Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry

Abstract

A multilevel transform is introduced to represent discretizations of integral operators from potential theory by nearly sparse matrices. The new feature presented here is to construct the basis in a hierarchical decomposition of the three-space and not, as in previous approaches, in a parameter space of the boundary manifold. This construction leads to sparse representations of the operator even for geometrically complicated, multiply connected domains. We will demonstrate that the numerical cost to apply a vector to the operator using the nonstandard form is essentially equal to performing the same operation with the fast multipole method. With a second compression scheme the multiscale approach can be further optimized. The diagonal blocks of the transformed matrix can be used as an inexpensive preconditioner which is empirically shown to reduce the condition number of discretizations of the single layer operator so as to be independent of mesh size.