## 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.