Open Access
June 2009 A fast directional algorithm for high frequency acoustic scattering in two dimensions
Björn Engquist, Lexing Ying
Commun. Math. Sci. 7(2): 327-345 (June 2009).


This paper is concerned with fast solution of high frequency acoustic scattering problems in two dimensions. We introduce a directional multiscale algorithm for the $N$-body problem of the two dimensional Helmholtz kernel. The algorithm follows the approach developed in Engquist and Ying, SIAM J. Sci. Comput., 29 (4), 2007, where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric conguration, the interaction between these two regions through the 2D Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank separated representation for the interaction between these regions. Based on this representation, the computation of the far field interaction is organized in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(NlogN)$ complexity for problems from two dimensional scattering applications. Finally, we combine this fast directional algorithm with standard boundary integral formulations to solve acoustic scattering problems that are of thousands of wavelengths in size.


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Björn Engquist. Lexing Ying. "A fast directional algorithm for high frequency acoustic scattering in two dimensions." Commun. Math. Sci. 7 (2) 327 - 345, June 2009.


Published: June 2009
First available in Project Euclid: 27 May 2009

zbMATH: 1182.65178
MathSciNet: MR2536441

Primary: 65N38 , 65R20

Keywords: fast multipole methods , Helmholtz equation , multidirectional computation , multiscale methods , N-body problems , oscillatory kernels

Rights: Copyright © 2009 International Press of Boston

Vol.7 • No. 2 • June 2009
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