A framework for studying the regularizing properties of Krylov subspace methods

Brianzi, Paola; Favati, Paola; Menchi, Ornella; Romani, Francesco

doi:10.1088/0266-5611/22/3/017

Cited by 14 publications

(16 citation statements)

References 10 publications

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“…It is important not to carry out too many iterations in order to avoid severe error propagation. This approach of determining a restored image is referred to as regularization by truncated iteration; see [2,4,5,6,11,14,18,19] for discussions. For many image restoration problems, LSQR and RRGMRES, and for some problems also GMRES, yield reasonable results within only a few iterations.…”

mentioning

confidence: 99%

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Morigi¹,

Reichel²,

Sgallari³

2010

Applied Numerical Mathematics

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Dedicated to Richard S. Varga on the occasion of his 80th birthday.Abstract. Image deblurring is a discrete ill-posed problem. This paper discusses cascadic multilevel methods designed for the restoration of images that have been contaminated by nonsymmetric blur and noise. Prolongation is carried out by nonlinear edge-preserving and noise-reducing operators, while restrictions are determined by weighted local least-squares approximation. The restoration problem is on each level solved by an iterative method, with the number of iterations determined by the discrepancy principle. The performance of several iterative methods is compared. Computed examples demonstrate the effectiveness of the image restoration methods proposed. The discrepancy principle requires that an estimate of the norm of the noise in the contaminated image be available. We illustrate how such an estimate can be computed with the aid of the nonlinear Perona-Malik diffusion equation.

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confidence: 99%

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Morigi¹,

Reichel²,

Sgallari³

2010

Applied Numerical Mathematics

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“…4 The preconditioner M was constructed as in section 3.2. In our computational examples we used a fixed λ for all regularization problems (3.13) at all steps of GMRES.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Thus in this paper we propose a method for solving numerically a 2D SPE with variable coefficients, discrete in space, using a preconditioned Krylov subspace method, GMRES (generalized minimum residual) [38]. The properties of Krylov methods applied to ill-posed problems have recently been studied in several papers [24,6,7,8,20,23,4]; also preconditioned GMRES has been proposed [20]. It has been reported [23] that without preconditioner GMRES fails to solve even 1D sideways parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method

Ranjbar¹,

Eldén²

2014

SIAM J. Sci. Comput.

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Abstract. The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed in an L 2 setting. We use a preconditioned generalized minimum residual method (GMRES) to solve a two-dimensional SPE with variable coefficients. The preconditioner is singular and chosen in a way that allows efficient implementation using the FFT. The preconditioner is a stabilized solver for a nearby problem with constant coefficients, and it reduces the number of iterations in the GMRES algorithm significantly. Numerical experiments are performed that demonstrate the performance of the proposed method.

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“…Due to their fast convergence behaviors, Krylov subspace methods (e.g., CGLS [8, pp. 288-293], LSQR [9], and LSMR [10]) are usually considered to be superior to Landweber method (refer to Brianzi et al [11] for an excellent overview of regularizing properties of Krylov subspace methods). However, the semiconvergence behavior of Krylov subspace methods is much more pronounced than that of Landweber method.…”

Section: Introductionmentioning

confidence: 99%

Vector Extrapolation Based Landweber Method for Discrete Ill‐Posed Problems

Zhao

Huang

et al. 2017

Mathematical Problems in Engineering

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Landweber method is one of the classical iterative methods for solving linear discrete ill-posed problems. However, Landweber method generally converges very slowly. In this paper, we present the vector extrapolation based Landweber method, which exhibits fast and stable convergence behavior. Moreover, a restarted version of the vector extrapolation based Landweber method is proposed for practical considerations. Numerical results are given to illustrate the benefits of the vector extrapolation based Landweber method.

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A framework for studying the regularizing properties of Krylov subspace methods

Cited by 14 publications

References 10 publications

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method

Vector Extrapolation Based Landweber Method for Discrete Ill‐Posed Problems

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