SAYAS NUMERICS SEMINAR     Register to attend talk
Feb. 23, 2021 at 3:30pm (Eastern Time)

Direct Sampling Algorithms for Inverse Scattering

Isaac Harris
Purdue University

In this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far-field data and a known function. It can be shown that the imaging functional is strictly positive in the scatterer and decays as the sampling point moves away from the scatterer. The analysis uses the factorization of the far-field operator and the Funke-Hecke formula. This method can also be shown to be stable with respect to perturbations in the scattering data. We will discuss the inverse scattering problem for acoustic waves.
This is joint work with A. Kleefeld.

Time Fractional Gradient Flows: Theory and Numerics

Wenbo Li
University of Tennessee, Knoxville

This talk is concerned with a generalization of the classical gradient flow problem to the case where the time derivative is replaced by the fractional Caputo derivative. We first define the energy solution for this problem and prove its existence and uniqueness, under suitable assumptions, via a generalized minimizing movements scheme. We then introduce and discuss a semi-discrete numerical scheme where we only discretize in time. We then present a posteriori error estimates, which allow to adaptively choose time steps. We also give convergence rates. Several numerical experiments are also presented in the end to help the understanding of the problem and the numerical scheme.

Quadrature by Zeta Correction

Bowei Wu
University of Texas at Austin

We consider the approximation of singular integrals on closed smooth contours and surfaces. Such integrals frequently arise in the solution of Boundary Integral Equations. We introduce a new quadrature method for these integrals that attains high-order accuracy, numerical stability, ease of implementation, and compatibility with the "fast" algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Our quadrature method exploits important connections between the punctured trapezoidal rule and the Riemann zeta function, leading to remarkably simple construction procedures.