Moments of a linear operator, with applications to the trace of the inverse of matrices and the solution of equations

Brezinski, Claude; Fika, Paraskevi; Mitrouli, Marilena

doi:10.1002/nla.803

Cited by 23 publications

(34 citation statements)

References 20 publications

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“…An extrapolation procedure for the estimation of the error norm when solving a linear system was initially introduced in . Extensions of this approach in least squares and regularization were presented in and in estimating the trace of powers of linear operators in Hilbert spaces in . In , the bilinear form x T A − 1 y was estimated for any real matrix A , and in , estimates for the bilinear form y ∗ f ( A ) x , for a Hermitian matrix

A \in {double-struckC}^{p \times p},

and

x, y \in {double-struckC}^{p}

were developed.…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

Estimating the diagonal of matrix functions

Fika

Mitrouli

Roupa

2016

Math Methods in App Sciences

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The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix f(A), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions f(A), where A∈double-struckRp×p is a symmetric matrix and f a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

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A \in {double-struckC}^{p \times p},

and

x, y \in {double-struckC}^{p}

were developed.…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

Estimating the diagonal of matrix functions

Fika

Mitrouli

Roupa

2016

Math Methods in App Sciences

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“…There are several other methods for computing an approximation of the trace of an implicitly defined large symmetric matrix; see, for example, Brezinski et al . and Tang and Saad . Some methods focus on computing the trace of particular matrix functions.…”

Section: Introductionmentioning

confidence: 99%

GCV for Tikhonov regularization via global Golub–Kahan decomposition

Fenu

Reichel

Rodriguez

2016

Numerical Linear Algebra App

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Generalized Cross Validation (GCV) is a popular approach to determining the regularization parameter in Tikhonov regularization. The regularization parameter is chosen by minimizing an expression, which is easy to evaluate for small-scale problems, but prohibitively expensive to compute for large-scale ones. This paper describes a novel method, based on Gauss-type quadrature, for determining upper and lower bounds for the desired expression. These bounds are used to determine the regularization parameter for large scale problems. Computed examples illustrate the performance of the proposed method and demonstrate its competitivenes

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“…In many cases, however, a low accuracy approximation is sufficient. Numerous methods have been presented to address this need for estimating the trace of the inverse of symmetric positive definite matrices through Gaussian bilinear forms [7,11], modified moments [12,13], and Monte Carlo (MC) techniques [7,11,14,15,12,6,16].…”

Section: Introductionmentioning

confidence: 99%

Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse

Laeuchli

Kalantzis

et al. 2016

Journal of Computational Physics

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A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse.Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method.

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Moments of a linear operator, with applications to the trace of the inverse of matrices and the solution of equations

Cited by 23 publications

References 20 publications

Estimating the diagonal of matrix functions

Estimating the diagonal of matrix functions

GCV for Tikhonov regularization via global Golub–Kahan decomposition

Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse

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