An efficient algorithm for the parallel solution of high-dimensional differential equations

TL;DR

This paper introduces an adaptive waveform relaxation (AWR) method that significantly accelerates the simulation of high-dimensional differential equations, achieving up to 16 times faster computation through novel heuristics and graph partitioning techniques.

Contribution

The paper presents a new adaptive waveform relaxation method and tailored graph partitioning heuristics to improve the efficiency of solving high-dimensional differential equations.

Findings

AWR coupled with graph partitioning yields 3 to 16 times speedup.

AWR effectively handles high-dimensional differential-algebraic equations.

Heuristics for graph partitioning enhance the performance of AWR.

Abstract

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16.