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 equation, 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(m 3), of the discretized Lippmann–Schwinger operator. With this matrix decomposition, each back substitution requires only O(m 2 log (m)); therefore, a scattering problem with m incident waves can be solved, altogether, in O(m 3 log (m)) flops.
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Chen, Y. A Fast, Direct Algorithm for the Lippmann–Schwinger Integral Equation in Two Dimensions. Advances in Computational Mathematics 16, 175–190 (2002). https://doi.org/10.1023/A:1014450116300
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DOI: https://doi.org/10.1023/A:1014450116300