The root condition decides whether a linear multistep method is stable on the scalar test equation, but it is purely asymptotic: it says nothing about the transient amplification a non-normal companion matrix can produce, and deciding whether a given stability parameter lies in the stability region ordinarily requires computing roots. I will show that the Euclidean division sequence behind the Sturm, or Schur–Cohn, root-location test removes both limitations at once. Applied to the Möbius image of the stability polynomial, it yields coefficients, rational in the stability parameter, that are all positive exactly when the root condition holds, which recovers Schur–Cohn, and that are simultaneously the weights of an exact discrete energy whose decrease over one step equals the squared solution increment. Equivalently, the sequence writes down, in closed form, a rational solution of the Stein equation for the companion matrix, giving a computable bound on all its powers: a Dahlquist G-matrix produced automagically, for every method and every parameter in the stability region, with no Lyapunov equation to solve and no roots to compute.
Meeting Information
The seminar typically meets on Mondays from 12:15pm–1:15pm in McBryde 455 (Commons Room).
The initial organizational meeting for the Fall Semester 2026 will take place August 24, 2026 at noon in McBryde 455 (Commons Room).
Contributions – Fall Semester 2026
August 31, 2026
Speaker: Daniel Appelö
Virginia Tech
Title: Long Division Is All You Need: G-Stability, Automagically
September 14, 2026
Speaker: Mirjeta Pasha
Virginia Tech
Title: Randomized Sketching for Practical Large-Scale Tensor Arithmetic
Adding tensors stored in Tucker format is a fundamental operation in scientific computing, arising naturally in PDE solvers, uncertainty quantification, and data-driven modeling. However, naive approaches suffer from rapid rank growth and exploding memory costs. In this talk, I will present a sketching-based framework that performs compressed arithmetic directly on the factor matrices and core tensors, avoiding the formation of large intermediates while controlling rank and preserving accuracy. By exploiting the algebraic structure of Khatri-Rao and Kronecker products, the method achieves substantial computational savings over direct summation and recompression. I will discuss the computational complexity of the approach and illustrate its effectiveness on problems including a parameter-dependent elliptic PDE solved via GMRES and a linear transport equation discretized with high-order discontinuous Galerkin methods. This talk will be mainly focused on this manuscript: https://arxiv.org/pdf/2603.13532.
September 21, 2026
(12:15pm–2:15pm)
(12:15pm–2:15pm)
Speaker: Anuj Abhishek
Case Western
(Host: Mirjeta Pasha)
Title: Efficient Uncertainty Quantification for Nonlinear Inverse Problems : some recent results
In this talk, we will discuss a computationally efficient framework for uncertainty quantification in nonlinear inverse problems, with particular emphasis on PDE-based imaging problems such as Quantitative Photoacoustic Tomography and Electrical Impedance Tomography. Standard Bayesian approaches typically rely on MCMC methods, which can become prohibitively expensive when each likelihood evaluation requires solving a PDE. Building on a recently proposed two-stage pushforward methodology, we instead formulate a Bayesian regression problem for an auxiliary variable whose posterior is available in closed form, and then push this posterior through a deterministic reconstruction map to obtain uncertainty estimates for the unknown parameter without MCMC sampling. We will discuss the mathematical interpretation of the resulting posterior, including posterior contraction results, as well as numerical performance. We will also indicate how this framework can be combined with learned reconstruction maps and operator-learning methods, providing a broader perspective on mathematically grounded uncertainty quantification for modern inverse problems.
Speaker: Jiangguo Liu
Colorado State
(Host: Tao Lin)
Title: Weak Galerkin Finite Element Solvers for Linear and Nonlinear Poroelasticity
This talk presents a family of any order finite element solvers for both linear and nonlinear poroelasticity problems. The backward differentiation formulas (BDFs) are used for temporal discretizations, whereas the weak Galerkin (WG) finite elements on quadrilaterals are utilized for spatial discretizations. For the latter, the discrete weak gradients of finite element shape functions are established in the vector- or matrix- Arbogast-Correa spaces. Such combinations of BDF and WG discretizations produce numerical solutions that have well-balanced spatial and temporal errors in all six quantities, namely, displacement, dilation (divergence of displacement), stress, pressure, velocity, and normal flux. For nonlinear poroelasticity problems in which permeability depends on solid dilation or the mean stress, Picard iterations are employed to solve the resulting nonlinear algebraic systems. Rigorous analysis together with numerical experiments demonstrate that these new solvers have optimal-order convergence and are free of locking. This talk is based on the joint work with several collaborators.
September 28, 2026
Speaker: Kun Huang
Virginia Tech
Title: A lifting–projection method for the homogeneous Landau equation based on tensor-train cross approximation
We introduce a tensor-train lifting–projection method for the homogeneous Landau equation. The nonlinear collision dynamics are lifted to a linear six-dimensional equation that reduces to spherical diffusion in relative velocity. By solving this diffusion through its exact semigroup, the method avoids a time-step stability restriction and remains stable for arbitrarily large time steps. Tensor-train cross approximation makes the lifted problem computationally feasible, while a low-rank correction enforces conservation of mass, momentum, and energy. Numerical examples for Maxwell molecules and Coulomb interactions demonstrate the method’s accuracy and efficiency.
Speaker: Camilo Arenas Mata
Virginia Tech
Title: Bayesian inverse problems with sparse priors and randomization
Sparse priors, such as the Laplace prior, are of considerable interest in Bayesian inverse problems because they promote sparsity and preserve edges in the solution, which are often more appropriate than the smooth reconstructions obtained with Gaussian priors. The resulting non-Gaussian posterior, however, can be difficult to sample from efficiently, especially in high dimensions. In this talk, we present a posterior sampling approach based on the randomize-then-optimize (RTO) framework and a variable transformation that maps the Laplace prior to a standard Gaussian. The method involves the solution of an optimization problem, which we solve with the Levenberg-Marquardt method. For the linear systems at every step, we compare a randomized direct approximation with an iterative solver accelerated by a randomized preconditioner. We evaluate the performance of the approach through numerical experiments on image deconvolution and diffusion inverse problems.
October 5, 2026
Speaker: Christopher Beattie
Virginia Tech
Title: A computational linear algebra view of the Vecchia approximation
The Vecchia approximation, named after the statistician Aldo Vecchia, is a common covariance approximation strategy used in the analysis of Gaussian processes such as occur in large scale settings found in geospatial regression and machine learning. As a computational linear algebra strategy, the Vecchia approximation has the capacity to represent accurately matrices having inverse Cholesky factors that are close to sparse and it is often able to compute these representations in sublinear time with substantial parallelism. While it does not necessarily replicate any individual entries for the original matrix, it does guarantee optimal approximation accuracy for a given sparsity pattern as measured both with respect to the Kullback-Leibler divergence and with respect to a boutique condition number, the Kaporin condition number. In a slightly different context, Vecchia approximations have also evolved under the name ``incomplete inverse Cholesky preconditioners” within the class of sparse approximate inverse preconditioners used in conjugate gradient iteration. I will discuss the motivation behind the Vecchia approximation in Gaussian process regression, the Nystrom method that it exploits, and the sense in which optimal approximations are produced.
Speaker: Shuang Liu
UNT
(Host: Yingda Cheng)
Title: Computational Moving Boundary Problems
Moving boundary (or often called "free boundary") problems are ubiquitous in nature and technology. A computational perspective of moving boundary problems can provide insight into the "invisible" properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. However, challenges lie in the numerical study of moving boundary problems. Examples include difficulties in solving PDEs in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. In this talk, I will discuss three specific topics of moving boundary problems, with applications plasma physics (ITER tokamak machine design), and cell biology (cell movement).
October 12, 2026
Speaker: Reetish Padhi
Virginia Tech
Title: Beyond linear subspaces: Nonlinear moment matching meets quadratic manifolds
Linear model order reduction (MOR) methods often struggle to efficiently approximate systems exhibiting slow Hankel singular value decay, such as those in transport-dominated problems. Quadratic manifolds (QMs) have been explored in the literature to overcome these challenges. These methods involve approximating the full-order state with a quadratic approximation characterized by linear and quadratic projection matrices. In contrast to other QM approaches, we offer a system-theoretic way of picking these projection matrices with the theoretical guarantees that the resulting reduced order model (ROM) interpolates the so-called nonlinear moments of full order model (FOM), i.e., the steady-state behavior of the FOM for specific input frequencies. Numerical results on the 1D advection equation demonstrate that a low-dimensional QM accurately reconstructs the state, outperforming linear subspaces of much higher dimension.
Speaker: Paul Cazeaux
Virginia Tech
Title: High-Dimensional Randomized Linear Algebra with Tensor Train Sketches, and Applications to Many-Body Physics
Quantum many-body problems, such as spin chains or quantum chemistry problems are posed on state spaces whose dimension grows exponentially with the number of sites, far beyond direct computation. Tensor trains (TT) make such problems tractable whenever entanglement is limited. However, every operation (applying a Hamiltonian, or performing standard linear algebra operations such as linear combinations of states) inflates the TT-ranks, and recompressing them by the standard SVD sweeps quickly becomes the bottleneck. Randomized numerical linear algebra offers an alternative: trace estimation and low-rank approximation from random sketches, with guarantees that do not depend on the ambient dimension. To realize better computational scaling, however, the sketch must itself be a tensor network, and the natural choice (Kronecker-shaped columns, i.e. the Khatri-Rao sketch), fails exponentially as the dimension increases. We introduce TTStack, a family of sketches built by stacking independent random matrix product states, and show that its injectivity is governed by a measure of subspace entanglement. This yields embedding guarantees with parameters linear in the tensor order, and, for states that conserve particle number, linear in the particle number instead. I will point to the resulting algorithms: randomized and adaptive tensor train rounding, trace estimation for partition functions, and a sketched Rayleigh–Ritz eigensolver. Depending on time, I will illustrate them on spin chains and on the ground state and thermodynamics of molecules.
Joint work with Mi-Song Dupuy (Sorbonne Université) and Rodrigo Figueroa Justiniano (Virginia Tech).
Joint work with Mi-Song Dupuy (Sorbonne Université) and Rodrigo Figueroa Justiniano (Virginia Tech).
October 19, 2026
Speaker: Matthew Walker
Virginia Tech
Title: An immersed discontinuous Galerkin method of wave propagation in nonhomogeneous media using high degree polynomials
We present an immersed discontinuous Galerkin (IDG) method for modeling wave propagation in two-dimensional nonhomogeneous acoustic and elastic media. The method allows for interface-independent meshes consisting of elements which are cut by the interface and composed of both media (interface elements) and elements which are composed of only one medium. For interface elements, we construct special piecewise polynomial interface finite element (IFE) shape functions to use as a basis for our discrete solution and test spaces. On noninterface elements, we use the standard two-dimensional Legendre polynomial tensor product basis functions. To illustrate the stability, accuracy, and effectiveness of the IDG method, we present computational results for several interface problems.
October 26, 2026
(12:15pm–2:15pm)
(12:15pm–2:15pm)
Speaker: Eric de Sturler
Virginia Tech
Title: Low rank preconditioner updates using iterative solver information
We discuss new methods to update preconditioners with low-rank updates using information that is generated by the solver. A theoretical analysis shows that this can be a promising approach for both multiple, consecutive right-hand sides as well as for sequences or linear systems where the matrix changes modestly. Numerical experiments demonstrate that the approach can substantially accelerate convergence.
Speaker: Achintya Sunil
Virginia Tech
Title: Preconditioner Updating with Lasso-Based Sparse Approximate Matrix Maps for Selected Columns
Design problems such as topology optimization and PDE-based inverse problems rely on efficient preconditioning strategies to solve sequences of linear systems. The sparse approximate mapping (SAM), which leverages a mapping between two matrices to update the preconditioner, relieves the computational bottleneck of computing a new one. However, identifying a good sparsity pattern for the SAM that combines fast convergence with an efficient map is a challenge. This problem has limited the utility of the SAM in settings where stiffness matrices have a relatively large number of nonzeros per column on average, such as in topology optimization and in high order finite elements. In this presentation, we introduce two methods to solve this problem: the lasso-based pattern sparsification method and the selective column computation strategy. These methods can be used separately or in tandem. The former solves the lasso problem to modest accuracy using the alternating direction method of multipliers (ADMM) to sparsify each column of the pattern. The latter computes only a subset of the total number of columns of the SAM and is useful in settings where the changes between matrices are confined to a small subset of columns. The storage overhead of our approaches remains negligible relative to the stiffness matrix. We demonstrate substantially lower computational costs than recomputing the preconditioner, while maintaining about the same number of GMRES iterations until convergence. Additionally, we discuss selection of the parameters for ADMM.
Speaker: Youngsoo Choi
Lawrence Livermore National Lab
(Host: Traian Iliescu)
Title: Data-Driven Finite Elements: Toward Foundation Models for Computational Science
How can data-driven models generalize beyond the geometries and scales used for training while remaining grounded in governing physics? This talk presents the data-driven finite element method (DD-FEM), a framework that combines learned local approximation spaces with the assembly principles of classical finite elements. The central idea is to learn reusable building blocks from simulations on small domains, then assemble and couple these building blocks through the governing equations to solve larger, previously unseen problems. I will describe how local linear bases and nonlinear representations can be constructed from simulation data and coupled across subdomains. Examples in anisotropic diffusion, linear and nonlinear structural mechanics, and fluid dynamics demonstrate how local models can be reused across domain sizes and configurations, achieving substantial computational savings while maintaining accuracy relative to full-order simulations. Selected examples also illustrate generalization across geometry, physical conditions, and time horizons. These developments motivate a compositional approach to foundation models for computational science: learning local representations that support a range of downstream problems through reuse and physics-based assembly. I will discuss both the demonstrated capabilities and the remaining challenges, including approximation and error analysis, scalable solvers, adaptive refinement, and the conditions under which learned elements can reliably transfer to new problems.
November 2, 2026
Speaker: Rudi Smith
Virginia Tech
Title: Randomized Tucker-Sketched GMRES
Solving large-scale tensor-structured linear systems in the Tucker format, using solvers such as GMRES, faces a fundamental bottleneck whereby the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences: RHOSVD-Tucker sGMRES, which uses randomized HOSVD with per-iteration rank selection, and MLN-Tucker sGMRES, which leverages the multilinear Nyström approximation with a fixed rank. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. We demonstrate also, applied to inverse problems, that the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.
Speaker: Peter DelMastro
Virginia Tech
Title: A Provably Stable Truncation Scheme for the Hierarchical Equations of Motion
We present a technique to provably stabilize the Hierarchical Equations of Motion (HEOM) for simulating non-Markovian quantum dynamics. Building on the recently established equivalence between HEOM and Lindblad pseudomodes, we first identify the non-unitary transformation in the Schrödinger picture that that maps an infinite-dimensional pseudomode system to its equivalent HEOM representation. We then propose to stabilize an HEOM simulation by formally starting with an equivalent pseudomode system, truncating the pseudomodes to finitely many dimensions, and then mapping them to the HEOM basis. In the case of uncoupled pseudomodes, we derive practical analytic expressions for the resulting dynamics in the HEOM basis, whose stability we demonstrate through numerical experiments in both full- and tensor-network arithmetic. For these examples, we also find that our stabilized HEOM scheme uses much lower bond dimensions than HEOM with a standard truncation method, even when this truncation does not cause instability. Our final numerical examples reveal that converting from coupled pseudomodes to HEOM through the same process is not as straightforward due to many couplings created across the HEOM hierarchy.
November 9, 2026
(DDS 450)
(DDS 450)
Speaker: Gergory Janes
Virginia Tech
Title: Toward ROMs for transport-dominated unequal strength and size counter-rotating vortex pairs
Unequal strength and size counter-rotating vortex pairs are dominant coherent structures often found in aero/hydrodynamic flows. Despite their streamwise coherence and centered origin, the orbital coupling of these vortex pairs produces a transport-dominated motion, commonly associated with the Kolmogorov n-width barrier, that are challenging for common mode-based model reduction approaches. In the presented work, the use of translation and rotation time-dependent shifts of our snapshot matrix is performed to remove the orbital energy of the vortex pairs. These shifts enable the characterization of finite core vortices with a reduced number of POD modes in comparison to the modes extracted on the computational frame. This will enable future modal analysis of streamwise vortex filament instabilities and evaluation of parametric dependencies.
November 16, 2026
Speaker: Aryeh Keating
Virginia Tech
Title: Robust forward-backward DMD methods for high-dimensional dynamics
The Dynamic Mode Decomposition (DMD) is a successful and popular approach for the inference of linear operators from discrete data pairs. However, the presence of measurement noise in the snapshot data quickly corrupts the accuracy of the results through statistical bias of the eigenvalues and modes of the inferred DMD operator. There have been different approaches proposed to compensate for noise corruption, ranging from regularization of the underlying DMD least-squares problem to the truncation of corrupted data. One such method that is based on the linearity of the operator of interest is the classical Forward-Backward DMD (FBDMD) method. Thereby, the method computes the geometrical average of the forward operator DMD operator A and the backward DMD operator B, allowing for the cancellation of the bias of the noise in the data. However, FBDMD comes with its own challenges that result from the non-uniqueness of the geometrical average for matrices and the potentially small number of given data samples compared to the dimensions of the snapshots. In this work, we propose an alternative FBDMD operator that achieves accuracy levels comparable to the classical approach. The performance of the method is demonstrated in several numerical experiments.
Speaker: Ghafirlia Istafa
Virginia Tech
Title: Bayesian Inversion of Viscoplastic Rheology in Plate-Boundary-Resolving Mantle Convection with Adaptive Sparse-Grid Surrogates
Earth's mantle convection is strongly influenced by rheological properties that govern the mantle's viscoplastic flows, and simulating these at plate-boundary-resolving scales requires large computational resources. Because these properties cannot be observed directly, they must be inferred from geophysical observations through inverse problems constrained by high-fidelity physical models. In this work, we formulate a Bayesian inverse problem for mantle rheology using observed plate velocities and the global mantle convection model implemented in Rhea. The nonlinear governing equations result in forward simulations with up to billions of degrees of freedom, making the repeated model evaluations required for direct Bayesian inference computationally prohibitive. To overcome this challenge, we construct a prior-informed adaptive sparse-grid surrogate of the data-misfit function. The adaptive procedure concentrates expensive high-fidelity model evaluations in the regions of parameter space most relevant to the Bayesian posterior, reducing the computational burden of exploring the inverse problem. The resulting surrogate enables efficient characterization of the full posterior distribution, including non-Gaussian structure, nonlinear parameter interactions, and correlations that cannot be captured by local or Gaussian approximations. In this way, adaptive sparse-grid surrogates provide a computationally tractable pathway to Bayesian inference for large-scale mantle convection models.
November 30, 2026
Speaker: Sam Bender
Virginia Tech
Title: TBA
TBA
Speaker: Kurt Stein
Virginia Tech
Title: TBA
TBA
December 7, 2026
Speaker: Mike Ackermann
Virginia Tech
Title: TBA
TBA
Speaker: Serkan Gugercin
Virginia Tech
Title: TBA
TBA