Some Properties of the Arnoldi-Based Methods for Linear Ill-Posed Problems

Novati, Paolo

doi:10.1137/16m106399x

Cited by 9 publications

(9 citation statements)

References 17 publications

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“…However, the use of GMRES as a regularization method is not widespread. Indeed, although some results on the SVD approximation properties of GMRES have been recently obtained (even in a continuous setting [37]), in some situations (eg, when A is highly non‐normal), GMRES performs worse than other Krylov subspace methods. One of the reasons for the bad performance of GMRES as an iterative regularization method may be that the approximation subspace for the solution explicitly contains the vector b , which is noise‐corrupted (indeed, this prompted the introduction of RR‐GMRES [26]).…”

Section: Regularizing Krylov Projection Methodsmentioning

confidence: 99%

“…and knowing that both  k+1 ( R , AA T , b, ) and  k+1 ( R , A T A, A T b, ) decrease with increasing k ≥ 2, one gets k+1 ( ) ≤  k ( ) (see Section 4.2 and equations (35), (37)). Similarly, since (2k) t R (t, ) > 0 for all k ≥ 1, t ≥ 0, > 0, Gauss quadrature rules can be used to compute lower bounds for P(x( )) in (56).…”

Section: Extensions To Other Parameter Choice Rulesmentioning

confidence: 99%

“…, if the derivatives of the function have constant sign on [ 1 , min {n.m} ], then upper or lower bounds for quadratic forms of the kind b T (AA T )b and b T A (A T A)A T b can be obtained by employing Gauss and Gauss-Radau quadrature rules of the form (34)-(37) …”

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confidence: 99%

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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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Section: Regularizing Krylov Projection Methodsmentioning

confidence: 99%

Section: Extensions To Other Parameter Choice Rulesmentioning

confidence: 99%

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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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“…Many numerical experiments available in the literature show that σ (m) 1 quickly approaches σ 1 (see also [17]), so that…”

Section: Setting the Regularization Parametersmentioning

confidence: 99%

Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

Gazzola

Novati

2020

Journal of Numerical Mathematics

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AbstractThis paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.

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“…Thus, solvers can be used “raw” (non‐preconditioned), which might appear as heresy for people familiar with forward problems. One may find, in the work of Novati and the associated references, the illustration of raw solvers successfully applied to compact systems. In the literature, SP methods were also most often tried without preconditioner, with certain success …”

Section: Solving the Sp Systemsmentioning

confidence: 99%

The Steklov‐Poincaré technique for data completion: Preconditioning and filtering

Ferrier

Kadri²,

Gosselet

2018

Numerical Meth Engineering

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Summary This paper presents a study of primal and dual Steklov‐Poincaré approaches for the identification of unknown boundary conditions of elliptic problems. After giving elementary properties of the discretized operators, we investigate the numerical solution with Krylov solvers. Different preconditioning and acceleration strategies are evaluated. We show that costless filtering of the solution is possible by postprocessing Ritz elements. Assessments are provided on a 3D mechanical problem.

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Some Properties of the Arnoldi-Based Methods for Linear Ill-Posed Problems

Cited by 9 publications

References 17 publications

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

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

Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

The Steklov‐Poincaré technique for data completion: Preconditioning and filtering

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