Numerical solution of a Cauchy problem for the Laplace equation

Berntsson, Fredrik; Eldén, Lars

doi:10.1088/0266-5611/17/4/316

Cited by 91 publications

(62 citation statements)

References 21 publications

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“…It is important to re-emphasis that nice analytical solution of the above problem exists for smooth Cauchy data ( f , g) only and arbitrarily small noise in Cauchy data can totally destroy the solution [7], [16]. In the following subsection a nice well-posed forward problem is presented.…”

Section: Problem Formulationmentioning

confidence: 99%

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

Majeed

Laleg‐Kirati

2015

2015 3rd International Conference on Control, Engineering &Amp; Information Technology (CEIT)

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Abstract-An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.

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Section: Problem Formulationmentioning

confidence: 99%

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

Majeed

Laleg‐Kirati

2015

2015 3rd International Conference on Control, Engineering &Amp; Information Technology (CEIT)

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“…One can introduce the cut off frequency N c and define a solution with the coefficients with k ≤ N c . The error estimate between exact solution and such a solution can be found in Berntsson and Eldén [12]. Yet, in the next section, we present a different approach based on the method of fundamental solution.…”

Section: Relation With the Tikhonov Regularization With Morozov Princmentioning

confidence: 99%

An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

Han

Ling²,

Takeuchi

2011

Commun. comput. phys.

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Abstract. Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

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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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Numerical solution of a Cauchy problem for the Laplace equation

Cited by 91 publications

References 21 publications

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

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

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