FlashMP: Fast Discrete Transform-Based Solver for Preconditioning Maxwell's Equations on GPUs

TL;DR

FlashMP introduces a GPU-accelerated preconditioning method using discrete transforms, significantly improving the efficiency and scalability of solving large-scale Maxwell's equations in electromagnetic simulations.

Contribution

It presents a novel discrete transform-based preconditioner, FlashMP, optimized for GPU architectures, enabling faster convergence and better scalability in electromagnetic problem solving.

Findings

Reduces iteration counts by up to 16x.

Achieves speedups of 2.5x to 4.9x over existing libraries.

Maintains parallel efficiency up to 84.1%.

Abstract

Efficiently solving large-scale linear systems is a critical challenge in electromagnetic simulations, particularly when using the Crank-Nicolson Finite-Difference Time-Domain (CN-FDTD) method. Existing iterative solvers are commonly employed to handle the resulting sparse systems but suffer from slow convergence due to the ill-conditioned nature of the double-curl operator. Approximate preconditioners, like Successive Over-Relaxation (SOR) and Incomplete LU decomposition (ILU), provide insufficient convergence, while direct solvers are impractical due to excessive memory requirements. To address this, we propose FlashMP, a novel preconditioning system that designs a subdomain exact solver based on discrete transforms. FlashMP provides an efficient GPU implementation that achieves multi-GPU scalability through domain decomposition. Evaluations on AMD MI60 GPU clusters (up to 1000 GPUs) show that FlashMP reduces iteration counts by up to 16x and achieves speedups of 2.5x to 4.9x compared to baseline implementations in state-of-the-art libraries such as Hypre. Weak scalability tests show parallel efficiencies up to 84.1%.