arXiv subjects

Igor Simunec

Publications and source records attributed to Igor Simunec.

11 recordsLinked to original sources

arXiv2026

Restoring similarity in randomized Krylov methods with applications to eigenvalue problems and matrix functions

The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard Arnoldi process. However, the resulting Hessenberg matrix is generally not similar to the one produced by the standard Arnoldi process, which can lead to delays or spike-like irregularities in convergence. In this paper, we introduce a modification of the randomized Arnoldi process that restores similarity with the Hessenberg matrix generated by the standard Arnoldi process. This is accomplished by enforcing orthogonality between the last Arnoldi vector and the previously generated subspace, which requires solving only one additional least-squares problem. When applied to eigenvalue problems and matrix function evaluations, the modified randomized Arnoldi process produces approximations that are identical to those obtained with the standard Arnoldi process. Numerical experiments demonstrate that our approach is as fast as the randomized Arnoldi process and as robust as the standard Arnoldi process.

math.NA↗

arXiv2025

Randomized orthogonalization and Krylov subspace methods: principles and algorithms

We present an overview of randomized orthogonalization techniques that construct a well-conditioned basis whose sketch is orthonormal. Randomized orthogonalization has recently emerged as a powerful paradigm for reducing the computational and communication cost of state-of-the-art orthogonalization procedures on parallel architectures, while preserving, and in some cases improving, their numerical stability. This approach can be employed within Krylov subspace methods to mitigate the cost of orthogonalization, yielding a randomized Arnoldi relation. We review the main variants of the randomized Gram--Schmidt and Householder QR algorithms, and discuss their application to Krylov methods for the solution of large-scale linear algebra problems, such as linear systems of equations, eigenvalue problems, the evaluation of matrix functions, and matrix equations.

math.NA↗

arXiv2025

Randomized biorthogonalization through a two-sided Gram-Schmidt process

We propose and analyze a randomized two-sided Gram-Schmidt process for the biorthogonalization of two given matrices $X, Y \in\mathbb{R}^{n\times m}$. The algorithm aims to find two matrices $Q, P \in\mathbb{R}^{n\times m}$ such that ${\rm range}(X) = {\rm range}(Q)$, ${\rm range}(Y) = {\rm range}(P)$ and $(\Omega Q)^T \Omega P = I$, where $\Omega \in\mathbb{R}^{s \times n}$ is a sketching matrix satisfying an oblivious subspace $\varepsilon$-embedding property; in other words, the biorthogonality condition on the columns of $Q$ and $P$ is replaced by an equivalent condition on their sketches. This randomized approach is computationally less expensive than the classical two-sided Gram-Schmidt process, has better numerical stability, and the condition number of the computed bases $Q, P$ is often smaller than in the deterministic case. Several different implementations of the randomized algorithm are analyzed and compared numerically. The randomized two-sided Gram-Schmidt process is applied to the nonsymmetric Lancozs algorithm for the approximation of eigenvalues and both left and right eigenvectors.

math.NA↗

arXiv2024

Estimation of spectral gaps for sparse symmetric matrices

In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method utilizes Hutchinson's stochastic trace estimator together with the Lanczos algorithm to approximate quadratic forms involving spectral projectors. Instead of focusing on determining the gap between two particular consecutive eigenvalues of $A$, we aim to find all gaps that are wider than a specified threshold. By examining the problem from this perspective, and thoroughly analyzing both the Hutchinson and the Lanczos components of the algorithm, we obtain error bounds that allow us to determine the numbers of Hutchinson's sample vectors and Lanczos iterations needed to ensure the detection of all gaps above the target width with high probability. In particular, we conclude that the most efficient strategy is to always use a single random sample vector for Hutchinson's estimator and concentrate all computational effort in the Lanczos algorithm. Our numerical experiments demonstrate the efficiency and reliability of this approach.

math.NA↗

arXiv2024

A low-memory Lanczos method with rational Krylov compression for matrix functions

In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Krylov subspaces that only involve small matrices. The cost of the compression procedure is negligible with respect to the cost of the Lanczos algorithm. This enables us to avoid storing the whole Krylov basis, leading to substantial reductions in memory requirements. This method is particularly effective when the rational Lanczos algorithm needs a significant number of iterations to converge and each iteration involves a low computational effort. This scenario often occurs when polynomial Lanczos, as well as extended and shift-and-invert Lanczos are employed. Theoretical results prove that, for a wide variety of functions, the proposed algorithm differs from rational Lanczos by an error term that is usually negligible. The algorithm is compared with other low-memory Krylov methods from the literature on a variety of test problems, showing competitive performance.

math.NA↗

arXiv2023

Error bounds for the approximation of matrix functions with rational Krylov methods

We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$ admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in [T. Chen, A. Greenbaum, C. Musco, C. Musco, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 787--811] (arXiv:2106.09806) for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.

math.NA↗

arXiv2023

A sketch-and-select Arnoldi process

A sketch-and-select Arnoldi process to generate a well-conditioned basis of a Krylov space at low cost is proposed. At each iteration the procedure utilizes randomized sketching to select a limited number of previously computed basis vectors to project out of the current basis vector. The computational cost grows linearly with the dimension of the Krylov space. The subset selection problem for the projection step is approximately solved with a number of heuristic algorithms and greedy methods used in statistical learning and compressive sensing.

math.NA↗

arXiv2022

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.

math.NA↗

arXiv2021

Computation of generalized matrix functions with rational Krylov methods

We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness of the algorithms and the accuracy of the bounds is illustrated with numerical experiments.

math.NA↗

arXiv2020

Rational Krylov methods for fractional diffusion problems on graphs

In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a non-analytic function involving fractional powers and $\boldsymbol{b}$ is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

math.NA↗

arXiv2019

Nonlocal network dynamics via fractional graph Laplacians

We introduce nonlocal dynamics on directed networks through the construction of a fractional version of a nonsymmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.

cs.SI↗