Regularization Methods for Ill-Posed Problems

Cheng, Jin; Hofmann, Bernd

doi:10.1007/978-3-642-27795-5_3-5

Cited by 3 publications

(2 citation statements)

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“…will be amplified infinitely by the unbounded symbol a(ξ) and lead to the integral (1.2) blow-up, therefore calculating the value for the pseudodifferential operator with unbounded symbol a(ξ) from the measured data g δ (x) is severely ill-posed [3]. Moreover, solving many ill-posed problems can lead to the numerical pseudo-differential operator, such as numerical differentiation [8], the inverse heat conduction problem [15,18,19] , the Cauchy problem of the Laplace equation [16], the backward heat conduction problem and so on [17], the details we can refer to [7,9].…”

Section: )mentioning

confidence: 99%

Wavelets and numerical pseudodifferential operator

Cheng

2016

Applied Mathematical Modelling

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Highlights• The NΨDO present a new general framework for solving the ill-posed problems.• The results for combination of the ΨDO and ill-posed problem is still very sparse.• Using the Meyer wavelet method, the order optimal error estimates are given.• Theoretical evaluation and numerical test validate the effectiveness of the method. AbstractCalculating the value for the pseudodifferential operator with unbounded symbol is an ill-posed problem. In the present paper we adopt a wavelet regularization method to solve this problem, error estimates of Hölder type between the regularized values and the exact values are derived. Applications of the general theory scheme and numerical test to the concrete problems are presented.

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Section: )mentioning

confidence: 99%

Wavelets and numerical pseudodifferential operator

Cheng

2016

Applied Mathematical Modelling

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“…Hence, one has to use effective regularization methods to obtain approximate and stable solutions of (1.1) when only noisy data are given. The two prominent regularization methods for solving equation (1.1) are Tikhonov-type regularization methods [4,5] and iterative regularization methods [4,[6][7][8].…”

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confidence: 99%

A projective two-point gradient Kaczmarz iteration for nonlinear ill-posed problems

Gao¹,

Han²,

Tong³

2021

Inverse Problems

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In this paper, we propose and analyze a fast Kaczmarz type method for solving multiple nonlinear ill-posed problems in Hilbert spaces. The method is the combination of the projective two-point gradient method and the Kaczmarz method. The key idea, in contrast to the standard two-point gradient method, is to use modified discrete backtracking search algorithm in each iteration in combination with metric projection of the step size to reduce the total number of performed steps and the computation time. Under reasonable conditions used in this work, we establish the strong convergence result of the method in the noise-free case. Moreover, we present the stability and regularity of the proposed method terminated by the discrepancy principle for the case of noisy data. Finally, some numerical experiments on a nonlinear parameter identification problem are presented, which exhibit that the effectiveness of reconstruction results and the acceleration effect of the method.

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