Two regularization methods for a Cauchy problem for the Laplace equation

Qian, Zhi; Fu, Chu-Li; Li, Zhenping

doi:10.1016/j.jmaa.2007.05.040

Cited by 67 publications

(35 citation statements)

References 20 publications

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“…This problem is taken from [49,53,59]. Problems of this kind are ill-posed and arise in several fields of physics and engineering such as hydrodynamics, tomography, theory of electronic signals, non-destructive testing, geophysics, seismology and others.…”

Section: Cauchy Problems For Elliptic Equationsmentioning

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: Cauchy Problems For Elliptic Equationsmentioning

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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“…They studied the effect of different boundary conditions of Dirichlet, Neumann and mixed boundary conditions on the solution. Qian et al [11] numerically solved Laplace's equation in a rectangle by using the Cauchy problem method. They proposed two different regularization methods on the ill-posed problem based on separation of variables.…”

Section: Introductionmentioning

confidence: 99%

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

Panahi¹,

Ghassemi²,

Nowruzi³

2017

J. Appl. Math. Comput. Mech.

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Abstract. In the current study, the Laplace equation is solved for rectangular and elliptical computational domains by using the boundary element method (BEM). For this accomplishment, 120 different aspect ratios in a rectangular and elliptical computational domains are designed. The Dirichlet and Neumann boundary conditions are used respectively for a rectangular and elliptical domains. Also, the Gaussian quadrature integral method is applied to solve the influence coefficient matrix in BEM. To assess a different aspect ratio on the potential solution, two different measurement positions are intended. According to our finding, with an increase of the aspect ratio, the potential value is increased for both rectangular and elliptical domains. However, a potential increment with aspect ratio enhancement is more visible in the elliptical domain.MSC 2010:76M15, 76B07

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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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Two regularization methods for a Cauchy problem for the Laplace equation

Cited by 67 publications

References 20 publications

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

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

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

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

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