Fall 2025 - Spring 2026

September 3, 2025

Organizational Meeting

October 8, 2025:

Speaker: Jeremiah Buenger

Title: Adaptive Look Ahead Meshing Methods and Nonlocal Observations for Ensemble Based Data Assimilation

Abstract: Adaptive spatial meshing has proven invaluable for the accurate and efficient computation of solutions of time-dependent partial differential equations. In a DA context the use of adaptive spatial meshes addresses several factors that place increased demands on meshing; these include the location and relative importance of observations and the use of ensemble solutions. To increase the efficiency of adaptive meshes for data assimilation, robust look ahead meshes are developed that fix the same adaptive mesh for each ensemble member for the entire time interval of the forecasts and incorporate the observations at the next analysis time. This allows for increased vectorization of the ensemble forecasts while minimizing interpolation of solutions between different meshes. The techniques to determine these robust meshes are based upon combining metric tensors or mesh density functions to define nonuniform meshes. We illustrate the robust ensemble look ahead meshes using traveling wave solutions of a bistable reaction-diffusion equation. A numerical experiment with different observation scenarios and different meshing methods is presented for a coupled system of two 1D Kuramoto-Sivashinsky equations.

Speaker: Jake Weaver

Title:  Domain and Spectral Localization for Square Root Filters and Their Applications

Abstract: This work develops projection-based localization for the ETKF. An overview of Kalman Filter techniques is presented, and the framework for dimension reduction and localization is then described. Through projections generated by matrices, identifications are made to the inputs of the ETKF algorithm to create a scheme with reductions in dimension. To showcase this generalization of localization, we consider the Lorenz '96 model to generate testing data. We test a domain localization approach, in order to showcase how the scheme can generalize standard Schur product-based localization schemes. In addition, we provide experiments in which the localization is spectral in nature. To do this, we employ a POD-based approach, in which we look at the SVD of a so-called "Snapshot Matrix". Extensive numerical results are given for these two approaches, showcasing various improvements in dimension reduction and accuracy as we change the experimental parameters. The findings highlight the generality of the algorithm, and provide a way to implement other schemes to reduce the dimension of the Kalman Filter.

October 15, 2025:

Speaker: Jon Tremblay

Title: Port-Hamiltonian Realizations of Nonminimal Discrete Time Linear Systems

Abstract: Port-Hamiltonian systems provide structure to a realization, making them ideal for modeling.  Although the equivalence of a general continuous time system and a Port-Hamiltonian system is known,  the discrete time case is largely unexplored.  The equivalence between the continuous and discrete time systems is discussed, along with problems arising in the context of a Port-Hamiltonian realization.  It is shown that having an equivalent Port-Hamiltonian realization is equivalent to solving a so-called discrete time KYP inequality.  For a general nonminimal discrete time system, a reduction is shown to explore the controllable and observable modes of the system and generate a solution for these modes. Using the spectrum of the corresponding symplectic pencil, a solution to the entire KYP inequality can be found, under the assumption that the (2, 2) block is positive definite.

November 12,  2025:

Speaker: Yi Wang (Chinese Academy of Sciences, Beijing, China)

Title: Stability of Riemann solutions

Abstract: I will talk about the recent developments on the time-asymptotic stability of generic Riemann profiles, containing viscous shock and rarefaction wave and even viscous contact wave, to several kinds of viscous conservation laws (compressible Navier-Stokes equations, Boltzmann equation and non-convex conservation laws).

November 19,  2025:

Speaker: Hongguo Xu

Title: Invariant subspace perturbations for defective eigenvalues of structured matrices

Abstract: For structured matrices, their eigenvalues and invariant subspaces have special symmetric patterns. These symmetric patterns play a fundamental role in applications. We consider two types of structured matrices, the matrices that are Hermitian with respect to an indefinite inner product and the Hamiltonian matrices. Using the recently developed general perturbation theory we provide structured fractional perturbation results for the invariant subspaces corresponding to the eigenvalues that are perturbed from a single defective eigenvalue of the same first fractional order. Other related results are also provided.

December 3,  2025:

Speaker: Zhuoran Wang

Title: Fluid-Solid Interaction with Poroelasticity: Numerical Modeling and Application

Abstract: Biot’s theory of poroelasticity provides a fundamental framework for modeling the mechanical behavior of fluid–solid interaction in porous media. This theory plays a central role in geomechanics, biomechanics, petroleum engineering, and hydrology. Despite its broad applicability, several major challenges persist in numerical modeling, including the design of stable finite element spaces, the treatment of heterogeneous and nonlinear physical parameters, and the efficient numerical solution of large, indefinite algebraic systems. In this talk, we present recent advances in numerical methods that address these challenges through the development of stable, parameter-free finite element methods and parameter-robust preconditioning strategies. We introduce flexible finite element spaces that are stable, locking-free and penalty-free, while achieving optimal-order convergence. In addition, we develop parameter-robust and efficient inexact block Schur complement preconditioners and domain decomposition methods for efficient solution of fluid–solid interaction problems. Finally, we verify the effectiveness of the developed methods through real-world applications, including biomechanical simulations of spinal cord dynamics relevant to the study of syringomyelia. These results demonstrate the potential of advanced poroelastic modeling techniques to provide reliable and computationally scalable tools for complex multiphysics systems.

TBD

Organizational Meeting

March 11, 2026

Speaker: Jens Lang (Technical Univeresity of Darmstadt, Germany)