Approximate solution of a Cauchy problem for the Helmholtz equation

Regińska, Teresa; Regiński, K.

doi:10.1088/0266-5611/22/3/015

Cited by 82 publications

(75 citation statements)

References 11 publications

(13 reference statements)

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“…In addition, a revised Tikhonov regularization method is also considered. Since the numerical implementation for our methods is similar to the method provided by [5], we only give some numerical results.…”

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

“…This aim of this paper is to give some regularization methods within the framework of general regularization theory, which are different from "Approximate solution of a Cauchy problem for the Helmholtz equation" by T. Reginska and K. Reginski [5], where Fourier regularization method is only an 'isolated' method. Moreover, we find that for the Cauchy problem Fourier regularization method is one of the considered spectral methods.…”

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

“…and Now we give a new interpretation for the Fourier regularization method provided by [5]. Applying the TSVD method for solving problem (2.1), we havê…”

Section: Error Estimates On the Cauchy Problem For The Helmholtz Equamentioning

confidence: 99%

“…In section 3, we have seen that the Fourier regularization method in [5] is also the Spectral method 3 (TSVD). Since some numerical results on Fourier regularization method has been investigated, here we only provide numerical results for the revised Tikhonov regularization method.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…However, these numerical methods are short of stability analysis and error analysis. Recently, in [5] many applications for a model of Helmholtz equation are introduced, a Fourier regularization method [7] (also is known as a method by cutting off high frequency directly) is applied for solving #675/06. Received: 21/VII/06.…”

Section: Introductionmentioning

confidence: 99%

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Two approximate methods of a Cauchy problem for the Helmholtz equation

Xiong

2007

Mat. apl. comput.

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Abstract.In this paper, we consider a Cauchy problem for the Helmholtz equation at fixed frequency, especially we give the optimal error bound for the ill-posed problem. Within the framework of general regularization theory, we present some spectral regularization methods and a modified Tikhonov regularization method to stabilize the problem. Moreover, Hölder-type stability error estimates are proved for these regularization methods. According to the regularization theory, the error estimates are order optimal. Some numerical results are reported. 35R25, 35R30. Mathematical subject classification:

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Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

“…and Now we give a new interpretation for the Fourier regularization method provided by [5]. Applying the TSVD method for solving problem (2.1), we havê…”

Section: Error Estimates On the Cauchy Problem For The Helmholtz Equamentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two approximate methods of a Cauchy problem for the Helmholtz equation

Xiong

2007

Mat. apl. comput.

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A regularization strategy to a backward problem for general nonlinear parabolic equations in Sobolev spaces

Hai

2024

Math Methods in App Sciences

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This article is devoted to solving a backward problem for general nonlinear parabolic equations in Sobolev spaces. The problem is hardly solved by computation since it is severely ill‐posed in Hadamard's sense. Using the Fourier transform and the truncation of high‐frequency components, we construct a regularized solution for the problem from the data given inexactly. As usual for ill‐posed problems, the rate of convergence can be obtained under additional smoothness assumptions on the regularity of the exact solution, the so‐called source conditions. By using a source condition in a generalized form, we prove a convergence estimate in a Sobolev space . Corresponding to different levels of the source condition, the convergence rates are improved gradually. We apply our theory to the backward problem with a convection–diffusion operator and a cubic nonlinearity. Finally, the numerical results are presented to test the validity of the proposed regularization method.

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Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

Chen

Wang

Cheng

et al. 2022

Int Journal of Mech Sys Dyn

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This paper proposes a semi‐analytical and local meshless collocation method, the localized method of fundamental solutions (LMFS), to address three‐dimensional (3D) acoustic inverse problems in complex domains. The proposed approach is a recently developed numerical scheme with the potential of being mathematically simple, numerically accurate, and requiring less computational time and storage. In LMFS, an overdetermined sparse linear system is constructed by using the known data at the nodes on the accessible boundary and by making the remaining nodes satisfy the governing equation. In the numerical procedure, the pseudoinverse of a matrix is solved via the truncated singular value decomposition, and thus the regularization techniques are not needed in solving the resulting linear system with a well‐conditioned matrix. Numerical experiments, involving complicated geometry and the high noise level, confirm the effectiveness and performance of the LMFS for solving 3D acoustic inverse problems.

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Approximate solution of a Cauchy problem for the Helmholtz equation

Cited by 82 publications

References 11 publications

Two approximate methods of a Cauchy problem for the Helmholtz equation

Two approximate methods of a Cauchy problem for the Helmholtz equation

A regularization strategy to a backward problem for general nonlinear parabolic equations in Sobolev spaces

Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

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