Fast directional multilevel computation for problems with oscillatory kernels

This paper describes a new directional multilevel algorithm for solving N -body or N -point problems resulted from integral formulations of wave scattering applications. These problems are difficult since the kernel involved is oscillatory. The starting point of our approach is the observation that the interaction between two point sets has a low rank separable representation as long as they follow an angular separation condition. In order to construct the separable representations numerically, we introduce an efficient procedure based on random sampling. Finally, the resulting low rank representations are embedded in a hierarchical multiscale and multidirectional framework to accelerate the far field computation in an optimal way. This new algorithm has O (N log N) complexity, and the numerical results demonstrate its effectiveness in engineering examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)