Challenges in computing matrix functions

TL;DR

This paper summarizes the outcomes of the f(A)b workshop focus groups on matrix functions and exponential integrators, outlining three core challenges: knowledge transfer, high-performance and energy-aware computing, and benchmarking. It discusses the distinction between computing $f(A)$ and evaluating $f(A)\bm{b}$, identifies gaps such as outdated surveys, curricular absence, and lack of robust stopping criteria, and proposes concrete next steps including benchmarking standards, matrix-free HPC strategies, and FAIR benchmarking practices. The authors advocate for standardized, language-agnostic benchmarks and test problem collections to enable fair comparisons and reproducibility, and emphasize open science to accelerate adoption. Overall, the work aims to guide the field toward exascale-ready algorithms and workflows, fostering cross-disciplinary impact and sustainable community development for $f(A)$-related methods.

Abstract

This manuscript summarizes the outcome of the focus groups at "The f(A)bulous workshop on matrix functions and exponential integrators", held at the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany, on 25-27 September 2023. There were three focus groups in total, each with a different theme: knowledge transfer, high-performance and energy-aware computing, and benchmarking. We collect insights, open issues, and perspectives from each focus group, as well as from general discussions throughout the workshop. Our primary aim is to highlight ripe research directions and continue to build on the momentum from a lively meeting.