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.

January 28, 2026

Organizational Meeting

February 4, 2026

Speaker

February 11, 2026

Speaker: Sheena Zeng

Title: BDDC Algorithms for Parabolic Problems

Abstract: This talk presents an application of Balancing Domain Decomposition by Constraints (BDDC) methods to parabolic problems discretized via the finite element and Backward Euler methods. Unlike standard sequential approaches, we utilize diagonalization-based parallel-in-time algorithms to solve linear systems across different time steps simultaneously. We show that a BDDC-preconditioned GMRES method provides a scalable solver for the resulting systems. To optimize performance, we also discuss some techniques to minimize total computational cost such as using inexact BDDC preconditioners. 

February 18, 2026

Speaker

February 25, 2026

Speaker

March 4, 2026

Speaker: Shuhao Cao (University of Missouri-Kansas City)

Title:  Transformer Neural Operators: Past and Present

Abstract: Transformers, introduced in "Attention is all you need" in 2017, have become the ubiquitous backbone architecture in modern deep learning across vision, language, and various scientific applications, such as GPT or AlphaFold. We formulate the heart and soul of Transformers, "Attention," as a Neural Operator, which can emulate maps between function spaces. In this talk, we offer unique perspectives on Attention through the lens of numerical analysis and approximation theory. In the meantime, we identify suitable scenarios for Neural Operators to enhance, rather than replace, classical reliable numerical schemes for various difficult problems, such as Electrical Impedance Tomography.

March 11, 2026

Speaker: 

March 25, 2026

Speaker

April 1, 2026

Speaker: Weishi Liu

Title:  Shear flow of nematic liquid crystals: Bifurcations for multiplicity and stability of steady states.

Abstract: In this talk, we will discuss a number of research activities for shear flow of nematic liquid crystals: (i) existence and multiplicity of steady states; (ii) bifurcation as mechanism for multiple steady states; (iii) implications of bifurcation specifics on stability of steady states. Nearly all these activities are in their early stages however a landscape for shear flow of nematic liquid crystal is ready to emerge.

April 8, 2026

Speaker: Liwei Cui

Title:  Nonlinear preconditioners for nonlinear PDEs.

Abstract: In this talk, we investigate nonlinear preconditioning techniques to enhance the robustness and efficiency of Newton’s method for solving nonlinear partial differential equations (PDEs). Standard Newton methods often struggle to converge in the presence of strong nonlinearities or when supplied with poor initial guesses. We compare the performance of the Newton–Krylov–Schur (NKS) method with that of nonlinear elimination (NE)–preconditioned Newton methods. Furthermore, we explore a two-level nonlinear preconditioning strategy that incorporates the Full Approximation Scheme (FAS) into the nonlinear elimination framework.

April 15, 2026

Speaker:  Sungmin Won

Title:  Scalable two-level domain decomposition methods for the Physics and Equality Constrained Artificial Neural Networks.

Abstract: Artificial neural networks have drawn significant attention as a powerful tool for approximating solutions to partial differential equations (PDEs) due to their key strengths: a mesh-free structure and computational feasibility for high-dimensional problems. Recent studies have explored various approaches, including Physics and equality constrained artificial neural networks (PECANN). PECANN aims to find data-driven solutions by minimizing loss functions based upon physical laws with constraints on boundary conditions. However, training a single neural network on large datasets spanning the entire domain can be inefficient, especially for complex problems. To address this, domain decomposition techniques are employed alongside PECANN. The computational domain is decomposed into overlapping or non-overlapping subdomains, and PECANN is applied locally on each subdomain to approximate the PDE solutions. A coarse neural network is used to ensure scalability so that the number of iterations does not depend on the number of subdomains. Numerical experiments demonstrate the efficiency and scalability of the proposed algorithms.

April 22, 2026

Speaker: Sumit Kumar

Title:  Effect of Stress Triaxiality and Lode Angle Parameters on Ductile Fracture using Phase Field Coupled Elasto-plasticity

Abstract: The effects of stress triaxiality and Lode angle parameters on crack initiation and propagation are investigated using a phase-field-coupled elasto-plasticity formulation. A multi-invariant, i.e., the first invariant of the stress tensor and the second and third invariants of the deviatoric stress tensor, dependent finite deformation hyperelasto-plasticity is coupled with the phase field theory of ductile fracture. A phase-field-coupled multi-invariant-dependent yield function is proposed by incorporating the Hosford equivalent stress into the Drucker–Prager yield function to accurately predict the ductile response prior to failure initiation. The measure of ductility is reported in the form of damage initiation surface in terms of threshold plastic energy as a function of stress triaxiality and normalized Lode angle parameters using the Mohr–Coulomb fracture initiation criterion. The crack initiation and propagation paths predicted by the proposed model for different states of stress triaxiality and Lode angle parameters, including axisymmetric tension, plane strain, and axisymmetric compression, match those reported experimentally in the literature.

April 29, 2026

Speaker:  Xu Zhang (Oklahoma State University)

Title:  Immersed Finite Element Methods for 3D Interface Problems: Theory and Applications

Abstract: Interface problems arise widely in science and engineering applications. They are often modeled by partial differential equations (PDEs), whose solutions typically exhibit low regularity, including kinks, singularities, discontinuities, and other non-smooth behaviors. Conventional finite element methods require meshes that align with the interface, which can be computationally expensive when the geometry is complex, especially in three dimensions. The immersed finite element (IFE) method provides an effective framework for solving such problems on interface-unfitted meshes. 

This talk presents recent advances in the development and analysis of IFE methods for three-dimensional interface problems. We construct IFE spaces on unfitted cuboidal or tetrahedral meshes that accommodate general interface geometries. Fundamental analytical results are established, including trace and inverse inequalities, as well as optimal approximation properties of the proposed spaces. These constructions are further applied to numerical schemes for solving PDE interface problems, for which optimal a priori error estimates are derived in both energy and L^2 norms. Numerical experiments are presented to validate the theoretical results and demonstrate the accuracy and robustness of the proposed methods, including their effectiveness for realistic three-dimensional interface geometries.

May 6, 2026

Speaker: Weizhang Huang