Abstract
The state-of-the-art, large-scale numerical simulations of the scattering problem for the Helmholtz equation in two dimensions rely on iterative solvers for the Lippmann-Schwinger integral equition, with an optimal CPU time O(m 3 log(m)) for an m-by-m wavelength problem. We present a method to solve the same problem directly, as opposed to iteratively, with the obvious advantage in efficiency for multiple right-hand sides corresponding to distinct incident waves. Analytically, this direct method is a hierarchical, recursive scheme consisting of the so-called splitting and merging processes. Algebraically, it amounts to a recursive matrix decomposition, for a cost of O(m3), of the discretized Lippmann-Schwinger operator. With this matrix decomposition, each back substitution requires only O(m2 log(m)); therefore, a scattering problem with m incident waves can be solved, altogether, in O(m3 log(m)) flops.
Original language | English (US) |
---|---|
Pages (from-to) | 175-190 |
Number of pages | 16 |
Journal | Advances in Computational Mathematics |
Volume | 16 |
Issue number | 2-3 |
DOIs | |
State | Published - 2002 |
Keywords
- Fast algorithm
- Lippmann-Schwinger-Helmholtz
- Scattering matrix
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics