A highly parallel algorithm for computing the action of a matrix exponential on a vector based on a multilevel Monte Carlo method

A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths which evolve through the indices of the matrix according to a suitable probability law. The computational complexity is proved in this paper to be significantly better than the classical Monte Carlo method, which allows the computation of much more accurate solutions. Furthermore, the positive features of the algorithm in terms of parallelism were exploited in practice to develop a highly scalable implementation capable of solving some test problems very efficiently using high performance supercomputers equipped with a large number of cores. For the specific case of shared memory architectures the performance of the algorithm was compared with the results obtained using an available Krylov-based algorithm, outperforming the latter in all benchmarks analyzed so far.The work has been performed under the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme; in particular, the authors gratefully acknowledge the support of the Computer Architecture Department at Universitat Politècnica de Catalunya (UPC), Spain and the computer resources and technical support provided by Barcelona Supercomputing Center (BSC), Spain. We acknowledge PRACE, Belgium for awarding us access to Marconi at CINECA, through grant 2010PA4246. This work was also supported by Fundação para a Ciência e a Tecnologia, Portugal under Grant No. UIDB/50021/2020, by the Spanish Ministry of Science and Technology through TIN2015-65316-P project and by the Generalitat de Catalunya, Spain (contract 2017-SGR-1414).Peer Reviewe