A Probabilistic Linear Solver Based on a Multilevel Monte Carlo Method

We describe a new Monte Carlo method based on a multilevel method for computing the action of the resolvent matrix over a vector. The method is based on the numerical evaluation of the Laplace transform of the matrix exponential, which is computed efficiently using a multilevel Monte Carlo method. E...

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Bibliographic Details
Published inJournal of scientific computing Vol. 82; no. 3; p. 65
Main Author Acebrón, Juan A.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.03.2020
Springer Nature B.V
Subjects
Accuracy
Algorithms
Boundary value problems
Computational Mathematics and Numerical Analysis
Decomposition
Iterative methods
Laplace transforms
Linear algebra
Markov chains
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Matrix algebra
Methods
Monte Carlo simulation
Multilevel
Parallel processing
Partial differential equations
Simulation
Statistical analysis
Theoretical
Monte Carlo method
High performance computing
Parallel algorithms
Multilevel
Network analysis
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Summary:We describe a new Monte Carlo method based on a multilevel method for computing the action of the resolvent matrix over a vector. The method is based on the numerical evaluation of the Laplace transform of the matrix exponential, which is computed efficiently using a multilevel Monte Carlo method. Essentially, it requires generating suitable random paths which evolve through the indices of the matrix according to the probability law of a continuous-time Markov chain governed by the associated Laplacian matrix. The convergence of the proposed multilevel method has been discussed, and several numerical examples were run to test the performance of the algorithm. These examples concern the computation of some metrics of interest in the analysis of complex networks, and the numerical solution of a boundary-value problem for an elliptic partial differential equation. In addition, the algorithm was conveniently parallelized, and the scalability analyzed and compared with the results of other existing Monte Carlo method for solving linear algebra systems.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-020-01168-2