CSE Distinguished Seminar | Leslie Greengard

Abstract

Over the last several decades, fast, robust, and high-order accurate methods have been developed for solving elliptic and parabolic partial differential equations in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for a source distribution within the domain (if any), followed by solving a boundary integral equation to impose the specified boundary conditions.
 
We present a new set of fast algorithms which are easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. 

Bio

Leslie Greengard received his B.A. degree in Mathematics from Wesleyan University in 1979, and his Ph.D. degree in Computer Science and M.D. degree from Yale University in 1987. He has been a faculty member at the Courant Institute of Mathematical Sciences, NYU since 1989 and served as the Institute’s director from 2006-2011. He is presently the Director of the Center for Computational Mathematics, at the Flatiron Institute, a division of the Simons Foundation. He is a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences.

Greengard and collaborators developed the Fast Multipole Method (FMM) for electromagnetics, the fast Gauss transform for diffusion, high order accurate methods for a variety of wave propagation problems, and spectral deferred correction methods for ODEs. He has also worked on methods for magnetic resonance image reconstruction, cryo-electron microscopy, and biophysical data analysis.