Adrianna Gillman: 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 a 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 speedup 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.