Scott MacLachlan : September 14, 2010

Multigrid Preconditioning Strategies for the Helmholtz Equation

Scott MacLachlan, Department of Mathematics, Tufts University, USA

Tuesday, September 14, 2:00 p.m. in the CERFACS conference room

Abstract:

Because of their wide potential applicability in geophysical and medical imaging algorithms, there has been a sustained interest in fast and accurate numerical techniques for the solution of the Helmholtz Equation in heterogeneous media. In this talk, I will present two preconditioning approaches for solving these equations. In the first approach, we extend the family of shifted-Laplace preconditioners first proposed by Erlangga, Oosterlee, and Vuik. In particular, we revisit the choice of discretization in order to examine questions about the accuracy of the discrete solution and, consequently, extend the multigrid approach to a fourth-order discretization; this is joint work with Oosterlee and Umetani. The second approach focuses, instead, on a continuous reformulation of the equations, allowing direct preconditioning of an auxiliary system; this is joint work with Haber. While these two approaches share many similar characteristics and challenges, the reformulation approach appears to offer an exciting new direction for research in this area.