Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

Wei Fan; Xindi Hu; Shengfeng Zhu

doi:10.3934/ipi.2023002

Article Contents

2023, Volume 17, Issue 4: 767-797. Doi: 10.3934/ipi.2023002

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Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

1.
Department of Mathematics, East China Normal University, Shanghai 200241, China
2.
Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

^*Corresponding author: Xindi Hu
^*Corresponding author: Xindi Hu

Received: July 2022

Revised: December 2022

Early access: February 15, 2023

Published online: February 15, 2023

Abstract / Introduction Full Text(HTML) Figure(21) Related Papers Cited by

Abstract

There exist research works on studying time-dependent integer-order and time-fractional constant-order geometric inverse source problems in the literature. The time-fractional variable-order geometric inverse source problems although also have important physical applications have not been studied mathematically and numerically in literature. The aim of this work is to study an inverse source problem associated with a variable-order time-fractional subdiffusion equation. We first build a mathematical model and show existence of the optimal shape for shape reconstruction of the source support. Then, shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations for numerically solving the model problem. In order to reconstruct the source support with topology unknown a priori, moreover, we build a phase-field model and propose a gradient algorithm allowing both shape and topological changes by a phase-field method. A variety of numerical examples are presented to demonstrate effectiveness of the two algorithms.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 35R11; Secondary: 49Q10, 65M60.

Citation:

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Figure 1. Illustration of domain settings

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Figure 2. Illustration of a phase-field function and its implicit representation

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Figure 3. Reconstruction process of source (black region on the first row) and results of computed $ u(T, x) $ (second row) in Example 1: $ \beta = 6\times 10^{-4} $

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Figure 4. Reconstruction comparisons with $ \beta = 6\times 10^{-4} $ (first row) and $ \beta = 2\times 10^{-4} $ (second row) in Example 1: initial guess (black), final result (red), and target (blue)

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Figure 5. Comparisons on the observed $ u(x, T) $ and reconstructed $ u(x, T) $ with $ \epsilon = 0 $ and $ \epsilon = 0.3 $ of Example 1: $ \beta = 2\times 10^{-4} $

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Figure 6. Convergence histories of the objectives of Example 1: $ \beta = 6\times 10^{-4} $

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Figure 7. Comparisons on $ \partial\Omega $ of Example 2: initial guess (black), final result (red), and target (blue)

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Figure 8. Convergence history of the objective for Example 2

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Figure 9. Comparisons on $ \partial\Omega $ for Example 3 with $ u_0 = 0 $ (first line) and $ u_0\not\equiv 0 $ (second line): initial guess (black), final result (red), and target (blue)

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Figure 10. Evolution process of the source for Case 1 of Example 4: square initial (The region with red color represents $ \Omega $)

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Figure 11. Evolution process of the source for Case 1 of Example 4: random initial

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Figure 12. Convergence histories of the objective for Case 1 of Example 4: square initial (left) and random initial (right)

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Figure 13. Reconstructed sources with random initial: Case 2 of Example 4

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Figure 14. Convergence histories of the objective of Case 2 of Example 4

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Figure 15. Comparisons of numerical and exact source of Example 5

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Figure 16. Convergence histories of the objective of Example 5

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Figure 17. Comparisons of numerical and exact sources of Example 6

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Figure 18. Convergence histories of the objective of Example 6

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Figure 19. Comparison of numerical and exact sources of Example 7

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Figure 20. Convergence histories of the objective of Example 7

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Figure 21. Comparison of numerical and exact sources of Example 8

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