Hardware-Oriented Krylov Methods for High-Performance Computing

TL;DR

This paper develops hardware-efficient Krylov subspace methods for high-performance computing, optimizing communication and computation to improve the solution of linear systems with multiple right-hand sides.

Contribution

It introduces a block Krylov framework, novel stabilization for CG and BiCGStab, and communication-optimized variants of GMRes tailored for distributed systems.

Findings

Optimized Krylov methods outperform traditional approaches in distributed settings.

New stabilization approach simplifies implementation of block Krylov methods.

Communication overlaps significantly reduce computation time.

Abstract

Krylov subspace methods are an essential building block in numerical simulation software. The efficient utilization of modern hardware is a challenging problem in the development of these methods. In this work, we develop Krylov subspace methods to solve linear systems with multiple right-hand sides, tailored to modern hardware in high-performance computing. To this end, we analyze an innovative block Krylov subspace framework that allows to balance the computational and data-transfer costs to the hardware. Based on the framework, we formulate commonly used Krylov methods. For the CG and BiCGStab methods, we introduce a novel stabilization approach as an alternative to a deflation strategy. This helps us to retain the block size, thus leading to a simpler and more efficient implementation. In addition, we optimize the methods further for distributed memory systems and the communication overhead. For the CG method, we analyze approaches to overlap the communication and computation and present multiple variants of the CG method, which differ in their communication properties. Furthermore, we present optimizations of the orthogonalization procedure in the GMRes method. Beside introducing a pipelined Gram-Schmidt variant that overlaps the global communication with the computation of inner products, we present a novel orthonormalization method based on the TSQR algorithm, which is communication-optimal and stable. For all optimized method, we present tests that show their superiority in a distributed setting.