Rescaling the GSVD with application to ill-posed problems

We consider Fredholm integral equation of the first kind with noisy data and use Landweber-type iterative methods as an iterative solver. We compare regularization property of Tikhonov, truncated singular value decomposition and the iterative methods. Furthermore, we present a necessary and sufficient condition for the convergence analysis of the iterative method. The performance of the iterative method is shown and compared with modulus-based iterative methods for the constrained Tikhonov regularization.

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“…The functions in (12) are the filter factors of the Landweber-type method (3). On the other hand, the obtained coefficients [see (12)]…”

Section: Landweber-type Methodsmentioning

confidence: 99%

“…These difficulties are inseparably connected with the compactness of the operator which is associated with the kernel K [28]. Several numerical methods have been employed to approximate the solution of (1), see [1,4,8,12,17,25,31,35,39].…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation of Fredholm integral equation of the first kind with noisy data

Mesgarani

Azari

2019

Math Sci

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“…The p first entries of Σ and Λ are ordered according to [2,3] for properties of the GSVD and its computation. Modifications of the GSVD have been described in [7,8]. The quotients σ i /λ i , 1 ≤ i ≤ p, are commonly referred to as generalized singular values of the matrix pair {A, L}.…”

Section: Iterated Tikhonov Regularization Based On the Gsvdmentioning

confidence: 99%

Generalized singular value decomposition with iterated Tikhonov regularization

Buccini

Pasha

Reichel

2020

Journal of Computational and Applied Mathematics

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Linear discrete ill-posed problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods try to reduce the sensitivity by replacing the given problem by a nearby one, whose solution is less affected by perturbations. This paper describes how generalized singular value decomposition can be combined with iterated Tikhonov regularization and illustrates that the method so obtained determines approximate solutions of higher quality than the more commonly used approach of pairing generalized singular value decomposition with (standard) Tikhonov regularization. The regularization parameter is determined with the aid of the discrepancy principle. This requires the application of a zero-finder. Several zero-finders are compared.

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“…where the orthogonal projectors P (i) are defined by (22) and the matrix V (i) i ∈ R n i × i has 1 ≤ i < n i orthonormal columns that span  (L (i) ) for i = 1, 2, … , d.…”

Section: Proposition 3 Let V (I)mentioning

confidence: 99%

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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Rescaling the GSVD with application to ill-posed problems

Cited by 17 publications

References 21 publications

Numerical investigation of Fredholm integral equation of the first kind with noisy data

Numerical investigation of Fredholm integral equation of the first kind with noisy data

Generalized singular value decomposition with iterated Tikhonov regularization

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

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