Inverse problems for regularization matrices

Noschese, Silvia; Reichel, Lothar

doi:10.1007/s11075-012-9576-8

Cited by 15 publications

(12 citation statements)

References 20 publications

(24 reference statements)

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“…see, for example, the work of Horn et al 20 for operations on matrices with a Kronecker product structure. We can apply this identity to express (13) in the form…”

Section: Corollarymentioning

confidence: 99%

“…In particular, V 1 = B/||B|| F . The use of the global Arnoldi method to the solution of (15) is mathematically equivalent to applying a standard Arnoldi method to (13). An advantage of the global Arnoldi method is that it can be implemented by using matrix-matrix operations, whereas the standard Arnoldi method applies matrix-vector and vector-vector operations.…”

Section: Corollarymentioning

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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“…see, for example, the work of Horn et al 20 for operations on matrices with a Kronecker product structure. We can apply this identity to express (13) in the form…”

Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

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

Dykes¹,

Huang

Noschese

et al. 2018

Numerical Linear Algebra App

Self Cite

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show abstract

“…For discussions on the importance of choosing suitable regularization matrices and on the design of many such matrices; see, e.g., [3,6,19,21].…”

Section: Computed Examplesmentioning

confidence: 99%

Simplified GSVD computations for the solution of linear discrete ill-posed problems

Dykes

Reichel

2014

Journal of Computational and Applied Mathematics

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Abstract. The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the context of Tikhonov regularization. Also, other regularization methods, including truncated GSVD, are considered. We compare the computational efforts required by the simplified GSVD method and the A-weighted generalized inverse introduced by Eldén.

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“…, ±n/2, and the scaling factor c chosen so that w is of unit length, is considered in [9, Example 3.1]. Many other examples of regularization matrices can be found in [8,9,19,22].…”

Section: A Rescaled Gsvdmentioning

confidence: 99%

Rescaling the GSVD with application to ill-posed problems

2014

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Abstract. The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by X T , is the same in both products. Available software for computing the GSVD scales the diagonal matrices and X T so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of X T . The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution.

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Inverse problems for regularization matrices

Cited by 15 publications

References 20 publications

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

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

Simplified GSVD computations for the solution of linear discrete ill-posed problems

Rescaling the GSVD with application to ill-posed problems

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