A Golub–Kahan-Type Reduction Method for Matrix Pairs

Hochstenbach, Michiel E.; Reichel, Lothar; Yu, Xuebo

doi:10.1007/s10915-015-9990-x

Cited by 13 publications

(20 citation statements)

References 24 publications

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“…and ⊗ denotes the Kronecker product, has frequently been used for this kind of problem; see, for example, other works. 11,[16][17][18][19] Various properties of the Kronecker product are described in the work of Horn et al 20 We note for future reference that  (L 1 ) = span{[1, 1, … , 1] T }. It may be attractive to replace the matrix (5) in (4) by the following tridiagonal matrix:…”

Section: Kx =Bmentioning

confidence: 99%

“…We briefly comment on the evaluation of the penalty term in (17). Let M i = P (1) V i P (2) , 1 ≤ i ≤ k. Then, the penalty term can be expressed as…”

Section: Corollarymentioning

confidence: 99%

“…Consider the special case of d = 2 space dimensions and let the matrix K be determined by discretizing an integral equation on a square n × n grid (i.e., N = n 2 ). The regularization matrix

L_{1, \otimes} = ([], \begin{matrix} center \\ array I & array \otimes & array L_{1} \\ array L_{1} & array \otimes & array I \end{matrix}),

where

I \in R^{n \times n}

is the identity matrix, L 1 is the following bidiagonal matrix:

L_{1} = \frac{1}{2} ([], \begin{matrix} array 1 & array - 1 & array & array & array & array 0 \\ array & array 1 & array - 1 & array & array & array \\ array & array & array 1 & array - 1 & array & array \\ array & array & array & array ⋱ & array ⋱ & array \\ array 0 & array & array & array & array 1 & array - 1 \end{matrix}) \in R^{false(n - 1 false) \times n},

and ⊗ denotes the Kronecker product, has frequently been used for this kind of problem; see, for example, other works . Various properties of the Kronecker product are described in the work of Horn et al We note for future reference that

scriptN false(L_{1} false) = span false{{false[1, 1, ⋯, 1 false]}^{T} false}

.…”

Section: Introductionmentioning

confidence: 99%

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Regularization matrices for discrete ill‐posed problems in several space dimensions

Dykes¹,

Huang

Noschese

et al. 2018

Numerical Linear Algebra App

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Summary Many applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right‐hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill‐posed problem by a penalized least‐squares problem, whose solution is less sensitive to the error in the right‐hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions.

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Section: Kx =Bmentioning

confidence: 99%

“…We briefly comment on the evaluation of the penalty term in (17). Let M i = P (1) V i P (2) , 1 ≤ i ≤ k. Then, the penalty term can be expressed as…”

Section: Corollarymentioning

confidence: 99%

“…Consider the special case of d = 2 space dimensions and let the matrix K be determined by discretizing an integral equation on a square n × n grid (i.e., N = n 2 ). The regularization matrix

L_{1, \otimes} = ([], \begin{matrix} center \\ array I & array \otimes & array L_{1} \\ array L_{1} & array \otimes & array I \end{matrix}),

where

I \in R^{n \times n}

is the identity matrix, L 1 is the following bidiagonal matrix:

L_{1} = \frac{1}{2} ([], \begin{matrix} array 1 & array - 1 & array & array & array & array 0 \\ array & array 1 & array - 1 & array & array & array \\ array & array & array 1 & array - 1 & array & array \\ array & array & array & array ⋱ & array ⋱ & array \\ array 0 & array & array & array & array 1 & array - 1 \end{matrix}) \in R^{false(n - 1 false) \times n},

scriptN false(L_{1} false) = span false{{false[1, 1, ⋯, 1 false]}^{T} false}

.…”

Section: Introductionmentioning

confidence: 99%

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Regularization matrices for discrete ill‐posed problems in several space dimensions

Dykes¹,

Huang

Noschese

et al. 2018

Numerical Linear Algebra App

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“…An orthonormal basis can be created with a generalization of Golub-Kahan-Lanczos bidiagonalization [13]. However, while the search space grows linearly as a function of the number of matrix-vector products, the dimension of the generalized Krylov subspace grows exponentially as a function of the total degree of a bivariate matrix polynomial.…”

Section: Subspace Expansion For Multiparameter Tikhonovmentioning

confidence: 99%

“…The first test image is also used in [13,23,25], and is shown in Figure 2. We use ρ = 0.075, 20 iterations for the first run, and 100 iterations for the second run.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Zwaan

Hochstenbach

2016

J Sci Comput

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Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.This work is part of the research program "Innovative methods for large matrix problems", which is (partly) financed by the Netherlands Organization for Scientific Research (NWO).

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Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

Gazzola

Landman

2020

GAMM-Mitteilungen

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Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms. K E Y W O R D S hybrid methods, imaging problems, Krylov subspace methods, large-scale linear inverse problems, regularization parameter choice rules, Tikhonov regularization 1 INTRODUCTION This paper considers linear, large-scale, discrete ill-posed problems of the form Ax true + e = b true + e = b , (1) where the matrix A ∈ R m×n represents a forward mapping that is ill-conditioned with ill-determined rank (ie, the singular values of A decay and cluster at zero without an evident gap between two consecutive ones), x true ∈ R n is the desired solution, and e ∈ R m is some unknown noise that affects the data b ∈ R m. Systems like (1) typically stem from the discretization of first-kind Fredholm integral equations, and model inverse problems arising in a variety of applications, such This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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A Golub–Kahan-Type Reduction Method for Matrix Pairs

Cited by 13 publications

References 24 publications

Regularization matrices for discrete ill‐posed problems in several space dimensions

Regularization matrices for discrete ill‐posed problems in several space dimensions

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

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