Title: Fast Direct Solvers for Boundary Integral Equations

Speaker: Adrianna Gillman, University of Colorado Boulder

Abstract: The numerical solution of linear boundary values problems play an important role in the modeling of physical phenomena.

As practitioners continue to want to solve more complicated problems, it is important to develop robust and efficient numerical methods. For some linear boundary value problems, it is possible to recast the problem as a boundary integral equation which sometimes leads to a reduction in dimensionality. The trade-off for the reduction in dimensionality is the need to solve a dense linear system. Inverting the dense N by N matrix via Gaussian elimination has a computational cost of O(N^3). When N is large, the cost in both memory and computational resources is too high for most practitioners. This talk presents solution techniques that exploit the physics in the boundary integral equation to invert the dense matrix for a cost that scales linearly (or nearly linearly) with N with

small constants. For example, on a laptop computer, a matrix with N=100,000 can be inverted in 90 seconds and applying the solver takes under a tenth of a second. The speed in which new boundary conditions can be processed makes these methods ideal applications involving many solves such as optimal design and inverse scattering. In these applications, fast direct solvers observe hundreds of times speed up over previously state of the art techniques. Examples of how these techniques can be used to create accelerated solvers for problems where the linear scaling solver is excessive will also be presented.

Bio: Adrianna Gillman is an Associate Professor in the Department of Mathematics at the University of Colorado, Boulder. In 2018, she was awarded an Alfred P. Sloan Research Fellowship. Her work has been supported by industrial contracts, NSF and, most recently, the Knut and Alice Wallenberg Foundation Grant, in cooperation with the Royal Swedish Academy of Sciences. Gillman's research lies in the intersection of numerical linear algebra and numerical partial differential equations with an emphasis on developing high order discretizations and efficient solvers for the linear systems that result from these discretizations. Her work has been applied to applications including scattering and Stokes flow.

Zoom: https://northwestern.zoom.us/j/96310557268

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