Fall 2024 - Spring 2025

September 4, 2024

Organizational Meeting

October 2, 2024:

Speaker: Jeremiah Buenger (University of Kansas)

Title: Robust Look Ahead Meshes in Data Assimilation on the Kuramoto-Sivashinsky Equation

Abstract: Adaptive spatial meshing has proven invaluable for the accurate, efficient solution of time dependent partial differential equations. In a data assimilation context several factors are present that place increased demands on meshing. These include the use of ensemble solutions and the location and relative importance of observations. 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 over the entire time interval of the forecasts. This increases the efficiency and allows for increased vectorization of the ensemble forecasts while minimizing interpolation of solutions between different meshes. These look ahead meshes support follow a small ensemble and also incorporate the observations at the next analysis time. The techniques to determine these robust meshes are based upon combining metric tensors or mesh density functions that are employed to define non-uniform meshes. We illustrate the robust ensemble look ahead meshes using traveling wave solutions of a bistable reaction-diffusion equation. Finally, numerical experiments with different observation scenarios are presented for a coupled system of two 1D Kuramoto-Sivashinsky equations.

November 6, 2024

Speaker: Jake Weaver (University of Kansas)

Title: Localization of the Kalman filter using a projection-based formulation

Abstract: Localization is a key part of many Data Assimilation (DA) schemes. For problems with complex behavior, we can leverage some sort of localization to both reduce the complexity of the calculations and also to improve the numerical results. The Kalman Filter in particular has seen various methods of localization used, from the LETKF to more advanced methods that are tailored for specific problems. This talk will discuss how to formulate localizations of many forms using projection matrices. By applying the theory of projections onto the Kalman Filter, we can then begin to alter how these projections are formed, and therefore propose many new types of localization. We will see how domain localization is handled using these methods, and then discuss and review some other localization methods, including spectral based methods and more. Then we look at some numerical experiments, showcasing that under this formulation, we still get accurate performance. Lastly, we discuss possible avenues in which this formulation may be used in the future.

November 13, 2024:

Speaker: Jiayue Han (University of Kansas Medical Center)

Title: Value-Gradient Based Formulation of Optimal Control Problem and Machine Learning Algorithm

Abstract: Optimal control problem is typically solved by first finding the value function through the Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value function (value-gradient) as a decoupled system of partial differential equations in the context of a continuous-time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the fact that they share the same characteristic curves as the HJE for the value function. For the theoretical part, we prove that this iterative scheme converges linearly for some suitable parameter in a weight function. For the numerical method, we combine a characteristic line method with machine learning techniques. Specifically, we generate multiple characteristic curves at each policy iteration from an ensemble of initial states and compute both the value function and its gradient simultaneously on each curve as the labelled data. Then supervised machine learning is applied to minimize the weighted squared loss for both the value function and its gradients. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates, particularly with less characteristic data or fewer training steps.

November 20, 2024:

Speaker: Hongguo Xu (University of Kansas)

Title: Invariant subspace perturbation for a non-Hermitian matrix with a single eigenvalue

Abstract: We show how invariant subspaces will change when a matrix with a single eigenvalue is perturbed. We focus on the case when an invariant subspace corresponds to the eigenvalues perturbed from the Jordan blocks of the same size. We characterize the perturbations in terms of fractional orders for the blocks of a matrix that defines an invariant subspace. We also provide explicit formulas for the coefficient matrices associated with the zero and first fractional orders.

December 4, 2024:

Speaker: Xuemin Tu  (University of Kansas)

Title: Overlapping domain decomposition methods for finite volume discretizations

Abstract: Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete L2 norm, an H1 norm, and an H1 semi-norm are introduced to connect the matrices resulting from the FVMs and related bi- linear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.

December 11, 2024:

Speaker: Yuan Liu (Wichita State University)

Title: Sparse grid discontinuous Galerkin methods for time dependent PDEs

Abstract: In this talk, we will introduce a class of adaptive multiresolution discontinuous Galerkin methods for several time dependent PDEs including reaction-diffusion equations, wave equations and Schrodinger equations. The main ingredients of the sparse grid discontinuous Galerkin methods include L2 orthonormal Alpert’s multiwavelets and the interpolatory multiwavelets. By exploring the inherent mesh hierarchy and the nested polynomial approximation spaces, multiresolution analysis can accelerate the computation and adjust the computational grid adaptively. Numerical examples in high dimensional space will be presented to demonstrate the effectiveness of the scheme. 

January 29, 2025

Organizational Meeting

March 12, 2025

Speaker: Weishi Liu (KU)

Title: Ion size effects on fluxes for 1-1 ionic mixtures

Abstract: We investigate effects of ion sizes on ionic flows through ion channels, using a quasi-one-dimensional Poisson-Nernst-Planck model with a hard-sphere potential accounting for ion sizes. For 1-1mixture without permanent charge as a starting point,  we are able to analyze to a great detail the effects and identify concrete problems for further studies toward a real understanding of this extremely rich topic of electrodiffusion with multiple interacting factors. 

March 26, 2025

Speaker: Qingguo Hong (Missuori University of Science and Technology)

Title: A priori error analysis and greedy training algorithms for neural networks solving PDEs

Abstract: We provide an a priori error analysis for methods solving PDEs using neural networks. We show that the resulting constrained optimization problem can be efficiently solved using greedy algorithms, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. This innovative greedy  algorithm is tested on several benchmark examples to confirm its efficiency and robustness.

April 2, 2025

Speaker:  Moon-Jin Kang (KAIST Korea)

Title: Long-time behavior for IBVP of barotropic Navier-Stokes system in 1D half space

Abstract: We will talk about the initial boundary value problem for compressible Navier-Stokes system in 1D half space, when we impose one of the three boundary conditions : impermeable, inflow and outflow at the origin, together with a constant state at far-field. We are interested in the case where the two constant states at the origin and the far-field are connected by a Hugoniot curve possibly together with Rarefaction curve and Boundary layer curve. In this talk, we will focus on the most complicated pattern, the superposition of the boundary layer solution, the 1-rarefaction wave, and the viscous 2-shock waves for the inflow problem when the boundary value is located at the subsonic regime. In this superposition, the boundary layer is degenerate and large. We prove that, if the initial data is a small perturbation of the superposition, then the solution asymptotically converges to the superposition up to a dynamical shift for the shock.

April 9, 2025

Speaker: Erik Van Vleck (KU)

Title: Dimension Reduction for Data Assimilation

Abstract: Data assimilation (DA) techniques combine models and data to make improved predictions often in a Bayesian context. Important challenges including addressing nonlinearity in models and data, detecting intermittent non-Gaussian behavior, and high dimensional models and data. The focus of the talk is on the adaptation of techniques to DA that have proven to be highly successful in the numerical solution of PDEs, adaptive spatial meshing (ASM) and reduced order modeling (ROM), and their potential for integration within DA techniques. After some motivation and discussion of challenges, current research results will be presented, followed by an outlook toward the future. 

April 16, 2025 (2:00pm, Snow Hall 306)

Speaker: Wolfgang Bangerth (Colorado State University)

Title: Simulating complex flows in the Earth mantle

Abstract: On long enough time scales, the Earth mantle (the region  between the rigid plates at the surface and the liquid metal outer core at depth) behaves like a fluid. While it moves only a few centimeters per year, the large length scales nevertheless lead to very large Rayleigh numbers and, consequently, very complex and expensive numerical simulations. At the same time, given the inaccessibility of the Earth mantle to direct experimental observation, numerical simulation is one  of the few available tools to elucidate what exactly is going on in the mantle, how it affects the long-term evolution of Earth's thermal and chemical structure, as well as what drives and sustains plate motion.

I will here review the approach we have taken in building the  state-of-the-art open source solver ASPECT (see http://aspect.geodynamics.org) to simulate realistic conditions in the Earth and other celestial bodies. ASPECT is built using some of the most  widely used and best software libraries for common tasks, such as  deal.II for mesh handling and discretization, p4est for parallel partitioning and rebalancing, and Trilinos for linear algebra. In this  talk, I will focus on the choices we have made regarding the numerical methods used in ASPECT, and in particular on the interplay between higher order discretizations on adaptive meshes, linear and nonlinear solvers, optimal preconditioners, and approaches to scale to thousands of processor cores. All of these are necessary for simulations that can answer geophysical questions.

Finally, I will discuss what mathematical questions have arisen out of applying numerical analysis tools to real-world problems. These questions illustrate that interdisciplinary collaboration is a two-way  street for knowledge exchange.

April 23, 2025

Speaker: Weizhang Huang  (KU)

Title: A regularization approach to parameter-free preconditioning for the efficient iterative solution of singular and nearly singular Stokes, elasticity, and poroelasticity flows

Abstract: While Schur complement preconditioning has been widely studied for saddle point systems, challenges remain when coming to singular and nearly singular systems that arise from Stokes flows and nearly incompressible elasticity and poroelasticity flows. For such systems, existing studies focus on the development of preconditioners spectrally equivalent to the underlying system. Those preconditioners are effective by design; however, they are not efficient in general since they are slated to solve nearly singular systems. In this talk we will present a new approach for developing effective and parameter-free block Schur complement preconditioners for those saddle point systems. A key of this approach is to regularize original systems with inherent identities and construct preconditioners based upon the regularized systems. It will be shown these preconditioners are straightforward to construct and implement. Moreover, bounds on the eigenvalues of the preconditioned systems will be derived. The convergence of MINRES and GMRES applied to those systems will be analyzed and shown to be independent of locking parameters and mesh size. Numerical results in 2D and 3D will be presented.

April 30, 2025

Speaker: Hongguo Xu (KU)

Title: An analytic method for combined eigenvalue and singular value perturbation bounds

 Abstract: We developed an analytic method for deriving a combined eigenvalue perturbation bound for Hermitian matrices and a combined singular value perturbation bound for general matrices. The method is based on elementary calculus techniques. It is simpler and more straightforward than the algebraic approach. The derived bounds are similar to the existing results. This is a joint work with Xiaoshan Chen from South China Normal University.

May 7, 2025

Speaker: Evan Olson (KU)

Title: Neural Networks for the Numerical Solutions of PDEs: PINN, PECANN, and Overlapping Domain Decomposition Methods

Abstract:  Neural networks are increasingly prevalent tools in modern society and can be used to approximate the solutions of partial differential equations (PDEs). This talk will focus on introducing two neural networks schemes for solving PDEs, the Physics Informed Neural Network (PINN) and the Physics and Equality Constrained Neural Network (PECANN). Then, to take advantage of potential parallel computing,  one-level and two-level  overlapping domain decomposition methods will be proposed for the PECANN and numerical experiments will show  the performance of these DD-based PECANN methods.