Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods

The method of moments is an important tool for solving boundary integral equations arising in a variety of applications. It transforms the physical problem into a dense linear system. Due to the large number of variables and the associated computational requirements, these systems are solved iteratively using methods such as GMRES, CG and its variants. The core operation of these iterative solvers is the application of the system matrix to a vector. This requires θ(n^2) operations and memory using accurate dense methods. The computational complexity can be reduced to O(nlog n) and the memory requirement to θ(n) using hierarchical approximation techniques. The algorithmic speedup from approximation can be combined with parallelism to yield very fast dense solvers. In this paper, we present efficient parallel formulations of dense iterative solvers based on hierarchical approximations for solving the integral form of Laplace equation. We study the impact of various parameters on the accuracy and performance of the parallel solver. We present two preconditioning techniques for accelerating the convergence of the iterative solver. These techniques are based on an inner-outer scheme and a block diagonal scheme based on a truncated Green´s function. We present detailed experimental results on up to 256 processors of a Cray T3D.