A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data

Feng, Xiaoli; Eldén, Lars; Fu, Chu-Li

doi:10.1515/jiip.2010.028

Cited by 35 publications

(28 citation statements)

References 14 publications

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“…In order to compare our preconditioner with the one in [27], we start with the 2D case , and then consider 3D problems.…”

Section: Numerical Implementation and Experimentsmentioning

confidence: 99%

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Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Feng¹,

Eldén²

2013

Inverse Problems

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An ill-posed Cauchy problem for a 3D elliptic PDE with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability error estimates are given. For the numerical implementation, a large sparse system is obtained from the discretizing the QBV problem using finite difference method (FDM). A LeftPreconditioner Generalized Minimum Residual (GMRES) method is used to solve the large system effectively. For the preconditioned system, a fast solver using the Fast Fourier Transform (FFT). Numerical results show that the method works well.

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“…In order to compare our preconditioner with the one in [27], we start with the 2D case , and then consider 3D problems.…”

Section: Numerical Implementation and Experimentsmentioning

confidence: 99%

“…We also made a preliminary numerical implementation for a slightly less general problem than the one in the present paper. Now we further develop the implementation from [27] and demonstrate that it is an efficient algorithm for solving the general problem (1.1)-(1.5) with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Feng¹,

Eldén²

2013

Inverse Problems

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“…Wei and Wang in [19] used the quasi-boundary value regularization method to deal with the backward problem. Now, this method is also studied for solving various types of inverse problems, such as parabolic equations [22][23][24], hyper-parabolic equations [25], and elliptic equations [26].…”

Section: Introductionmentioning

confidence: 99%

The quasi-boundary value regularization method for identifying the initial value with discrete random noise

Yang

Zhang

et al. 2018

Bound Value Probl

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In this paper, we study an inverse initial value problem for the fractional diffusion equation with discrete noise. This problem is ill-posed in the sense of Hadamard. We apply the trigonometric method in a nonparametric regression associated with the quasi-boundary value regularization method to deal with this ill-posed problem. The corresponding convergence estimate for this method is obtained. The numerical results show that this regularization method is flexible and stable. MSC: 35R25; 47A52; 35R30

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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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A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data

Cited by 35 publications

References 14 publications

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

The quasi-boundary value regularization method for identifying the initial value with discrete random noise

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

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