Date/Time: Thursday 4/17, 2:30-3:30pm

Location: FAB 1146

Speaker: Shidong Jiang (Center for Computational Mathematics, Flatiron Institute, Simons Foundation)

Title: Fast integral equation methods and the DMK framework

Abstract: In this talk, we will first present an overview of fast integral equation methods (FIEMs).

Fast algorithms, such as the fast multipole methods and their descendants, have made

profound impacts on many areas of scientific computing. Boundary integral equations of

the second kind are the natural formulation for boundary value problems of elliptic partial

differential equations due to the reduction of dimensionality by one, automatic satisfaction

of conditions at infinity for exterior and scattering problems, and well conditioning. The

computational bottleneck that arises because the resulting linear system is dense due to

the nonlocal and long-range nature of integral operators is largely removed by fast algorithms.

FIEMs have been highly successful in solving a large class of problems in fluid mechanics,

elasticity, acoustics, and electromagnetics.

Second, we introduce a new class of multilevel, adaptive, dual-space methods for computing

fast convolutional transforms. The DMK (dual-space multilevel kernel-splitting) framework

uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by

a sequence of corrections at finer and finer scales until the problem is entirely local, at which

point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits

the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting

the use of separation of variables without relying on the FFT. The DMK framework substantially

simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM,

Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work

per grid point, even in a fully adaptive context.