On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Plato, Robert

doi:10.1007/s002110050232

Cited by 33 publications

(36 citation statements)

References 18 publications

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“…We note that the rate result (16) was also mentioned by Tautenhahn in [20]. For similar rates in the case of an a posteriori parameter choice applied to an iterated version of Lavrentiev regularization we refer to [17].…”

Section: Definition 2 (Bias)mentioning

confidence: 86%

“…This inequality (10) with the factor 2 on the right-hand side follows as a consequence of a careful examination of the proof of Theorem 1.1.18 in [16] and of the fact that…”

Section: Definition 2 (Bias)mentioning

confidence: 99%

“…As general references for fractional powers of operators we refer to [7,11,16]. Below, some more properties of fractional powers considered in those references will be tacitly used.…”

Section: Definition 2 (Bias)mentioning

confidence: 99%

“…We introduce here as a typical non-selfadjoint accretive operator the Riemann-Liouville fractional integration operator V (for details see also [16,18]), sometimes called Volterra operator (cf. [7,Section 8.5]), defined as…”

Section: Definition 2 (Bias)mentioning

confidence: 99%

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Optimal rates for Lavrentiev regularization with adjoint source conditions

Plato¹,

Mathé²,

Hofmann³

2017

Math. Comp.

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There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247-274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

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Section: Definition 2 (Bias)mentioning

confidence: 86%

“…This inequality (10) with the factor 2 on the right-hand side follows as a consequence of a careful examination of the proof of Theorem 1.1.18 in [16] and of the fact that…”

Section: Definition 2 (Bias)mentioning

confidence: 99%

“…As general references for fractional powers of operators we refer to [7,11,16]. Below, some more properties of fractional powers considered in those references will be tacitly used.…”

Section: Definition 2 (Bias)mentioning

confidence: 99%

Section: Definition 2 (Bias)mentioning

confidence: 99%

See 2 more Smart Citations

Optimal rates for Lavrentiev regularization with adjoint source conditions

Plato¹,

Mathé²,

Hofmann³

2017

Math. Comp.

Self Cite

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“…Plato [14] We denote the termination index by m δ . We say that an iterative method for (1.3) equipped with this stopping rule is a regularization method if there is a constant α > 0, independent of δ, such that the iterates x This paper is organized as follows.…”

mentioning

confidence: 99%

On the regularizing properties of the GMRES method

Calvetti

Lewis

Reichel

2002

Numerische Mathematik

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Abstract. The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. However, little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill-posed problems with a right-hand side that is contaminated by errors. We show that when the associated error-free right-hand side lies in a finite-dimensional Krylov subspace, the GMRES method is a regularization method. The iterations are terminated by a stopping rule based on the discrepancy principle.

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Nonlinear Inverse Scale Space Methods for Image Restoration

Burger

Osher²,

Xu³

et al. 2005

Lecture Notes in Computer Science

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Abstract. In this paper we generalize the iterated refinement method, introduced by the authors in [8], to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image.The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known.The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation.

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On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Cited by 33 publications

References 18 publications

Optimal rates for Lavrentiev regularization with adjoint source conditions

Optimal rates for Lavrentiev regularization with adjoint source conditions

On the regularizing properties of the GMRES method

Nonlinear Inverse Scale Space Methods for Image Restoration

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