Scaling H2 Factorization to Billion-Unknown Three-Dimensional Problems

Abstract

Strong recursive skeletonization provides approximate direct solvers for dense matrices from integral equations and kernel methods, but distributed implementations face substantial communication costs and limited parallelism, especially in three dimensions. We present a distributed-memory $\mathcal{H}^2$ factorization solver for symmetric matrices that combines spatial box coloring, delayed Schur complement updates, streamed sketching-based interpolative decomposition, and BLAS-3 and shared-memory kernels. Spatial coloring enables every process to participate in each boundary stage, while delayed updates and streamed sketching reduce dense storage and communication. To our knowledge, this is the first distributed-memory strong recursive skeletonization factorization demonstrated for three-dimensional volume integral equations at billion-unknown scale. Experiments on up to 512 compute nodes demonstrate factorization of a three-dimensional Laplace system with 3.62 billion unknowns, alongside three-dimensional Helmholtz tests and a two-dimensional Matérn system with 268 million unknowns. Combining streamed sketching and fully delayed updates reduces factorization time by $6.9\times$ for Laplace and $8.5\times$ for Helmholtz relative to the unsketched, eager configuration, while reducing maximum per-process resident memory by 57\% and 54\%, respectively.