Numerical pseudodifferential operator and Fourier regularization

Fu, Chu-Li; Qian, Zhi

doi:10.1007/s10444-009-9136-5

Cited by 17 publications

(13 citation statements)

References 24 publications

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“…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]. Therefore the study of regularization…”

Section: )mentioning

confidence: 99%

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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%

“…As in [7,9] and [12], we will consider the pseudodifferential operator with unbounded symbols A(D) and A(D, η). * The project is supported by the NNSF of China (Nos.…”

Section: Introductionmentioning

confidence: 99%

Wavelets and numerical pseudodifferential operator

Cheng

2016

Applied Mathematical Modelling

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“…Problems of this kind have been considered, e. g., in [9,20,56,69]. They arise, e. g., in optoelectronics, and in particular in laser beam models, see [4,54,55,57].…”

Section: Cauchy Problem For the Helmholtz Equationmentioning

confidence: 99%

“…In the special case α = 1 this problem has been considered, e. g., in [20,21,22,51,60,70]. Time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model.…”

Section: Fractional Sideways Heat Conductionmentioning

confidence: 99%

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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Abstract. The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

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“…Therefore, it is necessary to study different highly efficient algorithms for solving it. In recent years, there are many special numerical methods to deal with this problem, such as the boundary element method [9], the method of fundamental solutions [10,17], the conjugate gradient method [11], the Landweber method [9], quasi-reversibility and truncation method [14], quasi-boundary and Tikhonov type regularization method [13,18], the Fourier regularization method [3,4] and so on [15]. In the present paper we will consider the following problem with inhomogeneous Dirichlet data in a strip domain:…”

Section: Introductionmentioning

confidence: 99%