SPARSE APPROXIMATION FOR SOLVING INTEGRAL-EQUATIONS WITH OSCILLATORY KERNELS

An integral equation formulation of Helmholtz's equation is considered as an example of a problem with an oscillatory kernel. For such a problem, the field due to a localized source contains a phase that is a function of position. This allows directional radiation patterns to be produced by using a phase cancellation when constructing "extended" sources. This directional property may be used in solving the integral equation. The region containing the sources is decomposed into subregions, each containing many such extended sources. The directional properties of these extended sources result in an N x N full matrix having many very small elements (which may be approximated by zero), and approximately Order [N] large elements. A matrix defining the transformation between localized and extended source formulations in two dimensions is introduced, and its condition number is calculated. This transformation is effective whenever a large enough number of the unknowns correspond to "smooth" regions containing the sources. Numerical examples for a formulation of scattering of waves obeying Helmholtz's equation are considered in two dimensions. A sample calculation illustrates that the sparse matrix that results allows a solution with Order [N] operations per iteration. A permutation of matrix rows and columns is introduced and an example is given in which it moves large matrix elements towards the diagonal. This suggests that incomplete LU preconditioning might be quite effective. Even without preconditioning, the resulting method is the most efficient available for general surface problems using the Helmholtz equation.