Lightning-fast Boundary Element Method

Abstract

Boundary element methods (BEM) for solving linear elliptic partial differential equations have gained traction in a wide range of graphics applications: they eliminate the need for volumetric meshing by solving for variables exclusively on the domain boundary through a linear boundary integral equation (BIE). However, BEM often generate dense and ill-conditioned linear systems that lead to poor computational scalability and substantial memory demands for large-scale problems, limiting their applicability and efficiency in practice. In this paper, we address these limitations by generalizing the Kaporin-based approach to asymmetric preconditioning: we construct a sparse approximation of the inverse-LU factorization of arbitrary BIE matrices in a massively parallel manner. Our sparse inverse-LU factorization, when employed as a preconditioner for the generalized minimal residual (GMRES) method, significantly enhances the efficiency of BIE solves, often yielding orders-of-magnitude speedups in solving times.