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Implementations of range restricted iterative methods for linear discrete ill-posed problems

linear ALGEBRA AND ITS APPLICATIONS 2012年第10期436卷 3974-3990页

作者： Neuman, A. Reichel, L. Sadok, H. Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Littoral Lab Math Pures & Appl Ctr Univ Mi Voix F-62228 Calais France

This paper is concerned with iterative solution methods for large linear systems of equations with a matrix of ill-determined rank and an error-contaminated right-hand side. The numerical solution is delicate, because the matrix is very ill-conditioned and may be singular. It is natural to require that the computed iterates live in the range of the matrix when the latter is symmetric, because then the iterates are orthogonal to the null space. Computational experience indicates that it can be beneficial to require that the iterates live in the range of the matrix also when the latter is nonsymmetric. We discuss the design and implementation of iterative methods that determine iterates with this property. New implementations that are particularly well suited for use with the discrepancy principle are described. (C) 2010 Elsevier Inc. All rights reserved.

关键词： linear discrete ill-posed problem Iterative method Truncated iteration Tikhonov regularization

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Arnoldi decomposition, GMRES, and preconditioning for linear discrete ill-posed problems

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APPLIED NUMERICAL MATHEMATICS 2019年 142卷 102-121页

作者： Gazzola, Silvia Noschese, Silvia Novati, Paolo Reichel, Lothar Univ Bath Dept Math Sci Bath BA2 7AY Avon England SAPIENZA Univ Roma Dipartimento Matemat Guido Castelnuovo Ple A Moro 2 I-00185 Rome Italy Univ Trieste Dipartimento Matemat & Geosci Via Valerio 12-1 I-34127 Trieste Italy Kent State Univ Dept Math Sci Kent OH 44242 USA

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as boundary value problems for elliptic partial differential equations. The method is also applied to the iterative solution of linear systems of equations that are obtained by discretizing linear ill-posed problems, such as many inverse problems. However, GMRES does not always perform well when applied to the latter kind of problems. This paper seeks to shed some light on reasons for the poor performance of GMRES in certain situations, and discusses some remedies based on specific kinds of preconditioning. The standard implementation of GMRES is based on the Arnoldi process, which also can be used to define a solution subspace for Tikhonov or TSVD regularization, giving rise to the Arnoldi-Tikhonov and Arnoldi-TSVD methods, respectively. The performance of the GMRES, the Arnoldi-Tikhonov, and the Arnoldi-TSVD methods is discussed. Numerical examples illustrate properties of these methods. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： linear discrete ill-posed problem Arnoldi process GMRES Truncated iteration Tikhonov regularization Truncated singular value decomposition

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FGMRES for linear discrete ill-posed problems

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APPLIED NUMERICAL MATHEMATICS 2014年 75卷 175-187页

作者： Morikuni, Keiichi Reichel, Lothar Hayami, Ken Res Org Informat & Syst Natl Inst Informat Chiyoda Ku Tokyo 1018430 Japan Kent State Univ Dept Math Sci Kent OH 44242 USA Grad Univ Adv Studies Sch Multidisciplinary Sci Dept Informat Chiyoda Ku Tokyo 1018430 Japan

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear discrete ill-posed problems. The FGMRES method, proposed by Saad, is a generalization of GMRES that allows larger flexibility in the choice of solution subspace than GMRES. This paper explores application of FGMRES to the solution of linear discrete ill-posed problems. Numerical examples illustrate that FGMRES with a suitably chosen solution subspace may determine approximate solutions of higher quality than commonly applied iterative methods. (C) 2013 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Iterative method linear discrete ill-posed problem Subspace selection

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Algorithms for range restricted iterative methods for linear discrete ill-posed problems

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NUMERICAL ALGORITHMS 2012年第2期59卷 325-331页

作者： Neuman, Arthur Reichel, Lothar Sadok, Hassane Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Littoral Lab Math Pures & Appl F-62228 Calais France

Range restricted iterative methods based on the Arnoldi process are attractive for the solution of large nonsymmetric linear discrete ill-posed problems with error-contaminated data (right-hand side). Several derivations of this type of iterative methods are compared in Neuman et al. (linear Algebra Appl. in press). We describe MATLAB codes for the best of these implementations. MATLAB codes for range restricted iterative methods for symmetric linear discrete ill-posed problems are also presented.

关键词： linear discrete ill-posed problem Iterative method Truncated iteration

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A Note on the GMRES Method for linear discrete ill-posed problems

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Advances in Applied Mathematics and Mechanics 2009年第6期1卷 816-829页

作者： Nao Kuroiwa Takashi Nodera Graduate School of Science and Technology Keio University3-14-1HiyoshiKohokuYokohamaKanagawaJapan Department of Mathematics Faculty of Science and TechnologyKeio University3-14-1HiyoshiKohokuYokohamaKanagawaJapan

In this paper,we are presenting a proposal for new modified algorithms for RRGMRES and *** is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed *** this paper we have focused on the residual norm and have come-up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance *** numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.

关键词： Numerical computation GMRES iterative method linear discrete ill-posed problem

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Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

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JOURNAL OF SCIENTIFIC COMPUTING 2017年第3期70卷 990-1009页

作者： Zwaan, Ian N. Hochstenbach, Michiel E. Eindhoven Univ Technol Dept Math & Comp Sci POB 513 NL-5600 MB Eindhoven Netherlands

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.

关键词： Tikhonov Multiparameter Tikhonov Generalized Krylov Multidirectional subspace expansion Subspace truncation Subspace method linear discrete ill-posed problem Regularization Regularization parameter

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Tensor Krylov subspace methods with an invertible linear transform product applied to image processing

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APPLIED NUMERICAL MATHEMATICS 2021年 166卷 186-207页

作者： Reichel, Lothar Ugwu, Ugochukwu O. Kent State Univ Dept Math Sci Kent OH 44242 USA

This paper discusses several transform-based methods for solving linear discrete ill-posed problems for third order tensor equations based on a tensor-tensor product defined by an invertible linear transform. linear transform-based tensor-tensor products were first introduced in Kernfeld et al. (2015) [16]. These tensor-tensor products are applied to derive Tikhonov regularization methods based on Golub-Kahan-type bidiagonalization and Arnoldi-type processes. GMRES-type solution methods based on the latter process also are described. By applying only a fairly small number of steps of these processes, large-scale problems are reduced to problems of small size. The number of steps required by these processes and the regularization parameter are determined by the discrepancy principle. The data tensor is a general third order tensor or a tensor defined by a laterally oriented matrix. A quite general regularization tensor can be applied in Tikhonov regularization. Applications to color image and video restorations illustrate the effectiveness of the proposed methods. (C) 2021 Published by Elsevier B.V. on behalf of IMACS.

关键词： Discrepancy principle Invertible linear transform linear discrete ill-posed problem Tensor Arnoldi process Tensor bidiagonalization process Tensor Tikhonov regularization

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A GCV based Arnoldi-Tikhonov regularization method

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BIT NUMERICAL MATHEMATICS 2014年第2期54卷 501-521页

作者： Novati, Paolo Russo, Maria Rosaria Univ Padua Dipartimento Matemat Padua Italy

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented.

关键词： linear discrete ill-posed problem Tikhonov regularization Arnoldi algorithm Generalized cross validation

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Adaptive Arnoldi-Tikhonov regularization for image restoration

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NUMERICAL ALGORITHMS 2014年第4期65卷 745-757页

作者： Novati, Paolo Russo, Maria Rosaria Univ Padua Dept Pure Math I-35121 Padua Italy

In the framework of the numerical solution of linear systems arising from image restoration, in this paper we present an adaptive approach based on the reordering of the image approximations obtained with the Arnoldi-Tikhonov method. The reordering results in a modified regularization operator, so that the corresponding regularization can be interpreted as problem dependent. Numerical experiments are presented.

关键词： linear discrete ill-posed problem Image restoration Tikhonov regularization Arnoldi algorithm Krylov methods

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A generalized LSQR algorithm

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NUMERICAL linear ALGEBRA WITH APPLICATIONS 2008年第7期15卷 643-660页

作者： Reichel, Lothar Ye, Qiang Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Kentucky Dept Math Lexington KY 40506 USA

LSQR is a popular iterative method for the solution of large linear system of equations and least-squares problems. This paper presents a generalization of LSQR that allows the choice of an arbitrary initial vector for the solution subspace. Computed examples illustrate the benefit of being able to choose this vector. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： linear system of equations least-squares problem iterative method linear discrete ill-posed problem

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