# A Monte Carlo method for computing the action of a matrix exponential on a vector

## TL;DR

This paper introduces a Monte Carlo approach for efficiently computing the action of a matrix exponential on a vector, leveraging random paths and Markov chains, and demonstrates its superior performance on large-scale problems.

## Contribution

It extends existing Monte Carlo methods by incorporating probabilistic path generation for matrix exponential actions, enabling efficient large-scale computations.

## Key findings

- The algorithm effectively computes single entries and full vectors.
- It outperforms Krylov-based methods on large-scale problems.
- Benchmarks show significant efficiency gains.

## Abstract

A Monte Carlo method for computing the action of a matrix exponential for a certain class of matrices on a vector is proposed. The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional. This representation extends the existing linear algebra Monte Carlo-based methods, and was used in practice to develop an efficient algorithm capable of computing both, a single entry or the full vector solution. Finally, several relevant benchmarks were executed to assess the performance of the algorithm. A comparison with the results obtained with a Krylov-based method shows the remarkable performance of the algorithm for solving large-scale problems.