GPU Optimizations for the Hierarchical Poincar\'e-Steklov Scheme

TL;DR

This paper introduces GPU optimizations for the 2D Hierarchical Poincaré-Steklov scheme, significantly enhancing computational speed while maintaining high accuracy for solving high-frequency Helmholtz equations.

Contribution

The work develops straightforward GPU optimizations for the HPS method, enabling faster local static condensation on high-order discretizations with minimal implementation complexity.

Findings

GPU optimizations reduce local static condensation cost.

Numerical experiments show high accuracy of HPS with GPU acceleration.

Significant speedup in solving high wavenumber Helmholtz problems.

Abstract

This manuscript presents GPU optimizations for the 2D Hierarchical Poincar\'e-Steklov (HPS) discretization scheme. HPS is a multi-domain spectral collocation method that combines high-order discretizations with direct solvers to accurately resolve highly oscillatory solutions. The domain decomposition approach of HPS connects domains directly via a sparse direct solver. The proposed optimizations exploit batched linear algebra on modern hybrid architectures, are straightforward to implement, and improve the solver's practical speed. The manuscript demonstrates that GPU optimizations can significantly reduce the traditionally high cost of performing local static condensation for discretizations with very high local order $p$. Numerical experiments for the Helmholtz equation with high wavenumbers on curved and rectangular domains confirm the high accuracy achieved by the HPS discretization and the significant reduction in computation time achieved with GPU optimizations.