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.
Highlights
Computing the action of a matrix function on a vector is experiencing these days a reborn interest
The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain
In the field of partial differential equations, numerically solving a boundary-value PDE problem by means of the method of lines requires in practice to compute the action of a matrix exponential over the initial condition
Summary
Computing the action of a matrix function on a vector is experiencing these days a reborn interest. For the specific case of computing the action of a Hermitian matrix exponential over a vector, which is of interest in Quantum Physics, it has been proposed in [32] an efficient algorithm based on a novel randomized linear algebra technique known in the literature as the Nystrom method These probabilistic methods offer important computational advantages. The purpose of this paper is to extend the existing aforementioned Monte Carlo methods for dealing with other functions of matrices, such as the matrix exponential, and for the problem of computing the action of a matrix exponential on a vector for a certain class of matrices This is done by resorting to a probabilistic representation of the vector solution based on generating random paths corresponding to samples of a suitable continuous-time Markov chain. To conclude we summarize the main results and discuss potential directions for future research
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