Speaker: Adrianna Gillman, University of Colorado, Boulder

Title: Fast direct solvers for boundary integral equations 

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 an 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 computational cost of O(N^3). 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 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. Extensions of the single body direct solver to select  applications will also be presented. In these applications, fast direct solvers observe hundreds of times speed up over previously state of the art techniques.  There will also be a peak at an algorithm that is efficient and accurate for variable media scattering problems.    

Bio: Adrianna Gillman is an Associate Professor in the Department of Mathematics at the University of Colorado, Boulder. Her work lies in the intersection of numerical linear algebra and numerical partial differential equations.  Specifically, Adrianna's work focuses 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.