WaveHoltz solver delivers 25x speedup for Helmholtz equation on GPUs

Researchers have long struggled to solve the indefinite Helmholtz equation iteratively on massively parallel hardware. Amit Rotem's new approach tackles both the numerical and architectural challenges head-on. The first contribution is a block-level domain decomposition: each subdomain maps to a single thread block, and all subdomain solves run concurrently in one kernel launch. The second is WaveHoltz, a fixed-point iteration tailored for GPUs. It has a minimal memory footprint, requires no reduction operations, and keeps subdomain data largely resident in shared memory, L1, and L2 caches. This design eliminates device-level synchronizations and replaces global memory traffic with register-level operations.

Benchmarked on an NVIDIA A100 with a custom CUDA implementation, WaveHoltz outperforms the standard MINRES solver by 2x to 25x, with the gap widening for larger subdomains. Crucially, evaluating WaveHoltz in single precision yields an additional 2x–10x speedup—an opportunity not available to MINRES because reduced precision destroys Krylov vector orthogonality. Rotem also explores two threading strategies: one degree of freedom per thread for small subdomains, and multiple degrees of freedom per thread for larger ones. This work, submitted to IEEE, promises to accelerate wave simulations in acoustics, electromagnetics, and seismology on GPU clusters.