Multigrid Method for Solving Inverse Problems for Heat Equation

Al-Mahdawi, H.K.; Abotaleb, Mostafa; Alkattan, Hussein; Tareq, Al-Mahdawi Zena; Badr, Amr; Kadi, Ammar

doi:10.3390/math10152802

Cited by 11 publications

(3 citation statements)

References 34 publications

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“…Inverse heat transfer problems encompass a spectrum of classifications, including inverse heat conduction problems (the most prevalent), inverse heat convection problems, inverse heat radiation problems, and inverse heat conjugation problems. Researchers have comprehensively explored inverse problems related to heat conduction [1][2][3][4][5][6][7][8][9], heat convection [10][11][12][13][14][15][16], and heat radiation [17][18][19], employing diverse algorithms. However, investigations into inverse conjugate heat transfer problems [20][21][22] remain notably scarce within the available literature.…”

Section: Introductionmentioning

confidence: 99%

An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques

Huang,

Fang

2024

Mathematics

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In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures used for capturing and analyzing thermal data, while thermal cameras are specific tools used within those techniques to capture thermal images. In the present study, the interface conditions of the plate and air domains are obtained using perfect thermal contact conditions, and therefore we define the problem studied as an inverse conjugate heat transfer problem. Achieving the simultaneous solution of the continuity, Navier–Stokes, and energy equations within the air domain, alongside the heat conduction equation in the plate domain, presents a more intricate challenge compared to conventional inverse heat conduction problems. The validity of our inverse solutions was verified through numerical simulations, considering various inlet air velocities and plate thicknesses. Notably, it was found that due to the singularity of the gradient of the cost function at the final time point, the estimated results near the final time must be discarded, and exact measurements consistently produce accurate boundary heat fluxes under thin-plate conditions, with air velocity exhibiting no significant impact on the estimates. Additionally, an analysis of measurement errors and their influence on the inverse solutions was conducted. The numerical results conclusively demonstrated that the maximum error when estimating heat flux consistently remained below 3% and higher measurement noise resulted in the accuracy of the heat flux estimation decreasing. This underscores the inherent challenges associated with inverse problems and highlights the importance of obtaining accurate measurement data in the problem domain.

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Section: Introductionmentioning

confidence: 99%

An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques

Huang,

Fang

2024

Mathematics

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“…The multigrid method has been a dynamic approach for solving forward problems of partial differential equations [22][23][24][25]. More recently, this method has been utilized to address inverse problems in various fields such as heat transfer [26,27], optical imaging [28][29][30][31], biomedical science [32][33][34], fluids in porous media [35], and economics [36,37].…”

Section: Introductionmentioning

confidence: 99%

Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation

Liu

Ouyang

Guo³

et al. 2023

Mathematics

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This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized and investigated. Additionally, to address the ill-posedness of the inverse problem, the Tikhonov regularization is incorporated. Numerical results demonstrate the computational performance of this method. The proposed combination strategy displays remarkable capabilities in reducing noise, avoiding local minima, and accelerating convergence. Moreover, this combination method performs better than any one method used alone.

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“…Multigrid methods [23] , [24] , [25] are frequently used to accelerate the convergence rate of iterative methods [26] . The main idea of this work is to apply parallel computing to the classical multigrid method and consider the cost of parallel and equational computing.…”

Section: Introductionmentioning

confidence: 99%