Speeding Up Krylov Subspace Methods for Computing \(\boldsymbol{{f}(A){b}}\) via Randomization

Cortinovis, Alice; Kressner, Daniel; Nakatsukasa, Yuji

doi:10.1137/22m1543458

SIAM J. Matrix Anal. Appl.

2024

DOI: 10.1137/22m1543458

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Speeding Up Krylov Subspace Methods for Computing \(\boldsymbol{{f}(A){b}}\) via Randomization

Alice Cortinovis,

Daniel Kressner,

Yuji Nakatsukasa

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2024

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Cited by 3 publications

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A Fast Monte Carlo Algorithm for Evaluating Matrix Functions with Application in Complex Networks

Guidotti,

Acebrón,

Monteiro

2024

J Sci Comput

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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.

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A Fast Monte Carlo Algorithm for Evaluating Matrix Functions with Application in Complex Networks

Guidotti,

Acebrón,

Monteiro

2024

J Sci Comput

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Sketched and truncated polynomial Krylov subspace methods: Matrix Sylvester equations

Palitta,

Schweitzer,

Simoncini

2024

Math. Comp.

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Thanks to its great potential in reducing both computational cost and memory requirements, combining sketching and Krylov subspace techniques has attracted a lot of attention in the recent literature on projection methods for linear systems, matrix function approximations, and eigenvalue problems. Applying this appealing strategy in the context of linear matrix equations turns out to be far more involved than a straightforward generalization. These difficulties include analyzing well-posedness of the projected problem and deriving possible error estimates depending on the sketching properties. Further computational complications include the lack of a natural residual norm estimate and of an explicit basis for the generated subspace. In this paper we propose a new sketched-and-truncated polynomial Krylov subspace method for Sylvester equations that aims to address all these issues. The potential of our novel approach, in terms of both computational time and storage demand, is illustrated with numerical experiments. Comparisons with a state-of-the-art projection scheme based on rational Krylov subspaces are also included.

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A Sketch-and-Select Arnoldi Process

Güttel,

Simunec

2024

SIAM J. Sci. Comput.

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Speeding Up Krylov Subspace Methods for Computing \(\boldsymbol{{f}(A){b}}\) via Randomization

Cited by 3 publications

References 21 publications

A Fast Monte Carlo Algorithm for Evaluating Matrix Functions with Application in Complex Networks

A Fast Monte Carlo Algorithm for Evaluating Matrix Functions with Application in Complex Networks

Sketched and truncated polynomial Krylov subspace methods: Matrix Sylvester equations

A Sketch-and-Select Arnoldi Process

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