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**A fast directional algorithm for high frequency acoustic scattering in two dimensions.**
*(English)*
Zbl 1182.65178

Summary: 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 [the authors, SIAM J. Sci. Comput. 29, No. 4, 1710–1737 (2007; Zbl 1180.65006)], 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 \(O(N\log N)\) 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.

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65R20 | Numerical methods for integral equations |