Regularization Technique for an Inverse Space-Fractional Backward Heat Conduction Problem

Karimi, Milad; Moradlou, Fridoun; Hajipour, Mojtaba

doi:10.1007/s10915-020-01211-2

Cited by 9 publications

(7 citation statements)

References 37 publications

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“…Then, we pass to the limit l → ∞ and afterwards we differentiate the result with respect to η. This gives that {h, u} solves problem ( 12)- (13). Finally, we prove the uniqueness of a solution by contradiction.…”

Section: Rothe Time Discretization Based On Graded Meshesmentioning

confidence: 70%

“…Now, we prove the uniqueness of a solution by contradiction. We suppose that two solutions {h 1 , u 1 } and {h 2 , u 2 } solve problem ( 12)- (13). Then, the differences h := h 1 − h 2 and u := u 1 − u 2 are satisfying…”

Section: Theorem 2 (Uniqueness)mentioning

confidence: 99%

“…In the same manner, a wavelet approach based on Meyer wavelet theory can be developed to regularize the source inverse problem under consideration. We refer to [12,13,15,14] for more details. Finally, it should be noted that the main purpose of this contribution is to show theoretically (based on energy estimates) and numerically (based on a novel formulated nonuniform Rothe scheme) how to reconstruct a time-dependent source from the knowledge of an integral measurement for a non-autonomous time Caputo fractional diffusion equation of order 0 < β < 1.…”

Section: Remarkmentioning

confidence: 99%

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On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Hendy

Bockstal

2021

J Sci Comput

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An inverse source problem for a non-automonous time fractional diffusion equation of order (0 < β < 1) is considered in a bounded Lipschitz domain in R d . The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe's method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from t = 0. Rothe's method on a uniform grid addresses the existence of a such a solution (non-smooth with t γ term where 1 > γ > β ) under low regularity assumptions, whilst Rothe's method over graded grids has the advantage to cope better with the behaviour at t = 0 (also here t β is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

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Section: Rothe Time Discretization Based On Graded Meshesmentioning

confidence: 70%

Section: Theorem 2 (Uniqueness)mentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Hendy

Bockstal

2021

J Sci Comput

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“…In contrast to the forward problem, the inverse problem is usually ill-posed and ill-conditioned [ 23 , 24 ]. Some regularization methods [ 25 , 26 ], such as the Levenberg–Marquardt [ 27 ] method and the conjugate gradient method [ 28 , 29 ], are commonly used to deal with such problems. However, it is difficult to obtain gradient information using these methods, and they easily lead to the dilemma of local optimization [ 30 ].…”

Section: Introductionmentioning

confidence: 99%

Improved Bayesian Optimization Framework for Inverse Thermal Conductivity Based on Transient Plane Source Method

Liu

Lyu

et al. 2023

Entropy

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In order to reduce the errors caused by the idealization of the conventional analytical model in the transient planar source (TPS) method, a finite element model that more closely represents the actual heat transfer process was constructed. The average error of the established model was controlled at below 1%, which was a significantly better result than for the analytical model, which had an average error of about 5%. Based on probabilistic optimization and heuristic optimization algorithms, an optimization model of the inverse heat transfer problem with partial thermal conductivity differential equation constraints was constructed. A Bayesian optimization algorithm with an adaptive initial population (BOAAIP) was proposed by analyzing the influencing factors of the Bayesian optimization algorithm upon inversion. The improved Bayesian optimization algorithm is not affected by the range and individuals of the initial population, and thus has better adaptability and stability. To further verify its superiority, the Bayesian optimization algorithm was compared with the genetic algorithm. The results show that the inversion accuracy of the two algorithms is around 3% when the thermal conductivity of the material is below 100 Wm−1K−1, and the calculation speed of the improved Bayesian optimization algorithm is three to four times faster than that of the genetic algorithm.

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“…The authors in [24] proposed a so-called logarithmic regularization method to solve this case when ρ ≡ 1 and G ≡ 0. The more general nonlinearity was considered in [9,10], where convergence rates in the n-dimensional case were obtained under a wavelet method and a modified Tikhonov regularization method, respectively. Now we turn out attention to the case A x,1 = x D α υ .…”

mentioning

confidence: 99%

Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator

Hai

2022

CPAA

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<p style='text-indent:20px;'>In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., <b>33</b> (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space <inline-formula><tex-math id="M2">\begin{document}$ H^q(\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>

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Regularization Technique for an Inverse Space-Fractional Backward Heat Conduction Problem

Cited by 9 publications

References 37 publications

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Improved Bayesian Optimization Framework for Inverse Thermal Conductivity Based on Transient Plane Source Method

Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator

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