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 and CG and its variants. The core operation of these iterative solvers is the application of the system matrix to a vector. This requires $\theta(n^2)$ operations and memory using accurate dense methods. The computational complexity can be reduced to $O(n \log n)$ and the memory requirement to $\Theta(n)$ 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 potential integral equations of the first kind. We study the impact of various parameters on the accuracy and performance of the parallel solver. We demonstrate that our parallel formulation incurs minimal parallel processing overhead and scales up to a large number of processors. 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. Our code achieves raw computational speeds of over 5 GFLOPS. When compared to the accurate solver, this corresponds to a speed of approximately 776 GFLOPS.