Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

Wei, Ting; Hon, Y.C.; Ling, Leevan

doi:10.1016/j.enganabound.2006.07.010

Cited by 160 publications

(86 citation statements)

References 40 publications

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“…The approximate representation of the MFS solution for this problem is written as [3,[15][16][17] u(…”

Section: Formulation Of Singular Boundary Methodsmentioning

confidence: 99%

A method of fundamental solution without fictitious boundary

Chen¹,

Wang²

2009

Mesh Reduction Methods

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This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solution (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of fictitious boundary. In order to avoid the singularity at origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the interpolation matrix. This study tests the SBM successfully to three benchmark problems, which shows that the method has rapid convergent rate and is numerically stable.

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“…The approximate representation of the MFS solution for this problem is written as [3,[15][16][17] u(…”

Section: Formulation Of Singular Boundary Methodsmentioning

confidence: 99%

A method of fundamental solution without fictitious boundary

Chen¹,

Wang²

2009

Mesh Reduction Methods

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“…In the potential theory, it is well known that the method of fundamental solutions (MFS) can be used to solve the Laplacian problems when a fundamental solution is known [28,29]. In the MFS the trial solution of u at the field point z = (r cos θ , r sin θ ) can be expressed as a linear combination of the fundamental solutions U(z, s j ):…”

Section: Example 64 We Solve the Cauchy Problem Of The Laplace Equamentioning

confidence: 99%

A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

Liu¹

2013

Communications in Numerical Analysis

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Starting from a quadratic invariant manifold in terms of the residual vector r = Bx − b for an n-dimensional ill-posed linear algebraic equations system Bx = b, we derive an ODEs system for x which is equipped with a state feedback controller to enforce the orbit of the state vector x on a specified manifold, whose residual-norm is exponentially decayed. To realize the above idea we develop a very powerful implicit scheme based on the novel GL(n, R) Lie-group method to integrate the resultant differential algebraic equation (DAE). Through numerical tests of inverse problems we find that the present Lie-group DAE algorithm can significantly accelerate the convergence speed, and is robust enough against the random noise.

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“…We present a numerical procedure for solving a Cauchy problem using the method of fundamental solution (MFS). In literature, the fundamental solutions and Green's functions appear to be some attractive alternatives for solving direct [32,33] and inverse problems [16,17,[34][35][36][37].…”

Section: Implementation With Mfsmentioning

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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Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

Cited by 160 publications

References 40 publications

A method of fundamental solution without fictitious boundary

A method of fundamental solution without fictitious boundary

A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

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

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