Steven Wise
University of Tennessee at Knoxville

Friday, May 10, 2019. 4pm-5:15pm in Rolla G5

Title: Convergence Analyses of some Nonlinear Multi-Level Algorithms for Non-Quadratic Convex Optimization Problems via Space Decomposition and Subspace Correction

Abstract: Nonlinear multi-level methods, such as the full approximation storage (FAS) multigrid scheme, are widely used solvers for nonlinear problems. In this presentation, a new framework to analyze FAS-type methods for convex optimization problems is developed. FAS can be recast as an inexact version of a nonlinear multigrid method based on space decomposition and subspace correction, namely the successive subspace optimization (SSO) method of Jinchao Xu and coauthors. The theory is quite general and is an abstraction of both SSO and the preconditioned steepest descent (PSD) method. In our algorithm, we show that the local problem in each subspace can be simplified to be linear and one gradient descent iteration is enough to ensure linear convergence of the FAS scheme.  This work is joint with Long Chen and Xiaozhe Hu.

Host: Han

Nicholas Wintz
Lindenwood University

Friday, May 3, 2019. 4pm-5:15pm in Rolla G5

Title: The Kalman filter on stochastic time scales (joint work with Dylan Poulsen, Washington College)

Abstract: In this paper, we discretize a stochastic linear time-invariant system to a dynamic system on a time scale. We then develop a Kalman filter to estimate the true state for the corresponding system. Here, the measurement-update and time-update equations account for the size of the time step when the time scale is generated randomly. Numerical examples are also provided.

Host: Bohner, Grow

Friday, April 26, 4pm-5:15pm
Rolla Building, Room G5

Leo Rebholz
Professor
Department of Mathematical and Statistical Sciences
Clemson University

Title: A proof that Anderson acceleration really does accelerate convergence in fixed point iterations, with application to incompressible flow

Abstract: We propose, analyze and test Anderson-accelerated Picard iterations for solving the incompressible Navier-Stokes equations (NSE). Anderson acceleration has recently gained interest as a strategy to accelerate linear and nonlinear iterations, based on including an optimization step in each iteration. We extend the Anderson-acceleration theory to the steady NSE setting and prove that the acceleration improves the convergence rate of the Picard iteration based on the success of the underlying optimization problem. The convergence is demonstrated in several numerical tests, with particularly marked improvement in the higher Reynolds number regime. Our tests show it can be an enabling technology in the sense that it can provide convergence when both usual Picard and Newton iterations fail.  Lastly, generalization of the theory to general fixed point iterations will be given.

Host: He

Friday, March 22, 4pm-5:15pm
Rolla Building, Room G5

Kenji Nakanishi
Professor
Research Institute for Mathematical Sciences
Kyoto University

Title:  Global dynamics of the nonlinear Schrödinger equation with potential

Abstract: The nonlinear Schrödinger equation is a typical nonlinear disperisve PDE, which describes various wave phenomena in terms of dispersion and nonlinear interactions of the waves. Such PDEs are known to generate various types of solutions, typically scattering, solitons and blow-up. The goal of this study is to classify the solutions in terms of their behavior forward and backward in time, and to predict it from the initial data. In this talk, we consider the equation with a cubic nonlinearity and a linear potential, so that we can characterize the ground state solitons and the first excited solitons for small mass. The main result is classification into 9 sets of solutions with small mass under an energy constraint slightly above the first excited solitons.

Host: Murphy

Friday, February 1, 4pm-5:15pm
Rolla Building, Room G5


Abner Salgado
Associate Professor
Department of Mathematics
University of Tennessee

Title: Regularity and rate of approximation for obstacle problems for a class of integro-differential operators

Abstract: We consider obstacle problems for three nonlocal operators: 

A) The integral fractional Laplacian

B) The integral fractional Laplacian with drift

C) A second order elliptic operator plus the integral fractional Laplacian

For the solution of the problem in Case A, we derive regularity results in weighted Sobolev spaces, where the weight is a power of the distance to the boundary. For cases B and C we derive, via a Lewy-Stampacchia type argument, regularity results in standard Sobolev spaces. We use these regularity results to derive error estimates for finite element schemes. The error estimates turn out to be optimal in Case A, whereas there is a loss of optimality in cases B and C, depending on the order of the integral operator.

Biographical Sketch:  Dr. Abner J. Salgado is an Associate Professor at the Department of Mathematics at the University of Tennessee, Knoxville. His research revolves around the Numerical Analysis of partial differential equations and related questions. He is interested in the design, analysis and implementation of approximation schemes for complex fluids, nonlocal problems, degenerate and singular diffusion problems, and nonlinear partial differential equations in general.

Host: He

Friday, January 25, 4pm-5:15pm
Rolla Building, Room G5

Ari Stern
Associate Professor
Department of Mathematics
Washington University in St. Louis

Title: Hybrid finite element methods for geometric PDEs

Abstract: Many geometric PDEs have local properties, such as symmetries and conservation laws, that one might wish a numerical method to preserve. With classical finite element methods, it is difficult to make sense of such properties, except in a weak or averaged sense. I will discuss how hybrid finite element methods, based on non-overlapping domain decomposition, provide a natural framework for talking about such properties. Specifically, I will discuss some recent results obtained by applying this approach to the multisymplectic conservation law for Hamiltonian PDEs (joint with Robert McLachlan), as well as to charge conservation in Maxwell's equations (joint with Yakov Berchenko-Kogan).

Biographical Sketch:  Dr. Ari Stern is Associate Professor of Mathematics and Statistics at Washington University in St. Louis, where he has been since 2012. Before this, he received a BA and MA in Mathematics from Columbia University and a PhD in Applied and Computational Mathematics from Caltech, and he did postdoctoral work at UCSD. His research is primarily focused on structure-preserving numerical methods for differential equations, particularly for systems with symmetries, conservation laws, or other geometric structures.

Host: He