About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains

Bourgeois, Laurent; Dardé, Jérémi

doi:10.1080/00036810903393809

Cited by 53 publications

(97 citation statements)

References 6 publications

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“…There are different techniques for deriving conditional stability functions β including the logarithmic convexity method, weighted energy methods, Gronwall's lemma and maximum principles [2,30,48], or methods based on Carleman estimates [8,37,38,75]. Let us explain by a simple model example the method of logarithmic convexity as one of the most traditional methods to obtain conditional stability functions β.…”

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

confidence: 99%

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%

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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show abstract

“…However, for β = 10 −6 the error could not be decreased further with the finest mesh (I = J = K = 511), hence, a higher mesh resolution should be adopted if we want to obtain a higher accuracy. In addition, Figure 2 shows the graphs of u β (x, y, M) for Examples 1 and 2 with the exact data {ϕ 0 i j } for the coarse mesh resolution I × J × K = 63 3 . For β too small such as 10 −6 , the instability phenomenon is manifested by the strongly oscillating contour lines.…”

Section: Calculation Proceduresmentioning

confidence: 99%

“…For β too small such as 10 −6 , the instability phenomenon is manifested by the strongly oscillating contour lines. In case of noisy data with ǫ > 0, to show the sensitivity of the computational accuracy to noise of the data, we repeated calculations with a variety of noise amplitudes ǫ = 10 −l for l = 2, 6, and illustrated the numerical results only with the finest mesh I × J × K = 511 3 , so that errors from mesh resolution do not contribute to δ ǫ,β . These results are shown in Tables 2 and 3 and Figures Table 3 1.0E-1 1.7E+0 3 and 4 for Examples 1 and 2, respectively.…”

Section: Calculation Proceduresmentioning

confidence: 99%

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On the Cauchy problem for semilinear elliptic equations

Tuan

Binh

Viet

et al. 2015

Journal of Inverse and Ill-Posed Problems

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We study the Cauchy problem for non-linear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new regularization method for stabilising the ill-posed problem. These new results extend some earlier works on Cauchy problems for nonlinear elliptic equations. Numerical results are presented and discussed.

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“…Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g. [4,5,7,9,12,13,16,21,23,29,33,35] and the references therein. Recently, a linear inhomogeneous version of Helmholtz equation (i.e.…”

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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About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains

Cited by 53 publications

References 6 publications

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

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

On the Cauchy problem for semilinear elliptic equations

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

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