Conditional Stability Estimates and Regularization with Applications to Cauchy Problems for the Helmholtz Equation

Regińska, Teresa; Tautenhahn, Ulrich

doi:10.1080/01630560903393170

Cited by 32 publications

(30 citation statements)

References 30 publications

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“…Conditional stability estimates by using interpolation [24,25]. Such inequalities which extend the classical interpolation inequality became a powerful tool in the analysis of regularization under general smoothness conditions, see, e.g., [7,26,28,40,41,42,45,46,47,56,61]. Variable Hilbert scale interpolation is sometimes also called interpolation with a function parameter, see [6,44].…”

Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning

confidence: 99%

“…Problems of this kind have been considered, e. g., in [9,20,56,69]. They arise, e. g., in optoelectronics, and in particular in laser beam models, see [4,54,55,57].…”

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

“…That is, both A and A −1 are unbounded. For deriving conditional stability estimates, we use the decomposition idea as outlined in [56,69]. We decompose the real axis R into R = I ∪ W and call I = (−∞, 0] the ill-posed part and W = [0, ∞) the well-posed part.…”

Section: Proof (Sketch)mentioning

confidence: 99%

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Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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Abstract. The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

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Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning

confidence: 99%

“…Problems of this kind have been considered, e. g., in [9,20,56,69]. They arise, e. g., in optoelectronics, and in particular in laser beam models, see [4,54,55,57].…”

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

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Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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“…For instance, Elden and Berntsson [14] used the logarithmic convexity method to obtain a stability result of Hölder type. Alessandrini et al [1] provided optimal stability results under minimal assumptions, whilst Reginska and Tautenhahn [32] presented some stability estimates and a regularization method for a Cauchy problem for Helmholtz equation. Many methods have been proposed to solve the Cauchy problem for linear homogeneous elliptic equations, such as the method of successive iterations [10], the alternating method [26], the conjugate gradient method [11,24], the iterative regularization method [15], the quasi-reversibility method [23,28], the fourth-order modified method [30], the Fourier truncation regularized (or spectral regularized method) [17,35], etc.…”

Section: Introductionmentioning

confidence: 99%

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

Tuan

Thang

Lesnic

2016

Journal of Mathematical Analysis and Applications

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Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the souce term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane-Emden, Allen-Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates.

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“…For their stable approximate solution such equations require regularization when the given data are noisy. The mathematical theory and practice of regularization (see, e.g., the textbooks [1,4,7,11,20,25] and the papers [2,5,9,22,24,26,28,33,35]) takes advantage of some knowledge concerning the nature of ill-posedness of the underlying problem. This nature regards available a priori information and the degree of ill-posedness from which conclusions with respect to appropriate regularization methods and efficient regularization parameter choices can be drawn.…”

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confidence: 99%

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Hofmann¹,

Kindermann²

2010

Methods and Applications of Analysis

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Abstract. In inverse problems it is quite usual to encounter equations that are ill-posed and require regularization aimed at finding stable approximate solutions when the given data are noisy. In this paper, we discuss definitions and concepts for the degree of ill-posedness for linear operator equations in a Hilbert space setting. It is important to distinguish between a global version of such degree taking into account the smoothing properties of the forward operator, only, and a local version combining that with the corresponding solution smoothness. We include the rarely discussed case of non-compact forward operators and explain why the usual notion of degree of ill-posedness cannot be used in this case.Key words. Degree of ill-posedness, regularization, linear operator equation, Hilbert space, modulus of continuity, spectral distribution, source condition.AMS subject classifications. 47A52, 65J20.1. Introduction. It is an intrinsic property of a wide class of inverse problems that small perturbations in the data may lead to arbitrarily large errors in the solution. Hence, abstract models of inverse problems are frequently associated with operator equations formulated in infinite dimensional spaces that are ill-posed in the sense of Hadamard. For their stable approximate solution such equations require regularization when the given data are noisy. The mathematical theory and practice of regularization (see, e.g., the textbooks [1,4,7,11,20,25] and the papers [2,5,9,22,24,26,28,33,35]) takes advantage of some knowledge concerning the nature of ill-posedness of the underlying problem. This nature regards available a priori information and the degree of ill-posedness from which conclusions with respect to appropriate regularization methods and efficient regularization parameter choices can be drawn.We restrict our considerations here on ill-posed linear operator equations

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Conditional Stability Estimates and Regularization with Applications to Cauchy Problems for the Helmholtz Equation

Cited by 32 publications

References 30 publications

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

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