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

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.

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

“…Finding a probabilistic representation for this problem requires in practice [7] to use a splitting method for approximating the action of the matrix exponential over the vector u as follows,…”

“…However, in practice this does not occur. Through the analysis of the intersection similarity of several networks [7] it was shown that the chosen value of β does not affect significantly the results, being in all cases the differences well below the typical error tolerances, and even becoming smaller for increasingly larger network sizes. Consequently, and to ensure fast convergence of the method, in the simulations below we have used β = 1/λ max , where λ max is the maximum eigenvalue of A.…”

“…The value of β was chosen to be 1 for the small-world network and 1/d max for the scale-free network. The last one was chosen specifically to ensure the convergence of the method, as it was pointed out in [7]. Note that, for both networks, the computational time attains a minimum at a specific value of l 0 , which is well approximated by Eq.…”

“…To prevent such a computationally costly procedure, a reasonable alternative relies on computing a generalization of the communicability, that is e βA , where β is typically interpreted as an effective "temperature" of the network (see [21], e.g.). Essentially the idea that was exploited in [7] was to use the inverse of the maximum eigenvalue as the value of the parameter β, which in practice will control the rapid growth of the norm of the matrix A with the size of the network. Different values of β could have a direct impact not only on the entries of the communicability vector, but also on the ranking of the nodes according to their communicability values.…”

“…One of the main contributions of this paper is precisely to develop a multilevel method for the problem of computing the action of a matrix exponential over a vector. This is done by conveniently adapting the probabilistic representation of the solution derived in [7] to the multilevel framework. In addition, the convergence of the method is analyzed, as well as the computational cost estimated.…”