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: TBA
TBA
September 28, 2026
Speaker: Hengyi Shi
Virginia Tech
Title: TBA
TBA
Speaker: Camilo Arenas Mata
Virginia Tech
Title: TBA
TBA
October 5, 2026
Speaker: Christopher Beattie
Virginia Tech
Title: TBA
TBA
Speaker: Kun Huang
Virginia Tech
Title: TBA
TBA
October 12, 2026
Speaker: External Visitor
TBA
(Host: Daniel Appelö)
Title: TBA
TBA
Speaker: Paul Cazeaux
Virginia Tech
Title: TBA
TBA
October 19, 2026
Speaker: Reetish Padhi
Virginia Tech
Title: TBA
TBA
Speaker: Slimane Adjerid
Virginia Tech
Title: TBA
TBA
October 26, 2026
Speaker: Eric de Sturler
Virginia Tech
Title: TBA
TBA
Speaker: Achintya Sunil
Virginia Tech
Title: TBA
TBA
November 2, 2026
Speaker: Rudi Smith
Virginia Tech
Title: TBA
TBA
Speaker: Peter DelMastro
Virginia Tech
Title: TBA
TBA
November 9, 2026
(DDS 450)
(DDS 450)
Speaker: Agnieszka Miedlar
Virginia Tech
Title: TBA
TBA
Speaker: Gergory Janes
Virginia Tech
Title: TBA
TBA
November 16, 2026
Speaker: Aryeh Keating
Virginia Tech
Title: TBA
TBA
Speaker: Ghafirlia Istafa
Virginia Tech
Title: TBA
TBA
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