvements when solving linear systems of equations with asynchronous Jacobi (ASJ) method. In order to restore the convergence of the original ASJ method in the presence of data corruption or communication delay, we derive the ADMM (Alternating Direction Method of Multipliers) and average consensus push-sum inspired rejection and update criteria. Joint work with Zachary R. Atkins, Alyson L. Fox, Colin V. Ponce, and Christopher J. Vogl. Funded by LLNL LDRD projects 21-FS-007 and 22-ERD-045. Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-831426.
Speaker: Yingda Cheng
Virginia Tech
Title: Dynamic Low Rank Approximation (DLRA) and How to Fix It
In recent years, the Dynamic Low Rank Approximation (DLRA) has received lots of attention for computing time-dependent low rank solution of matrix/tensor differential equations. The method is based on the orthogonal projection of the original equation onto the tangent space of the low rank manifold. While this method enjoys many good properties, such as stability and simplicity in implementation, it suffers from the tangent projection error (modeling error) which cannot be eliminated by mesh refinement. In this talk, I will show a simple approach to remove this error to enhance the convergence property. The idea is to combine with the step truncation schemes. We propose an adaptive strategy that can accelerate the computation for moderately stiff problems. This is a recent joint work with Daniel Appelö.
March 1, 2024
Speaker: Max Heldman
Virginia Tech
Title: Coupling Earthquake Cycles with Plate Tectonics: Toward Scalable Numerical Methods for Multiscale Visco-Elastic-Plastic PDEs
We discuss the extension of the Rhea code, which simulates the instantaneous coupling of mantle convection to plate tectonics, to the dynamic coupling of the slow, creeping motion of large-scale plate tectonics with the episodic slip events associated with great earthquakes. In the quasi-static regime, this involves solving a viscoelastic-plastic constitutive relation for the time evolution of the stress tensor, constrained by the incompressible Stokes equations. After time discretization, we derive a visco-plastic problem with an effective viscosity and strainrate to be solved on each timestep using Rhea's scalable solvers.
Speaker: Steffen Werner
Virginia Tech
Title: Structure-Preserving Interpolation of Quadratic-Bilinear Systems
For accurate modeling of real-world phenomena, high-dimensional nonlinear dynamical systems are indispensable. Thereby, many physical properties are encoded in the internal differential structure of these systems. Some examples for differential structures are second-order time derivatives in mechanical systems or time-delay terms. When using such models in computational settings such as in numerical simulations, the high-dimensional nature of the models represented by large numbers of differential equations describing the dynamics becomes the main computational bottleneck. A remedy to this problem is model order reduction, which is concerned with the construction of cheap-to-evaluate surrogate models that are described by a significantly smaller number of differential equations while accurately approximating the input-to-output behavior of the original high-dimensional system. It has been shown that many nonlinear phenomena can be equivalently modeled using quadratic-bilinear systems. Therefore, many established model order reduction approaches such as transfer function interpolation and balanced truncation have been extended to the class of quadratic-bilinear systems in recent years. However, in most cases, these approaches cannot be applied to systems with internal differential structures. In this work, we present a structure-preserving interpolation framework for the multivariate transfer functions of quadratic-bilinear systems. This new approach allows the simulation-free construction of reduced-order models for quadratic-bilinear systems with internal structures, and it is able to preserve differential structures in the reduced-order model.
March 15, 2024
(DDS 366)
(DDS 366)
Speaker: Ning Wan
Virginia Tech
Title: On the Convergence of CROP-Anderson Acceleration Method
The Anderson Acceleration (AA) is a well-established method that allows to speed up or encourage convergence of fixed-point iterations. It has been successfully used in a variety of applications, in particular within the Self-Consistent Field (SCF) iteration method for quantum chemistry and physics computations. In recent years, the Conjugate Residual with OPtimal trial vectors (CROP) algorithm was introduced and shown to have a better performance than the classical Anderson Acceleration with less storage needed. In this talk, we will delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm. Our objectives include a comprehensive study of their convergence properties, explaining the underlying relationships, and substantiating our findings through some numerical examples. Through this exploration, we contribute valuable insights that can enhance the understanding and application of the mixing acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.
Speaker: Paul Cazeaux
Virginia Tech
Title: Randomized Algorithms and Krylov Solvers in the Tensor Train Format
The Tensor-Train (TT) format stands out as a highly compact, low-rank representation for high-dimensional tensors, with numerous applications including the computation of many-body ground states with limited entanglement in spin models of condensed matter physics or quantum chemistry. The efficiency of this format hinges on the rounding operation, a process that trims the internal ranks of tensors to ensure feasible computational times and memory usage. In this talk, I will present a randomized algorithm specifically designed for the TT rounding task, building upon and expanding existing randomized low-rank matrix approximation methodologies. This algorithm not only preserves the TT format's integrity but also provides a computational advantage, particularly evident when rounding a sum of TT-tensors or the result of a matrix-vector product—a frequent bottleneck in numerous applications, in particular when using Krylov linear or eigenvalue solvers. I will further discuss potential applications of the randomized algorithms in particular to a problem from quantum chemistry, where certain symmetries are encoded as an additional structure for the tensor train format.
March 22, 2024
Speaker: Linus Balicki
Virginia Tech
Title: Introducing Low-Rank Structure to the p-AAA Algorithm
Rational approximation is a widely used tool for data-driven modeling of linear dynamical systems. In recent years the adaptive Antoulas-Anderson (AAA) algorithm has established itself as a successful method for computing rational approximations from a set of sampled data. The p-AAA algorithm extends the original AAA framework to multivariate rational functions which allows for effectively capturing the dynamics of parameter-dependent systems. As with many approaches for computing multivariate approximations, the computational cost of p-AAA increases significantly when considering functions of many variables. In order to overcome this drawback we suggest imposing a low-rank structure on the approximations computed by p-AAA. We discuss implications for the algorithm and investigate in which cases our proposed approach is particularly effective.
Speaker: Yichen Li
Virginia Tech
Title: Phase-field simulation of freezing water droplet
When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this talk, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The non-negative entropy production is ensured by the design of the governing equations, which include the energy equation, the Navier-Stokes equations, and the two phase-field equations. The open-source deal.ii package is then used to solve these equations using finite-element techniques. The Gibbs-Thomson and Clausius-Clapeyron equations, which demonstrate how the melting temperature depends on interface curvature and pressure, respectively, are reproduced in our model. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment. In the end, we will discuss the effects of the water-ice-air tri-junction conditions and other parameters on the final shape of the frozen droplet.
March 29, 2024
Speaker: Hamza Adjerid
Virginia Tech
Title: Nonlinear Feedback Control for Stokes-type DAEs
We study optimal feedback control for a class of differential algebraic equations (DAEs) with quadratic nonlinearity. Systems of this specific form would arise in the discretization of control problems associated with the Navier-Stokes equations, but here we focus our attention on finite-dimensional systems with this form. Our approach is to use the strangeness framework for DAEs to appropriately recast the control problem as a system of differential equations with polynomial drift terms. At this stage it is appropriate to develop standard approximations to the feedback control laws. We describe this process and note the differences in our implementation to other approaches that rely on projections (which can destroy sparsity in the original DAEs). To simplify the exposition, we describe the optimal control for cases when the control does not appear in the algebraic constraints though we describe how this can be generalized.
Speaker: Amit Rotem
Virginia Tech
Title: The Semidiscrete WaveHoltz Iteration
Due to the indefinite nature of the Helmholtz equation, solutions of this equation are difficult to approximate. Classical algorithms for definite elliptic equations (such as Krylov iteration, domain decomposition, multigrid, etc.) are less effective, and may fail to converge entirely. The WaveHoltz algorithm solves the Helmholtz equation by filtering the wave equation in the time domain. The resulting fixed point iteration can be accelerated via Krylov space methods, and the linear system is positive definite for problems with energy conserving boundary conditions. In this presentation, we show that for any stable discretization of the wave equation, the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem.
April 5, 2024
(DDS 220)
(DDS 220)
Speaker: Yichen Guo
Virginia Tech
Title: Sparse Approximate Inverse Smoother for Multigrid Preconditioning in GPU-Accelerated Poisson Solvers
Two classes of preconditioners have been proved as effective for high-order finite element discretizations: the \(p\)-multigrid with a Chebyshev-accelerated Jacobi smoother and the low-order refined preconditioner. However, both require a significant number of iterations when applied to highly deformed meshes. The Sparse Approximate Inverse (SAI) preconditioner has been shown to effectively accelerate the convergence of iterative methods, with its construction being inherently parallel. In this work, we apply the SAI as a smoother within the \(p\)-multigrid preconditioning. Given the dense element matrices and the costly matrix-vector operations of high-order finite element discretizations, we use a matrix-free method for construction. We focus on three aspects: choosing an appropriate sparsity pattern for the SAI and implementing a thresholding strategy to reduce the number of unknowns; coloring nodes to segregate them into several sets and solving numerous least squares problems simultaneously using the matrix-free operator; and applying multiple precision to decrease the matvec operation costs. We evaluate the performance of the smoother on highly deformed meshes using GPUs. Our numerical results demonstrate the effectiveness of this approach.
Speaker: Eric de Sturler
Virginia Tech
Title: Sketching and Recycling GMRES for a Sequence of PDE Solves
The iterative linear solver GMRES, the generalized minimum residual method, is among the most popular solvers for general linear systems. Unfortunately, its reliance on full orthogonalization of the search space makes it expensive. For \(m\) iterations the cost is \(O(N m^2)\), where \(N\) is the, usually, very large linear system size. Therefore, GMRES is often 'restarted' every \(m\) iterations, where \(m\) is kept modest. Unfortunately, this may lead to relatively slow convergence. Recent randomization techniques, called sketching, offer the potential to solve the least squares problem in GMRES much more efficiently, reducing the cost to \(O(N m \log m)\) for \(m\) iterations, but may suffer from numerical instability due to the generation of the search space basis without orthogonalization or with selective orthogonalization. We further combine sketching with subspace recycling, reusing selected subspaces from previous linear systems or restarts to accelerate convergence.
This is joint work with Fei Xue (Clemson University).
The talk will focus on the underlying mathematical ideas.
This is joint work with Fei Xue (Clemson University).
The talk will focus on the underlying mathematical ideas.
April 12, 2024
Speaker: Ionuț Farcaș
Virginia Tech
Title: Structure-exploiting data-driven modeling at scale: reduced modeling, uncertainty propagation and sensitivity analysis
Recent advances in algorithms, computer architectures, and high-performance computing enable high-fidelity simulations of complex real-world phenomena with high accuracy. However, making quantitative predictions over long time horizons or performing many-query tasks such as design optimization and uncertainty quantification remains computationally challenging even on large supercomputers. In this presentation, I will discuss how structure-exploiting data-driven methods can be used to accelerate or enable these tasks in large-scale simulations. I will start with the inner-loop and show how Operator Inference can be used to construct structure-preserving reduced models of large-scale systems from data. I will also briefly discuss a recent development which integrates domain decomposition into the data-driven reduced modeling procedure. The capabilities of this method will be demonstrated in a realistic rotating detonation rocket engine simulation with 75 million degrees of freedom and a sparse training data set. I will then move to outer-loops and show how structure-exploiting sparse grid approximations can be used to quantify uncertainty and perform sensitivity analysis in complex applications with computationally expensive forward models. I will demonstrate the effectiveness of this framework in a turbulent transport simulation in the edge of fusion reactors with more than 264 million degrees of freedom and eight uncertain inputs. I will also briefly show how this method can be used to construct accurate surrogate models for quantities of interest. Lastly, I will discuss context-aware multi-fidelity Monte Carlo sampling for efficient uncertainty propagation. This method optimally balances the costs of training low-fidelity models with the costs of Monte Carlo sampling by taking into account the context in which the learned low-fidelity models will be used, namely for variance reduction in Monte Carlo estimation. Time permitting, I will demonstrate the capabilities of the context-aware algorithm in a plasma micro-turbulence simulation with 12 uncertain inputs.
April 19, 2024
Speaker: Cankat Tilki
Virginia Tech
Title: Analysis of Wavelet-based Dynamic Mode Decomposition Method
Dynamic Mode Decomposition (DMD) is an extensively used method in data-driven modeling of dynamical systems with no forcing. The analytical connection between DMD and the Koopman theory has paved the way for the Extended DMD algorithm (EDMD). In simplest terms, EDMD constructs an approximation of the Koopman operator on a finite-dimensional subspace. This subspace is predetermined, mostly by designating the observables: measurement functions. In practice choosing good observables is a big challenge. To overcome this problem, one can employ strategies from the signal processing domain and use wavelets as a candidate for these observables, as done in Wavelet-based DMD (WDMD). In this talk, we will discuss that the EDMD algorithm with wavelets coincides with a specific case of WDMD, up to an error introduced by the approximate Wavelet Transform, thus establishing the analytical connection to the Koopman theory. Moreover, we will investigate some extensions of the WDMD method when the data is noisy and prove converge rates depending on the DWT decomposition level: number of wavelets used in EDMD.
Speaker: Haroun Meghaichi
Virginia Tech
Title: A High Order Geometry Conforming Immersed Finite Element for Elliptic Interface Problems
We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions exactly and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet-Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Mathematical analysis and numerical examples are provided to showcase the convergence properties of the method under \(h\) and \(p\) refinements.
April 26, 2024
Speaker: Matthias Voigt
UniDistance Suisse
Title: Numerical Methods for Large-Scale Nonlinear Eigenvalue Problems
Nonlinear eigenvalue problems arise in a large variety of applications such as the stability analysis of delay differential equations, the treatment of differential equations with nonlinear boundary conditions, or the modal analysis of vibroacoustic problems. In this talk, we will discuss numerical methods for nonlinear eigenvalue problems that are described by matrices of large dimension. Two techniques will be presented. The first one considers the eigenvalue problem in a Schur complement form and is based on projecting the large matrices in an interpolatory framework in order to obtain a reduced nonlinear eigenvalue problem that can be solved more efficiently. Based on the eigenpair residuals, new interpolation points and corresponding projection matrices can be computed. In this way, we obtain an iterative method for computing a few eigenvalues close to a desired target point. The second method we consider is the NLFEAST algorithm that is designed to compute all eigenvalues that are enclosed by a given contour. We will adapt this method to the particular application case of a nonlinear eigenvalue problem originating from a modal analysis in vibro-acoustics and we will illustrate that this method is feasible for efficiently computing approximations to the resonance frequencies in a frequency band of interest. This is joint work with Rifqi Aziz and Emre Mengi (Koc University, Istanbul) as well as Suhaib Koji Baydoun, Benedikt Goderbauer, Christopher Jelich, and Steffen Marburg (TU Munich).
Contributions – Fall Semester 2023
September 1, 2023
Speaker: Justin Krometis
Virginia Tech
Title: What Lies Beneath: Dodging Pitfalls in Complicated Models
Recent years have seen an explosion in the application of artificial intelligence and machine learning (AI/ML) to practical problems from computer vision to game playing to algorithm design. This growth has been mirrored and, in many ways, been enabled by the development and maturity of publicly-available software packages such as PyTorch and TensorFlow that make model building, training, and testing easier than ever. While these packages provide tremendous power and flexibility to users, and greatly facilitate learning and deploying AI/ML techniques, they and the models they provide are extremely complicated and as a result can present a number of subtle but serious pitfalls. This talk will present three examples from the presenter's recent experience where obscure settings or bugs in these packages dramatically changed model behavior or performance-one from classic deep learning regression, one from image classification, and one from reinforcement learning. I will also show an example from (non-AI/ML) parameter estimation where a similarly issue buried deep in a codebase seriously undermined the accuracy of the results. These examples illustrate the importance of thinking carefully about the results that a model is producing and carefully checking each step in its development before trusting its output.
Speaker: Sean Reiter
Virginia Tech
Title: \(\mathcal{H}_2\)-optimal model reduction of linear systems with quadratic outputs
In this work, we investigate the \(\mathcal{H}_2\) model reduction problem for linear dynamical systems with quadratic outputs (LQO). Our primary contributions are twofold: First, we derive Wilson first-order optimality conditions for \(\mathcal{H}_2\) model reduction. We show that these constraints are equivalent to rational interpolation of a univariate and multivariate transfer function that characterize the input-output map of the underlying model; these describe the linear and quadratic parts of the output, respectively. Secondly, we adapt the iterative rational Krylov algorithm as an efficacious approach for computing reduced-order LQO models that satisfy the necessary optimality conditions upon convergence. Numerical experiments validate the effectiveness of the proposed framework.
September 8, 2023
Speaker: Steffen W. R. Werner
Virginia Tech
Title: Learning mechanical systems using structured barycentric forms
Data-driven reduced-order modeling is an essential tool in constructing high-fidelity compact models to approximate physical phenomena when explicit models, such as state-space formulations with access to internal variables, are not available yet abundant input/output data are. When deriving models from theoretical principles, the resulting models typically contain differential structures that lead to certain system properties. A common example are dynamical systems with second-order time derivatives arising in the modeling of mechanical or electro-mechanical processes. In this case, data are often available in the frequency domain, where the systems' input-to-output behavior is described by rational functions rather than differential equations. Classical frequency domain approaches like the Loewner framework, vector fitting and AAA are available and can be used to learn unstructured (first-order) models from data. However, these models in their original formulation do not reflect the structure-inherited properties. The key element in the derivation of frequency domain approaches is the barycentric form of rational functions. In this work, we present structured extensions of the barycentric form for the case of mechanical systems. Building on these structured barycentric forms, we develop new algorithms for learning mechanical phenomena in the frequency domain, while enforcing the mechanical system structure in the model description.
September 15, 2023
Speaker: Daniel Appelö
Virginia Tech
Title: A numerical methodists adventure in the land of quantum computing
In this talk we will: 1. Introducing the most basic concepts in quantum computing. 2. Briefly highlight examples of quantum algorithms where the skills of a numerical analyst can be of use. 3. Describe one type of quantum computing hardware (a transmon) and how it is modeled. 4. Take you on a comprehensive journey through a real-world example involving characterization, control, and experimental validation, showcasing our experiences with a qutrit device within the Lawrence Livermore QUDIT testbed.
September 22, 2023
Speaker: Yingda Cheng
Virginia Tech
Title: Kinetic models and high dimensional scientific computing
In this talk, I will first introduce kinetic models, which is a class of models that provide statistical descriptions of particle systems. I will discuss a few prominent applications of such models, ranging from rarified gas, radiation transport in nuclear engineering and Vlasov dynamics in plasmas. We will discuss properties of kinetic models, including the underlying structure of the solutions and the multi scale behavior which bridges such models to fluid (macroscopic) equations. Kinetic models are prominent examples of high dimensional partial differential equations (PDEs), which motivates the second part of the talk. I will discuss several approaches for solving high dimensional PDEs, focusing on our work on sparse grid discontinuous Galerkin schemes that are built upon multiwavelet basis functions. I will also outline some potential future directions and research opportunities.
September 29, 2023
Speaker: Art Pelling
TU Berlin
Title: Challenges in Data-driven Reduced Order Modelling for Acoustics
Modelling the complex dynamics of acoustical systems constructively can be challenging due to imprecise knowledge of material properties and domain geometry. Thus, measurements are often unavoidable in practice and, in turn, high-fidelity measurement data is abundantly available in the field. Recent advances in the mathematical community of model order reduction, specifically data-driven methods, can help to obtain efficient and meaningful models from the high-dimensional data. So far, reduced order modelling methods are not widely used in acoustics. At the other end of the spectrum in mathematics, new methods are oftentimes only validated by means of synthetic data and toy models. Consequently, a noticeable gap between theory and practice remains. Our aims are twofold: Firstly, clearing obstructions that arise when employing recent data-driven reduced order modelling methods in acoustical engineering practice. Secondly, providing measurement data of real-world systems that can be used as benchmarks by the mathematicians that develop new data-driven methods. In this talk, we showcase some 'roadblocks' that arise when applying the so-called Eigensystem Realization Algorithm by means of real examples. Amongst others, we consider head-related transfer functions that are used for auralization in virtual reality applications.
Speaker: Johann Rudi
Virginia Tech
Title: Inverse Maps Based on Deep Neural Networks
Machine learning algorithms have successfully been able to approximate nonlinear maps under weak assumptions on the structure and properties of the maps. This talk presents deep neural networks to solve inverse problems, where we seek to estimate parameters of certain physical models. We employ the neural networks to approximate inverse or reconstruction maps for model parameter estimation from observational data. The models we target exhibit computational challenges in an inference setting, namely, having a highly nonlinear and nonconvex data misfit term and permitting only weakly informative priors on parameters. These challenges typically cause traditional optimization to fail and alternative algorithms to exhibit large computational costs, therefore deep learning can become attractive. We quantify the prediction errors of model parameters obtained from the neural networks and investigate the effects of network architectures with and without the presence of noise in observational data. Furthermore, this talk investigates new techniques for quantifying uncertainties in the predictions.
October 13, 2023
Speaker: Ning Wan
Virginia Tech
Title: Numerical Study of Anderson Acceleration Method
Anderson Acceleration is a method that allows to speed up the convergence of the fixed point iteration, which is widely used within the SCF (Self-Consistent Field Iteration) method in quantum chemistry/physics computations. In this talk, we aim to explore Anderson Acceleration under perturbations and various truncation approaches. By studying the effects of perturbations and different truncation methods, we can gain insights into the behavior and performance of Anderson Acceleration in practical applications.
Speaker: Petar Mlinarić
Virginia Tech
Title: What is the Optimal Damper Position for the Wave Equation?
Mechanical structures can break catastrophically due to external forces introducing resonances. One way to address this is by adding dampers to suppress the effect of disturbances. This leads to the question of how to find optimal damper viscosities and positions for a given mechanical structure. There is significant prior work on finite-dimensional second-order systems, but remains challenging because of the discrete nature of damper positions. In this talk, we investigate the influence of a single damper on an infinite-dimensional system, in particular, the wave equation.
October 20, 2023
Speaker: Dan Folescu
Virginia Tech
Title: Bounding Ho-Kalman Realization Errors Due to Noisy Data
Contour integral methods solve nonlinear eigenvalue problems by performing, as the name suggests, contour integration of a domain in the complex plane. As formulated in the system-theoretic framework of Brennan, Embree, and Gugercin, exact contour integration yields the external (or "black-box") description of a linear, time-invariant dynamical system. In practice, inexact numerical quadrature is used to construct underlying system moments, presenting stability concerns when solving the subsequent Realization problem: given a system's external description, construct an internal (or state-space) description of the system. A seminal approach to the realization problem was developed by Ho and Kalman in 1966, which has since been extended to data-driven approaches based on the Loewner framework, among others. Concerning stability, Oymak and Ozay have recently revisited the Ho-Kalman approach, providing upper bounds on the norm of the difference of state-space coefficient matrices realized from exact and noisy external system descriptions, respectively. In this talk, we will first give a brief overview of the relevant systems theory, along with a description of the Ho-Kalman realization algorithm. Then, we will present an upper bound from the recent work of Oymak and Ozay. Finally, we will discuss additional approaches to bounding the normed difference of state matrix realizations via perturbation theory. Numerical examples will be provided.
Speaker: Mike Ackermann
Virginia Tech
Title: Analysis of the Hermite Loewner Framework Towards a Robust Implementation of TF-IRKA
Model reduction via transfer function interpolation is frequently used to produce high-fidelity reduced-order models of large-scale dynamical systems. These interpolatory models can be produced via projection of the full-order model matrices or constructed directly from input-output data. The data-driven approach is desirable since it does not require knowledge of a realization of the full-order model, instead only a way to evaluate values and derivatives of its associated transfer function. However, while theoretically equivalent to explicit projection-based formulation, the data-driven approach is plagued by numerical issues and sensitivity to interpolation point selection. We will present our work thus far in analyzing the Hermite Loewner framework, which produces reduced-order models that interpolate the full-order transfer function and its derivative at specified points in the complex plane. We first examine the case where the poles of the full-order model are known, and then examine the connection to \(\mathcal{H}_2\)-optimal model reduction. We conclude by presenting an algorithm to produce locally \(\mathcal{H}_2\)-optimal reduced-order models from data that is robust to initial interpolation point selection.
October 27, 2023
Speaker: Eric de Sturler
Virginia Tech
Title: Mixed Precision Recycling Conjugate Gradients on the GPU
For difficult problems, in our context, problems on highly deformed meshes, the (preconditioned) Conjugate Gradient (PCG) method may converge slowly, because the linear system is very poorly conditioned, and expensive preconditioners are needed. To reduce the runtime cost, we can accelerate PCG further by 'recycling' the search spaces from previous linear system solves to increase the rate of convergence. We will discuss how to select and reuse approximate invariant subspaces from previous search spaces to accelerate the convergence of PCG\@. However, on the GPU, recycling is memory-wise expensive, as we store a basis for the approximate invariant subspace and typically also for its image under the matrix, as well as a window of CG search direction vectors to periodically update/improve the recycle space. As fast memory is quite limited on the GPU, we reduce the memory requirements by storing all these vectors in single precision and we avoid the image of the recycle space at the cost of an extra matrix-vector product. However, as the problem is ill-conditioned, we may need to solve to double precision accuracy. This presentation brings together an interesting blend of CG and high performance computing related topics, demonstrating how well-known algorithms need to be adapted for high performance architectures. We discuss a convergence result for CG when 'recycling' an approximate invariant subspace in a sequence of linear systems and demonstrate that the convergence is not sensitive to small perturbations (such as storing the basis vectors for the space in single precision). We also discuss how to solve to double precision accuracy while using single precision intermediate results and vectors stored in single precision without iterative refinement using the theory for inaccurate matrix-vector products in iterative linear solvers and how to control that accuracy.
Speaker: Max Heldman
Virginia Tech
Title: Methods for studying dense interacting particle systems: mean field limits and fluctuations
In this talk we discuss asymptotic methods for studying interacting reaction-drift-diffusion particle systems as the number of particles goes to infinity. First, we look at mean field models for reaction-diffusion systems and compare the result to standard mass action reaction-diffusion partial differential equations (PDEs). Then, we present a stochastic PDE model which captures the fluctuations of the particle concentrations about their deterministic mean-field limit. Finally, we show preliminary work on a drift-diffusion model.
November 3, 2023
Speaker: Heike Faßbender
TU Braunschweig
Title: On a new family of low-rank algorithms for large-scale algebraic Riccati equations
Finding the unique stabilizing solution \(X = X^H\) of a large-scale continuous-time algebraic Riccati equation (CARE) \(0 = R(X) := A^H X + X A + C^H C - X B B^H X\) with a large, sparse \(n \times n\) matrix \(A\), an \(n \times m\) matrix \(B\) and an \(p \times n\) matrix \(C\) is of interest in a number of applications. Here, \(B\) and \(C\) are assumed to have full column and row rank, resp., with \(m, p \ll n\). The unique stabilizing solution \(X = X^H\) is positive semidefinite and makes the closed-loop matrix \(A - B B^H X\) stable. Even so \(A\) is large and sparse, the solution \(X\) will still be a dense matrix in general. But our assumptions on \(B\) and \(C\) often imply that the sought-after solution \(X\) will have a low numerical rank (that is, its rank is \(\ll n\)). This allows for the construction of iterative methods that approximate \(X\) with a series of low rank matrices stored in low-rank factored form. To be precise, we focus on Hermitian low-rank approximations \(X_j\) to \(X\) of the form \(X_j = Z_j Y_j Z_j^H\), where \(Z_j\) is an \(n \times k_j\) matrix with only few columns and \(Y_j\) is a small square \(k_j \times k_j\) Hermitian matrix. There are several methods which produce such a low-rank approximation. Our approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by \(A^H\) and \(C^H\). The approximations \(X_j\) as well as \(\lVert R(X_j) \rVert_F\) can be computed fast and efficiently. In particular, our approach gives a whole new family of algorithmic descriptions of the same approximation sequence \(X_j\) to the Riccati solution as four other algorithms for CARE previously known in the literature.
November 10, 2023
(DDS 366)
(DDS 366)
Speaker: Volker Mehrmann
TU Berlin
Title: Hypocoercivity, hypocontractivity and short-time decay of solutions to linear evolution equations
For linear evolution equations (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time decay behavior of the norm of the fundamental solution matrix for linear continuous-time and discrete-time systems is characterized by an integer called hypocoercivity index or hypocontractivity index, respectively. The results extend to operators in Hilbert spaces and can be applied to the analysis of anisotropic flows.
December 1, 2023
Speaker: Sam Bender
Virginia Tech
Title: Transfer Function and \(\mathcal{H}_2\) Norm for Linear Time-Periodic Systems
Time-periodic systems are ubiquitous both naturally and as engineered systems. This is often as a consequence of periodic forcing due to rotation (e.g., the Earth's rotation generates both tidal gravity forces and diurnal temperature gradients that cyclically drive atmospheric and ocean flows; gyroscopic forces can generate significant vibration and noise in vehicles). More broadly, periodic phenomena can occur through the emergence of a dynamic balance between inertial and various restoring forces. For example, a structure exposed to an otherwise steady wind or current flow can experience large oscillations caused by vortex shedding or flutter. This can be destructive (the Tacoma Narrows bridge failure is a famous example); but there can be positive effects as well (e.g., high-efficiency windmills may take advantage of these effects). Linear time-periodic (LTP) systems play a fundamental role in the analysis, simulation, and control of such phenomena even when the underlying models reflect fundamentally nonlinear dynamics, since by their character the periodic phenomena of interest emerge as components of a "center manifold" and must themselves be stable at least when subjected to small perturbations. Were this not the case, say for a natural system, oscillatory phenomena would not generally be observed, while for an engineered system, they would not generally be desired. Beyond this, computational strategies for extracting periodic solutions of nonlinear systems necessitate repeated solution of linear periodic systems, and this leads (naturally) to the question of effective model order reduction for such systems.
Speaker: Hayden Ringer
Virginia Tech
Title: Shape From Sound: The Inverse Laplacian Eigenvalue Problem
The frequencies of a vibrating planar membrane correspond to the Dirichlet Laplacian eigenvalues on the planar domain. In 1966, Mark Kac posed the famous question "Can One Hear the Shape of a Drum?". In other words: is the shape of a planar domain uniquely determined by its Dirichlet Laplacian spectrum? In 1992, Gordon, Webb, and Wolpert constructed a pair of non-isometric domains with identical Laplacian eigenvalues, answering Kac's question in the negative. In this talk, we present a perspective on the inverse eigenvalue problem as an ill-posed inverse problem, and present some preliminary work in applying Bayesian inference techniques to this question. In particular, we will present the perturbation theory for Dirichlet Laplacian eigenvalues, as well our modifications to the state-of-the-art algorithm for computing Dirichlet Laplacian eigenvalues: Trefethen and Betcke's improved Method of Particular Solutions.
Contributions – Spring Semester 2023
February 10, 2023
Speaker: Max Heldman
Virginia Tech
Title: A convergent reaction-diffusion master equation with pair potentials
Particle-based stochastic reaction-diffusion (PBSRD) systems are used in computational biology to model the dynamics of particles which react, diffuse, and interact in space. In this talk, we describe a numerical method for mesoscale simulations of PBSRD systems based on the convergent reaction-diffusion master equation (CRDME) framework. The CRDME generates approximate samples from the probability density for the particle locations by discretizing the forward Kolmogorov equation for the density in space using a quadrature-modified finite element method, and then solving the resulting continuous-time Markov chain using a variant of Gillespie's stochastic simulation algorithm. Our work extends previous versions of the CRDME by adding pair potentials, which can be used to model, e.g., electrostatic forces and soft-core repulsion. The new method exhibits optimal convergence rates for the spatial discretization and exactly preserves important physical properties such as detailed balance.
February 24, 2023
Speaker: Michal Outrata
Virginia Tech
Title: Domain Truncation, Absorbing Boundary Conditions, Schur Complements, and Padé Approximation
In 1655, an Oxford professor John Wallis published his book Artihmetica Infinitorum - The Arithmetic of Infinitesimals - introducing the term continued fraction, which has later been used as the tool of choice in approximation theory for nearly 300 years, until the notation now bearing the name of Henri Padé became the one commonly used in the field. Around the time when Padé introduced his work - namely in 1870 - Hermann Schwarz introduced an iterative process for solving the Laplace problem, nowadays called the alternating Schwarz method. It was introduced as a theoretical tool but has since evolved into a broad family of associated methods - the Schwarz methods. One of the important aspects of these methods is choosing the interface conditions through which the method propagates the subdomain solution information and one common class of these interface conditions is called absorbing boundary conditions (ABCs). In this talk we marry these two areas - continued fractions and ABCs - and show for a model problem that the truncation of an unbounded domain by an artificial Dirichlet boundary condition placed far away from the domain of interest is equivalent to a specific absorbing boundary condition at the boundary of the domain of interest. We prove that the absorbing boundary condition thus obtained is a spectral Padé approximation about infinity of the transparent boundary condition. We also propose two additional ABCs motivated by this result and numerically show their efficiency for our test problem.
March 17, 2023
Speaker: Agnieszka Międlar
Virginia Tech
Title: Randomized FEAST Algorithm for Generalized Hermitian Eigenvalue Problems with Probabilistic Error Analysis
Randomized NLA methods have recently gained popularity because of their easy implementation, computational efficiency, and numerical robustness. We propose a randomized version of a well-established FEAST eigenvalue algorithm that enables computing the eigenvalues of the Hermitian matrix pencil \((A, B)\) located in the given real interval \(I \subset [\lambda_{\min}, \lambda_{\max}]\). In this talk, we present the analysis of this randomized variant of the subspace iteration method using a rational filter and propose several modifications of the original FEAST algorithm. First, we derive some new structural as well as probabilistic error analysis of the accuracy of approximate eigenpair and subspaces obtained using the randomized FEAST algorithm, i.e., bounds for the canonical angles between the exact and the approximate eigenspaces corresponding to the eigenvalues contained in the interval, and for the accuracy of the eigenvalues and the corresponding eigenvectors. Since this part of the analysis is independent of the particular distribution of an initial subspace, we denote it as structural. In the case of the starting guess being a Gaussian random matrix, we provide more informative, probabilistic error bounds. Our new algorithm allows to improve the accuracy of orthogonalization when \(B\) is ill-conditioned, efficiently apply the rational filter by using MPGMRES-Sh Krylov subspace method to accelerate solving shifted linear systems and estimate the eigenvalue counts in a given interval. Finally, we will illustrate numerically the effectiveness of presented error bounds and proposed algorithmic modifications. This is joint work with E. de Sturler (VT) and A. K. Saibaba (NC State).
Speaker: Mark Embree
Virginia Tech
Title: The Interior Structure of the Numerical Range: Inverse Problems and Projection Methods
The numerical range (or field of values) of a matrix is a closed, bounded, convex set in the complex plane that contains the eigenvalues. When analyzing various algorithms involving nonsymmetric matrices, the numerical range provides a convenient proxy for the eigenvalues. Unfortunately, the set is often inconveniently large. In this talk we will describe two attempts to refine our understanding of the numerical range: inverse numerical range problems, and a kind of nonsymmetric interlacing theorem due to Russell Carden. We will describe the implications of this interlacing for projection-based model reduction: If we perform Galerkin projection of a stable model, how many unstable modes can the reduced order model have?
March 24, 2023
Speaker: Sean Reiter
Virginia Tech
Title: Structure-preserving Extensions of the Quad-BT Framework
In this work we present extensions of the Quadrature-based Balanced Truncation (Quad-BT) framework of Gosea, Gugercin, and Beattie, to other types of balancing. Quad-BT is a "non-intrusive" reformulation of balanced truncation; a classical projection-based model order reduction technique for linear systems. Quad-BT builds accurate reduced-order models entirely from system response data (e.g., transfer function measurements) without the need to reference an explicit state-space realization of the underlying system. We extend this data-driven framework to include other types of balancing; namely, balanced stochastic truncation, positive-real balancing, bounded-real balancing, and frequency-weighted balanced truncation. This is accomplished by sampling certain spectral factors associated with the system of interest. We verify this approach with several numerical examples.
Speaker: Petar Mlinarić
Virginia Tech
Title: Optimal Reduced-order Modeling of Structured Linear Systems
There are various approaches to \(\mathcal{H}_2\)-optimal reduced-order modeling of (unstructured) linear time-invariant systems, such as the iterative rational Krylov algorithm that is based on interpolation. Interpolatory model order reduction was also extended to structured linear systems. We are interested in extending \(\mathcal{H}_2\)-optimal reduced-order modeling to structured linear systems and investigating whether it necessitates interpolation. Interpolatory \(\mathcal{H}_2\)-optimality conditions have been established for certain second-order systems and port-Hamiltonian systems, as well as special time-delay systems. Furthermore, for a special class of parametric linear time-invariant systems, interpolatory \(\mathcal{H}_2 \otimes \mathcal{L}_2\)-optimality conditions are known. In this talk, we will present some generalizations of these results using the work on \(\mathcal{L}_2\)-optimal reduced-order modeling and present them on a few numerical examples.
March 31, 2023
Speaker: Yichen Guo
Virginia Tech
Title: Stopping Criteria for the Conjugate Gradient Algorithm in High-order Finite Element Methods
The discretization of partial differential equations using finite element methods leads to the formulation of large-scale linear systems. Researchers often employ iterative techniques to solve these systems, which give rise to two distinct errors: algebraic error induced by the linear solver, and discretization error depending on the discretization. While algebraic error converges to zero as iterations progress, discretization error remains constant. Consequently, determining the optimal point to terminate iterations in order to maintain overall precision and prevent over-solving becomes a critical question. In this presentation, we will describe a stopping criterion based on the residual for the Poisson equation. First, we will review error estimation in the conjugate gradient method, followed by an exploration of polynomial-degree robust a posteriori error estimators for high-order finite element methods. Subsequently, we will introduce an alternative stopping criterion based on the linear residual and two associated terms, and also consider the Poisson equation with a heterogeneous coefficient. Finally, we will demonstrate the robustness and efficiency of this criterion by applying it to several benchmark problems and comparing its performance with other criteria.
Speaker: Ashlyn McDonald
Virginia Tech
Title: Learning Hyperparameters for Inverse Problems by Deep Neural Networks
Inverse problems arise in a wide variety of applications including biomedicine, environmental sciences, astronomy, and more, but computing reliable solutions to these problems requires the inclusion of prior knowledge, in a process that is often referred to as regularization. Most regularization techniques require suitable choices of regularization parameters, so I will describe new approaches that use deep neural networks to obtain regularization parameters. We can train multiple networks to approximate mappings from observation data to individual regularization parameters, and once the networks are trained, regularization parameters for newly obtained data can be computed by efficient forward propagation of the DNNs. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods.
April 7, 2023
Speaker: Jonathan Baker
Virginia Tech
Title: How much controllability is enough?
It is well known that a linear time-invariant system is (state) controllable if and only if the associated controllability matrix (CM) is full rank. To find the CM rank, it may be sufficient to examine a submatrix. This is especially true for multi-input systems for which the controllability matrix may have many more columns than rows; the rank of the entire CM may be the same as the submatrix formed by the first few blocks. The number of blocks needed to reach the maximum rank may be called the grade of the system. I will briefly explain how several areas benefit from understanding how the system grade is affected by the properties of the system and control matrices. I will present some widely known results, new results, and examples.
April 14, 2023
Speaker: Jonas Nicodemus
University of Stuttgart
Title: A New Perspective on Model Reduction for Port-Hamiltonian Systems
State of the art structure-preserving model order reduction (MOR) methods for port-Hamiltonian (pH) systems focus only on approximating the input-output dynamics by (approximately) minimizing classical system norms. Nevertheless, a pH system consists of two objects: the input-output dynamics and an energy function that is typically called the Hamiltonian. If we thus measure the approximation quality only with respect to the input-output dynamics, then the approximation of the Hamiltonian is not reflected at all. In this talk, we take a first step towards the dual-objective minimization problem of the optimal approximation of the input-output dynamics and the Hamiltonian by noticing that in the pH reduced-order model, we can modify the Hamiltonian without changing the system dynamics. Thus, for a given pH reduced-order model, we can search for the reduced Hamiltonian that best approximates the full Hamiltonian. Moreover, we propose a numerical algorithm to solve the minimization problem and demonstrate its applicability with numerical examples. This talk describes joint work with Paul Schwerdtner (TU Berlin) and Benjamin Unger (U Stuttgart).
Speaker: Linus Balicki
Virginia Tech
Title: Energy-Based Approximation of Linear Systems with Polynomial Outputs
Balanced truncation is a popular model order reduction technique that reduces the order of a system based on its state variables' contribution to the controllability and observability energy. For linear time-invariant systems, this approach has been thoroughly investigated and scalable computational frameworks exist. In this talk, I will focus on systems that are linear in the state equation but whose output is a polynomial of the state variables. The underlying energy functions of such systems turn out to be polynomials as well and the coefficients of these polynomials can explicitly be computed by solving large-scale tensor-structured linear systems. I will talk about the computational aspects of solving these tensor-structured systems and point out possible ways to use these tensor-based energy function representations in the context of model order reduction.
April 19, 2023
(10:15-11:15am)
(10:15-11:15am)
Speaker: Alice Cortinovis
Stanford University
Title: Speeding up Krylov subspace methods for matrix functions via randomization
In this talk we consider the computation of the action of a matrix function \(f(A)\), such as the matrix exponential or the matrix square root, on a vector \(b\). For a general matrix \(A\), this can be done by computing the compression of \(A\) onto a suitable Krylov subspace. Such compression is usually computed by forming an orthonormal basis of the Krylov subspace using the Arnoldi method. In this talk, we propose to compute (non-orthonormal) bases in a faster way and to use a fast randomized algorithm for least-squares problems to compute the compression of \(A\) onto the Krylov subspace. We present some numerical examples which show that our algorithms can be faster than the standard Arnoldi method while achieving comparable accuracy.
April 21, 2023
Speaker: Tobias Breiten
TU Berlin
Title: On the approximability of Koopman-based operator Lyapunov equations
Computing the Lyapunov function of a system plays a crucial role in optimal feedback control, for example when the policy iteration is used. This talk will focus on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman operator. Since this infinite-dimensional system has the structure of a weak-\(*\) continuous semigroup in a specially weighted \(L_p\)-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay, which justifies finite rank approximations with numerical methods. The usefulness for numerical computations will also be demonstrated with two short examples. This is joint work with Bernhard Höveler (TU Berlin).
April 28, 2023
Speaker: Cankat Tilki
Virginia Tech
Title: Wavelet-DMD in the Context of Extended-DMD
Dynamic Mode Decomposition and its variants have been successfully used in data-driven modeling of dynamical systems in diverse applications. DMD has been generalized so that it admits external forcing and outputs, giving rise to the input-output (ioDMD). DMD and ioDMD work with full state measurements. However in practice one does not always have access to full measurements. This leads to versions of DMD, such as delay-DMD, that use partial state measurements rather than full ones. One of the recent techniques that works with partial state information in a DMD setting is Wavelet-DMD (WDMD). The main idea behind this method is to create observables based on the stationary wavelet coefficients using the available measurements and apply ioDMD with these observables. Specifically, WDMD uses maximal overlap discrete wavelet transform to generate an auxiliary state variable from partial measurements, and then uses ioDMD to analyze dynamics with respect to this auxiliary variable and creates a data-driven model for the underlying dynamics. In this talk, we demonstrate that WDMD can be represented formally in the context of Extended DMD with a specific choice of observables. Thus we establish the connection between WDMD and the extensions of DMD. We also provide numerical examples to illustrate the theory.
Speaker: Sam Bender
Virginia Tech
Title: Reduction of Time-Periodic Systems
Time-periodic systems are ubiquitous both naturally and as engineered systems. This is often a consequence of periodic forcing due to rotation (e.g., the Earth's rotation generates both tidal gravity forces and diurnal temperature gradients that cyclically drive atmospheric and ocean flows; gyroscopic forces can generate significant vibration and noise in vehicles). One particular example of interest is tidal currents, which are modeled with nonlinear PDEs. Linearization around a periodic solution leads to linear time-periodic PDEs, then further discretization in space leads to large systems of time-periodic ODEs. In this presentation, I investigate computational strategies for model order reduction techniques that further simplify these time-periodic systems of ODEs into constant coefficient systems.
May 1, 2023
Speaker: Perfect Y. Gidisu
TU Eindhoven
Title: Generalized CUR-type factorizations for improved data analysis
A CUR factorization is a type of low-rank matrix decomposition that approximates a given matrix using a small subset of its rows and columns as bases for its row and column spaces. Unlike traditional low-rank decompositions, which use orthonormal bases, a CUR factorization offers advantages such as preserving sparsity and non-negativity, improved data interpretation, and reduced storage requirements. Mathematically, we aim to solve a combinatorial problem where we select a subset of rows and columns from a data matrix to construct thin matrices C and R. The goal is to minimize the approximation error over all possible choices for C and R. An exact solution to this problem is intractable, so researchers seek efficient mathematical methods that yield a solution not worse than the optimal by some factor. However, finding deterministic algorithms that result in a modest factor is still an open problem. We have developed deterministic algorithms that seek to minimize this factor while also extending the CUR factorization to multiple matrices.
Contributions – Fall Semester 2022
September 9, 2022
Speaker: Matthew Brown
Virginia Tech
Title: Advanced Research Computing (ARC) Overview and Getting Started
ARC is Virginia Tech's centralized resource for research computing at scale. ARC hosts clusters of computational equipment designed to be a platform for high performance computing (HPC) and based on standard models for HPC in academic and industrial settings. In addition, ARC provides data storage, software, and application interfaces to lower barriers to entry along with support, instruction, and consultation services. This talk will provide an overview of ARC's systems, a brief introduction to some standard parallelization concepts which are fundamental to computational system design, and some "getting started" information for those interested in using ARC systems.
September 16, 2022
Speaker: Linus Balicki
Virginia Tech
Title: Data-driven model order reduction with the p-AAA algorithm
Rational approximation represents a powerful tool for accurately capturing complex system dynamics via low-order models. The adaptive Antoulas-Anderson (AAA) algorithm combines the benefits of interpolatory methods and least-squares approximation to compute univariate rational approximants based on a set of function samples. A generalization to the AAA algorithm which enables approximations via multivariate rational functions has been introduced as the parametric AAA (p-AAA) algorithm. The method has been successfully used in order to obtain low-order surrogate models which precisely capture the behaviour of parametric dynamical systems and stationary models. I will present recent modifications to p-AAA which increase its viability for practical use cases amongst others by computing real-valued state-space representations of the p-AAA approximant and improving the conditioning of the numerical computations in various steps in the algorithm.
September 23, 2022
Speaker: Ning Wan
Virginia Tech
Title: Numerical Study of Anderson Acceleration Method
Anderson Acceleration is a method which allows to speed-up the convergence of the fixed point iteration, which is widely used within the SCF (Self-Consistent Field Iteration) method in quantum chemistry/physics computations. In this talk, we will discuss the convergence analysis and numerical behavior of Anderson Acceleration method.
September 30, 2022
Speaker: Jonathan Baker
Virginia Tech
Title: Nonlinear eigenvalue veering
Eigenvalue veering is a well-known phenomenon affecting most problems involving spectra of parameterized symmetric matrices. As a symmetric matrix changes, its (real) eigenvalues may change, but they do not typically combine into a double eigenvalue. When the eigenvalues are plotted against the matrix parameter, the eigenvalues may approach each other, but they appear to "veer" away to avoid each other. What's more, the eigenvectors of the veering eigenvalues seem to "jump" from one eigenvalue to the other. It turns out that veering and jumping tend to occur in parameterized nonlinear eigenvalue problems as well. In my presentation, I'll introduce this topic more thoroughly with examples and some theory.
October 14, 2022
Speaker: Eric de Sturler
Virginia Tech
Title: Golub-Kahan Bidiagonalization for Streaming Problems
We use the Golub-Kahan bidiagonalization algorithm with hybrid projection to solve inverse problems with regularization with only a limited part of the matrix available at a time and using only limited memory. The algorithm allows accessing the matrix one block at a time, this can be a block of rows or a block of columns (and a mix of these in the future). This is useful for extremely large problems, for fast processors with limited (local) memory, and for problems where we want to compute or update approximate solutions while data is coming in. It also allows us to combine randomization with iterative solvers. The possibility of solving a linear system with only a subset of columns of the matrix available at a time also allows a range of interesting solution strategies for general matrices.
Speaker: Petar Mlinarić
Virginia Tech
Title: Is IRKA a Riemannian Optimization Method?
The iterative rational Krylov algorithm (IRKA) is a well-known method for \(\mathcal{H}_2\)-optimal model order reduction (MOR) of linear time-invariant systems. The \(\mathcal{H}_2\)-optimal MOR problem is in finding the distance between a point and a set in the Hardy \(\mathcal{H}_2\) space, where the point represents the full-order model and the set a subset of reduced-order models. We consider the Riemannian embedded submanifold of reduced-order models of fixed McMillan degree in the \(\mathcal{H}_2\) space and explore whether IRKA can be interpreted as a Riemannian optimization method over this manifold.
October 17, 2022
(10-11am)
(10-11am)
Speaker: Daniel Appelö
MSU
Title: WaveHoltz: How the wave equation can be used to solve Helmholtz equation
We discuss the WaveHoltz iteration, for solving the Helmholtz equation. Our method makes use of time domain methods for wave equations to design frequency domain Helmholtz solvers. We show that the WaveHoltz iteration we propose results in a positive definite linear system even though we are solving the Helmholtz equation. A unique “free lunch” property that WaveHoltz possesses allows us to solve for multiple frequencies at the cost of a single solve. We will present numerical examples, using various discretization techniques, that show that our method can be used to solve fairly large problems with rather high wave numbers.
October 21, 2024
Speaker: Christian Frederiksen
Tulane University
Title: A Framework for Posterior Consistency in PDE Inverse Problems
One is often interested in estimating functional parameters in a partial differential equation given sparse and noisy observations of the solution. A Bayesian statistical methodology provides a comprehensive approach for such problems and establishing posterior consistency is an important step in validating this approach. In this talk I will introduce both posterior consistency and the Bayesian approach to inverse problems and present a newly developed abstract framework for establishing posterior consistency in PDE inverse problems. The abstract nature of the framework makes it easily adaptable to new problems unlike existing results which focus on specific problems. Additionally, and quite significantly, it allows for the use of fixed Gaussian priors and different observation types which are both absent in existing literature on PDE inverse problems. The talk should be readily accessible to anyone with a decent understanding of basic probability and PDE theory. This is joint work with Nathan Glatt-Holtz.
Speaker: Ashlyn McDonald
Virginia Tech
Title: Hyperparameter Learning by Deep Neural Networks (DNNs)
Inverse Problems have a wide variety of applications in medicine, environmental sciences, astronomy, and beyond, but these problems are often ill-posed for a number of reasons. In order to combat this, we can use prior knowledge in the form of hyperparameters to find approximate solutions to inverse problems. To begin, I'll provide some basic information about inverse problems and discuss previous work that has been done using machine learning to find hyperparameters. From there, I'll discuss our approach to using deep neural networks to learn multiple hyperparameters and demonstrate a few numerical results via an example in tomography.
October 28, 2024
Speaker: Serkan Güğercin
Virginia Tech
Title: A data-driven formulation for balanced truncation of bilinear dynamical systems
We introduce a non-intrusive data-driven time-domain formulation of balanced truncation (BT) for bilinear control systems. We build on our recent method [Gosea, Gugercin, Beattie, SIAM SISC, 2022] that recasts the classic BT method for linear time invariant systems as a data-driven method requiring only evaluations of either transfer function values or impulse responses. We extend the domain of applicability of this non-intrusive data-driven method to bilinear systems, a class of weakly nonlinear systems. This is a joint work with Ion Victor Gosea (MPI Magdeburg), Igor Pontes Duff (MPI Magdeburg), and Christopher Beattie (VT).
Speaker: Johann Rudi
Virginia Tech
Title: Computational Methods for Viscoplastic Stokes Flow
The main driver for the presented computational methods is coming from the goal to simulate Earth's mantle convection, which models 10,000-years-scale motion of rock from the surface down to ~3000km depth. While Earth's mantle flow is a fundamental geophysical process, enormous knowledge gaps remain. The reasons: Realistic mantle models pose computational challenges due to highly nonlinear rheologies, severe heterogeneities, anisotropies, and wide ranges of spatial scales. We present advances in nonlinear implicit solvers for Earth's instantaneous mantle flow governed by nonlinear instantaneous Stokes PDEs: (1) Heterogeneity-robust Schur complement preconditioning; (2) Advancing Newton's method for the computational challenges of viscoplastic Stokes Flow.
November 4, 2024
Speaker: Yichen Guo
Virginia Tech
Title: Stopping Criteria for the Conjugate Gradient Algorithm in High-order Finite Element Methods
In finite element methods for partial differential equations, the computed solution is to approximate the exact solution of the partial differential equation in discretization spaces. It is obtained by solving a linear system of equations using iterative methods. A natural question is when to stop the iterations to get a sufficiently accurate solution and avoid overly solving the linear system. In this talk, I will introduce several methods to evaluate the energy norm of the difference between the exact solution and the approximate solution obtained when stop the iteration by some criteria. We will start with error estimation in the conjugate gradient algorithm linked with Gauss quadrature, and then discuss a method to estimate the total error using flux reconstructions which is polynomial-degree-robust.
Speaker: Sean Reiter
Virginia Tech
Title: Integral-Driven Computation of Hamiltonian Matrix Elements for Parallel Implementations of Selected-CI Methods
Quantum chemistry provides powerful tools for understanding the rich electronic structure of molecular systems. One major challenge faced by standard implementations of traditional wave function methods is that they incur significant computational costs, e.g. scaling on with complexity \(\mathcal{O}(N^5)\) and higher, where \(N\) is proportional to the size of the chemical system of interest. This necessitates efficient implementations of these methods that take advantage of modern parallelism in order to study problems of even modest sizes. In this talk, we discuss a new integral-driven implementation of a selected configuration interaction method, CIPSI (Configuration Interaction using a Perturbation Selection made Iteratively). Our contributions include an integral-driven framework for implementing Slater-Condon rules. This provides an alternative means for deriving Hamiltonian matrix elements in a particular basis of Slater determinants. We also discuss the overall parallelization scheme, as well as a distributed "matrix free" implementation of Davidson's diagonalization algorithm.
November 11, 2024
Speaker: Dan Folescu
Virginia Tech
Title: Contour Integral Methods: A Room Filled with Knobs and Dials
Nonlinear operators present unique obstacles to the stable, yet efficient characterization of their spectra. In this setting, Contour Integral Methods (CIMs) are particularly appealing; the trapezoid rule applied to conformations of the unit circle yields exponentially convergent numerical quadratures and a great potential for parallelization. Further, one can interpret the nonlinear eigenvalue problem (NLEVP) as one of data realization for a corresponding transfer function. These qualities provide a fertile foundation for the exploration of optimal data sampling strategies - an aspect which will be showcased in this talk. We will begin with a motivating example and brief systems-theoretic description of the NLEVP. Care will be taken to delineate Hankel and Single-point Loewner strategies, with investigations via robust visual codes to follow. Finally, we will envision near-future tools of the trade, and end with a question: what features would help you find a rank drop in a \(\sigma\)-stack?
Speaker: Sam Bender
Virginia Tech
Title: Reduction of Time-Periodic Systems: the Floquet-Arnoldi Decomposition
Time-periodic systems are ubiquitous both naturally and as engineered systems. This is often as a consequence of periodic forcing due to rotation (e.g., the Earth's rotation generates both tidal gravity forces and diurnal temperature gradients that cyclically drive atmospheric and ocean flows; gyroscopic forces can generate significant vibration and noise in vehicles). One particular example of interest is tidal currents, which are modeled with nonlinear PDEs. Linearization around a periodic solution leads to linear time-periodic PDEs, then further discretization in space leads to large systems of time-periodic ODEs. In this presentation, I investigate computational strategies for model order reduction techniques that further simplify these time-periodic systems of ODEs into constant coefficient systems.
November 18, 2024
Speaker: Arvind K. Saibaba
NC State
Title: Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte Carlo estimators based on random Rademacher, sparse Rademacher, normalized and unnormalized Gaussian vectors. The novel use of matrix concentration inequalities in our proofs represents a systematic model for future analyses. Our bounds mostly do not depend on the matrix dimension and imply that the accuracy of the estimators increases with the diagonal dominance. I will also present LanczosMC a method for estimating the diagonals of the inverse of a symmetric positive definite matrix that combines a preconditioned Lanczos low-rank approximation for the inverse with a Monte Carlo estimator. The accuracy of the methods is demonstrated by numerical experiments on synthetic test matrices and applications to inverse problems, sensitivity analysis, and network science.
December 2, 2024
Speaker: Alan Garcia
Virginia Tech
Title: Dynamic recovery for power networks using DMD
Electrical power grids are one of the biggest and most important industrial systems, and so we need proper mathematical models (aka Swing Equations) to understand their dynamics and make predictions about their performance. We can use a DMD approach to recover matrices that reproduce the measurements but this is not enough to estimate the dynamical parameters of the model. We need to recover matrices that preserve the sparsity pattern and underlying properties as being symmetric Laplacian, diagonal, identity matrix, etc. We will present a DMD framework that includes a regularization step where we impose the underlying structure of the model.