Seminar is regularly scheduled on Wednesday, 2:00 - 3:00 pm in 306 Snow
Seminar is regularly scheduled on Wednesday, 2:00 - 3:00 pm in 306 Snow
Fall 2023
August 30, 2023
Organizational Meeting
September 6, 2023
Speaker: Weizhang Huang (KU)
Title: A grid-overlay finite difference method for the fractional Laplacian on arbitrary bounded domains
Abstract: In this talk I will represent a grid-overlay finite difference method for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domain and constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantages of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries and mesh adaptation. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. I will present numerical examples to demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented.
September 13, 2023
Speaker: Zhuoran Wang (KU)
Title: Full Weak Galerkin FEMs for linear and nonlinear poroelasticity on quadrilateral meshes
Abstract: In this talk, we present full weak Galerkin (WG) schemes for the poroe-
lasticity, which couples the Darcy 's law for fluid pressure and the law for
displacement of porous media. Both Darcy pressure and elasticity are
discretized using the WG finite element method. Discrete weak gradient
and numerical velocity are established in the Arbogast-Correa space. We
formulate the fully-discrete system using the backward differentiation for-
mula (BDF) method. Numerical experiments are presented for validating
the accuracy and the locking-free property of the new solvers. This is
a joint work with Dr. James Liu, Dr. Simon Tavener, and Dr. Ruishu
Wang.
September 20, 2023
Speaker: None Scheduled
September 27, 2023
Speaker: None Scheduled
October 4, 2023
Speaker: Weishi Liu (KU)
Title: Gouy-Chapman Layer and Beyond
October 11, 2023
Speaker: None Scheduled
October 25, 2023 (Joint with Probability and Statistics Seminar)
Speaker: Fraydoun Rezakhanlou (University of California at Berkeley)
Title: Kinetic Theory for Laguerre Tessellations
Abstract: In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.
Talk will be only on Zoom at 4:00PM!
https://kansas.zoom.us/j/93561034799
Meeting ID: 935 6103 4799
Passcode:1025
November 1, 2023
Speaker: Volker Mehrmann (TU Berlin)
Title: Hypocoercivity, hypocontractivity and short-time decay of solutions to linear evolution equations
Abstract: For linear evolution equations (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time decay behavior of the norm of the fundamental solution matrix for linear continuous-time and discrete-time systems is characterized by an integer called hypocoercivity index or hypocontractivity index, respectively. The results extend to operators in Hilbert spaces and can be applied to the analysis of anisotropic flows.
November 8, 2023 (CAM seminar at 2PM in 306 Snow)
Speaker: James Liu (Colorado State)
Title: Numerical Simulations for Subdiffusive Transport in Poroelastic Media
Abstract: Many biological and geological problems can be modeled as transport in porous media, e.g., gas and oil extraction from petroleum reservoirs and drug delivery to cancer sites. Moreover, the media could be poroelastic and transport is subdiffusive. In this talk, we present results from our on-going efforts on development of efficient and robust numerical solvers for time-fractional convection-diffusion problems and (linear and nonlinear) poroelasticity problems. We pay special attention to positivity-preserving, local mass conservation, and free of Poisson-locking. The talk is based on a couple of collaborative projects with researchers at Jilin University (China), KU, and ColoState.
November 8, 2023 (Joint with Probability and Statistics Seminar 4PM Central time on Zoom)
Speaker: Xiang Zhou (City University of Hong Kong)
Title: Value-Gradient Formulation for Optimal Control and Machine-Learning Algorithm: Eulerian and Lagrangian Viewpoint
Abstract: Optimal control problem is typically solved by first finding the value function through Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control.
In this work, 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 continuous―time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the properties that they share the same characteristic curves as the HJE for the value function. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates. This example will highlight the importance of unifying Eulerian and Lagrangian viewpoints for designing numerical schemes for high dimensional equation in computational math. This talk is mainly based on the joint work published at https://epubs.siam.org/doi/10.1137/21M1442838.
Bio: Professor Xiang Zhou received his BSc from Peking University (School of Mathematical Sciences) and PhD from Princeton University (PACM). He holds the associate professor at School of Data Science, City University of Hong Kong now. His major research focus is the study of rare event and computational methods for stochastic models, and has recent interests in machine learning algorithms for control, sampling and rare events.
Zoom info:
Time: Nov 8, 2023 04:00 PM Central Time (US and Canada)
https://kansas.zoom.us/j/93986919169
Meeting ID: 939 8691 9169
Passcode: 1108
November 15, 2023
Speaker: Geng Chen (KU)
Title: The unique and stable inviscid limit from compressible Navier-Stokes to Euler equations
Abstract: Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data.
I will discuss the recent result on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing artificial viscosity limit result for BV solutions of hyperbolic conservation laws. We use the relative entropy to prove the L^2 stability and uniqueness of this limit. This is a join work with Moon-Jin Kang and Alexis F. Vasseur.
November 29, 2023
Speaker: Hongguo Xu (KU)
Title: Speeding up QR algorithm with two-sided Rayleigh quotient shifts
Abstract: We improve the local convergence of the nonsymmetric QR algorithm to cubic rate by incorporating two-sided Rayleigh quotient shifts. For efficiency we propose a truncated version of the shift. With the truncated shift cubic local convergence rate is still observed and the CPU time can be saved up to 20% - 25%. We also propose the two-sided 2D Grassmann-Rayleigh quotient double-shifts for the doubly shifted QR algorithm. This is a joint work with Xiao-Shan Chen from South China Normal University.
December 6, 2023
Speaker: Erik Van Vleck (KU)
Title: Multi-Scale Data Assimilation Techniques via Projected Physical and Data Models
Spring 2024
January 24, 2024
Organizational Meeting
February 7, 2024
Not scheduled
February 14, 2024
Not scheduled
February 21, 2024
Not scheduled
February 28, 2024
Not scheduled
March 6, 2024
Ruishu Wang (Jilin University)
Title: A penalty free weak Galerkin finite element method on quadrilataral meshes
Abstract: The weak Galerkin finite element methods are non-standard finite element methods. The newly defined weak functions are considered as the approximate functions, which have two parts, inner and boundary, on each element. Weak derivatives are correspondingly defined. Appropriate spaces should be used when no penalty term is employed. We use the Arbogast-Correa element to define the weak gradient and obtain a penalty-free weak Galerkin scheme, which is then employed to solve problems related to Stokes flow, linear elasticity, and poroelasticity.
Ruishu Wang is an Associate Professor in the School of Mathematics at Jilin University.
Her research is centered on the numerical solution of partial differential equations, with a particular emphasis on the weak Galerkin finite element method and the numerical calculation of elasticity problems.
March 13, 2024
Spring Break
March 20, 2024
Not scheduled
March 27, 2024
Matthias Morzfeld (UCSD/ IGPP Scripps)
Title: High dimensional covariance estimation: Ad hoc tricks and mathematical explanations
Abstract: I consider the problem of estimating a n x n covariance matrix from a set of m sampled vectors, each of dimension n. The caveat is that m is much less than n, i.e., the dimension is huge and the number of samples is small. I will explain why this seemingly mundane problem is so important across all of Earth science and then report how this problem has been partially solved in numerical weather prediction (NWP) via a procedure called "covariance localization." We will then construct a theory of optimal localization. A numerical and analytical comparison of optimal and practical approaches reveals that (i) the practical approach, used for decades in NWP, very much mimics an optimal approach, implying that there is mathematical support for the practical, ad hoc solution; and (ii) the details of how covariance matrices are localized have only a second order effect, implying that "even bad localization is good localization." The latter point is quite subtle, but has practical implications which I will discuss. This talk summarizes work done in collaboration with Dr. Daniel Hodyss of the Naval Research laboratory and is supported by an ONR grant (N00014-21-1-2309).
April 3, 2024
Joonha Park (KU)
Title: Sampling from high-dimensional, multimodal distributions using automatically tuned, tempered Hamiltonian Monte Carlo
Abstract: Hamiltonian Monte Carlo (HMC) is widely used for sampling high-dimensional target distributions with known probability density up to proportionality. Despite its favorable dimension scaling properties, HMC encounters challenges when applied to strongly multimodal distributions. Traditional tempering methods that are often used to tackle multimodal distributions can be challenging to tune, particularly in high dimensions. In this work, we propose a method that combines tempering strategies with Hamiltonian Monte Carlo, facilitating efficient sampling of high-dimensional, strongly multimodal distributions with unknown mode locations. Our approach involves proposing candidate states for the Markov chain by solving Hamiltonian equations of motion with time-varying mass. Compared to simulated tempering or parallel tempering methods, our approach offers a distinctive advantage in scenarios where the target distribution changes at each iteration, such as in the Gibbs sampler. We further develop an automatic tuning strategy for our method, resulting in an automatically-tuned, tempered Hamiltonian Monte Carlo (ATHMC). We demonstrate the efficacy of ATHMC in constructing Markov chains with frequent transitions between isolated modes, using mixtures of log-polynomial densities and Bayesian posterior distributions for a sensor network self-localization problem.
April 10, 2024
Not scheduled
April 17, 2024
Hongguo Xu (KU)
Title: Solutions of a class of Riccati inequalities
Abstract: We consider a class of Riccati inequalities that has important applications in systems and control. We study the solvability of the Riccati inequalities in the most general setting. One main idea is to turn a Riccati inequality to a Riccati equation by adding a term that is treated as a perturbation. We then use the Hamiltonian spectral perturbation theory to characterize all the positive definite solutions of the Riccati equalities.
This is a joint work with Volker Mehrmann from TU Berlin.
April 24, 2024
Yannan Shen (KU)
Title: Global solution and singularity formation for the supersonic expanding wave of compressible Euler equations with radial symmetry
Abstract
In this talk, we will discuss the definition of rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that solutions with rarefaction initial data will not form shock in finite time, i.e. exist global-in-time as classical solutions. On the other hand, singularity forms in finite time when the initial data include strong compression somewhere. Several useful invariant domains will be also given. This is joint work with Geng Chen, Faris A. El-Katri, Yanbo Hu.
May 1, 2024
Not scheduled