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

## TL;DR

This paper introduces a highly parallel multilevel Monte Carlo algorithm for efficiently computing the matrix exponential action on a vector, demonstrating superior performance over traditional methods on supercomputers.

## Contribution

It presents a novel multilevel Monte Carlo algorithm that significantly improves computational complexity and scalability for matrix exponential actions.

## Key findings

- Outperforms classical Monte Carlo in accuracy and efficiency.
- Achieves high scalability on supercomputers with many cores.
- Outperforms Krylov-based methods in benchmark tests.

## Abstract

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.