Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

Dou, Fangfang; Fu, Chu-Li; Yang, Fenglian

doi:10.1016/j.cam.2009.01.008

Cited by 59 publications

(20 citation statements)

References 21 publications

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“…This is an order optimal Hölder stability estimate and the numerical experiment can be found in [7].…”

Section: Applicationsmentioning

confidence: 99%

Numerical pseudodifferential operator and Fourier regularization

Qian

2009

Adv Comput Math

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The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.

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“…This is an order optimal Hölder stability estimate and the numerical experiment can be found in [7].…”

Section: Applicationsmentioning

confidence: 99%

Numerical pseudodifferential operator and Fourier regularization

Qian

2009

Adv Comput Math

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“…But to the author's knowledge there are few papers, using the regularization method, with strict theoretical analysis, on identifying the heat source. Recently, in [17,18], the authors identified the heat source depending only on a spatial variable by the wavelet dual least squares method and the Fourier regularization method, respectively.…”

Section: Introductionmentioning

confidence: 99%

The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem

Yang

2010

Computers & Mathematics with Applications

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a b s t r a c tThis paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location. The problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. A simplified Tikhonov regularization method is given and an order optimal stability estimate is obtained. A numerical example shows that the regularization method is effective and stable.

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“…The uniqueness and conditional stability results for these problems can be found in [2][3][4][5][6][7]. Some numerical reconstruction schemes can be found in [1,[8][9][10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

“…In [1], the optimal error bound of the problem (1.1) has been obtained and a Fourier regularization method with an a prior parameter choice rule has been presented. It is well known that the ill posed problem is usually sensitive to the regularization parameter and the a priori bound is difficult to be obtained precisely in practice.…”

Section: Introductionmentioning

confidence: 99%

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Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales

Zhao¹,

Xie²,

Meng³

et al. 2014

JAMP

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In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.

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Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

Cited by 59 publications

References 21 publications

Numerical pseudodifferential operator and Fourier regularization

Numerical pseudodifferential operator and Fourier regularization

The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem

Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales

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