arXiv subjects

Nicolas L. Guidotti

Publications and source records attributed to Nicolas L. Guidotti.

3 recordsLinked to original sources

arXiv2026

Accelerating a restarted Krylov method for matrix functions with randomization

Many scientific applications require the evaluation of the action of a matrix function on a vector, and Krylov subspace methods are the most common ones for this task. Since the orthogonalization cost and the memory requirements can quickly become overwhelming as the basis grows, the Krylov method is often restarted after a few iterations. This paper proposes a new acceleration technique for restarted Krylov methods based on randomization. The numerical experiments show that the randomized method greatly outperforms the classical approach at the same level of accuracy. In fact, randomization can actually improve the convergence rate of restarted methods in some cases. The paper also compares the performance and stability of several existing randomized methods for solving very large, ill-conditioned problems, complementing the numerical analyses from previous studies.

math.NA↗

arXiv2024

A Fast Monte Carlo algorithm for evaluating matrix functions with application in complex networks

We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function using the power series expansion. This contrasts with existing Monte Carlo methods, which only work with one entry at a time, resulting in a significantly better convergence rate than the original approach. To assess the applicability of our method, we compute the subgraph centrality and total communicability of several large networks. In all benchmarks analyzed so far, the performance of our method was significantly superior to the competition, being able to scale up to 64 CPU cores with remarkable efficiency.

cs.DS↗

arXiv2024

A Stochastic Method for Solving Time-Fractional Differential Equations

We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.

math.NA↗