On regularization and error estimates for non-homogeneous backward Cauchy problem

Denche, M.; Abdessemed, Ala

doi:10.1016/j.ajmsc.2012.03.002

Cited by 2 publications

(5 citation statements)

References 9 publications

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“…Hence, we expect that the conv ergence results corresponding to the homogeneous case also hold similarly for the nonhomogeneous cases, by taking the f into the consideration. We refer readers to [11] for some related analysis based on an interesting extension of the quasi-boundary methods via regularized eigenfunction expansions of f and g. Our proposed OCM does not require any extra treatment in solving the nonhomogeneous cases from the numerical point of view, which appears to be more general than QBVMs.…”

Section: The Optimal Control Methods (With W = I)mentioning

confidence: 99%

“…f = 0), which otherwise will require some tedious analysis. The corresponding optimality system ( 9) is suitable for non-homogeneous cases and the corresponding conv ergence analysis will involve an integral term due to a nonzero f [11], which may deviate from our focus here. Thus we focus on f = 0 in this paper without loss of generality.…”

Section: Stability and Convergence Analysismentioning

confidence: 99%

“…We mention that the convergence results of the quasi-boundary value methods (i.e. QBVM, MQBVM, and PQBVM) can not be directly applied to solve such non-homogeneous backward parabolic equations, although there are several recent extensions [11,[55][56][57][58]. Such extensions are, however, mostly based on the idea of the truncation method [44] that requires the knowledge of full spectral information (i.e.…”

Section: Example 3 [55 56]: 1d With Non-homogeneous Source Termmentioning

confidence: 99%

“…For solving the non-homogeneous backward heat equation (i.e. f = 0), several different nontrivial extensions of such quasi-boundary methods were discussed in [11,35,39,44,49,[55][56][57], which involved more sophisticated approximations based on the truncated series . Such quasi-boundary methods were also generalized for solving backward time-dependent and nonlinear parabolic equations, backward time-fractional diffusion problems [30,59,62], and backward time-fractional inverse source problem [61].…”

Section: Introductionmentioning

confidence: 99%

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Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework

Liu

Xiao

2019

Inverse Problems

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In this paper, we propose to solve the classical backward parabolic equation under the optimal control framework associated with the Tikhonov regularization formulation, where the initial condition is set to be the control variable while the misfit of the final condition is formulated as the Tikhonov objective functional to be minimized. The corresponding first-order necessary optimality system is discretized in a one-shot manner by a second-order finite difference scheme in space and time. The proposed optimal control setting, as discussed in the paper, provides more general framework in terms of solving this type of ill-posed inverse problem. In particular, several existing nonlocal quasi-boundary value methods appear to be the special cases of our proposed optimal control framework with appropriately chosen generalized regularization setting respectively. The relationship between our Tikhonov regularization approach and the quasi-boundary value methods is discussed in detail. The optimal control approach based on Tikhonov regularization is shown to deliver the known optimal order convergence rate. Numerical examples are provided to demonstrate both effective accuracy of approximation as well as excellent stability under the optimal control realization, comparing with three recent quasi-boundary value methods in literature.

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Section: The Optimal Control Methods (With W = I)mentioning

confidence: 99%

Section: Stability and Convergence Analysismentioning

confidence: 99%

Section: Example 3 [55 56]: 1d With Non-homogeneous Source Termmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework

Liu

Xiao

2019

Inverse Problems

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“…Most of these methods are used to solve one-dimensional homogeneous situations; there are a few papers on the nonhomogeneous case in higher dimensional space. Scientists M. Denche and A. Abdessemed [22] gave extensions of the quasi-boundary methods to the nonhomogeneous case. Paper [23] regularized the two-dimensional nonhomogeneous backward heat problem by perturbing the final value.…”

Section: Introductionmentioning

confidence: 99%

Numerical Method for Solving Nonhomogeneous Backward Heat Conduction Problem

Jiang

2018

International Journal of Differential Equations

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In this paper we consider a numerical method for solving nonhomogeneous backward heat conduction problem. Coupled with the likewise Crank Nicolson scheme and an intermediate variable, the backward problem is transformed to a nonhomogeneous Helmholtz type problem; the unknown initial temperature can be obtained by solving this Helmholtz type problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several problems in both two and three dimensions. The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise.

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On regularization and error estimates for non-homogeneous backward Cauchy problem

Cited by 2 publications

References 9 publications

Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework

Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework

Numerical Method for Solving Nonhomogeneous Backward Heat Conduction Problem

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