Simultaneously identifying piecewise smooth conductivity and initial value for a heat conduction equation

Shuli Chen; Gen Nakamura; Haibing Wang

doi:10.3934/ipi.2024029

Article Contents

2025, Volume 19, Issue 1: 142-173. Doi: 10.3934/ipi.2024029

This issue Previous Article Stability estimates for the inverse source problem with passive measurements Next Article An inverse problem with partial Neumann data and $ L^{n/2} $ potentials

Simultaneously identifying piecewise smooth conductivity and initial value for a heat conduction equation

1.
School of Mathematics, Southeast University, Nanjing 210096, China
2.
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
3.
Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan
4.
Nanjing Center for Applied Mathematics, Nanjing 211135, China

^*Corresponding author: Haibing Wang
^*Corresponding author: Haibing Wang

Received: February 2024

Revised: June 2024

Early access: June 2024

Published: February 2025

The first author was supported by National Natural Science Foundation of China (No. 12071072) and China Scholarship Council (No. 202106090240). The second author was supported by the JSPS KAKENHI (Grant Numbers JP19K03554 and 22K03366). The third author was supported by National Natural Science Foundation of China (Nos. 12071072, 12241102). This work was also supported by the Jiangsu Provincial Scientific Research Center of Applied Mathematics under Grant No. BK20233002.

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Abstract

The paper considers an inverse problem of simultaneously identifying the piecewise smooth conductivity and initial value of a heat conduction equation defined over a domain $ \Omega $ from the measurement at a fixed time and the Cauchy data on a part of the boundary of $ \Omega $. First, we analyze the regularity of the solution for the forward problem, especially the regularity of the solution near the interface where the conductivity is discontinuous. Then, based on this regularity, we prove the conditional stability by combining the logarithmic convexity theory and the conditional stability of identifying the conductivity. Upon having the uniqueness and conditional stability, we study the inverse problem numerically, which is to transform it into a minimization problem. The existence and stability results of the minimizer are rigorously analyzed. By decomposing the minimization problem into a solve-mark-refine-looping scheme, we propose an iterative algorithm with an adaptively matching regularization to preserve the smoothness and discontinuity of the conductivity, respectively. This algorithm is efficient and easy to implement numerically. Finally, we present two-dimensional and three-dimensional numerical examples to show the performance of the proposed algorithm.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 35K05.

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Figure 1. Geometric situation of the piecewise smooth conductivity

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Figure 2. The exact solutions of Example 1

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Figure 3. Numerical results of Example 1 for different noise levels (from the first row to the last row, $ \epsilon = 0.001, \, 0.005, \, 0.01, \, 0.05 $, respectively): the first and second columns show the reconstructions of $ q^{k+1} $ and $ \varphi^{k+1} $, respectively, the third column displays the adaptive indices $ \chi_{\Omega^s}^{k+1} $ defined by (64), and the last column shows the slices of $ q^{k+1} $ along $ x_2 = 0 $

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Figure 4. The exact solutions of Example 2

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Figure 5. Numerical results of Example 2 for different noise levels (from the first row to the last row, $ \epsilon = 0.001, \, 0.005, \, 0.01, \, 0.05 $, respectively): the first and second columns show the reconstructions of $ q^{k+1} $ and $ \varphi^{k+1} $, respectively, the third column displays the adaptive indices $ \chi_{\Omega^s}^{k+1} $ defined by (64), and the last column shows the slices of $ q^{k+1} $ along $ x_2 = 0 $

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Figure 6. The exact solutions and the reconstructions of Example 3 at three slices along different directions for $ \epsilon = 0.001 $

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Table 1. Numerical results of Example 1 for different noise levels

$\epsilon$	$ (\beta, \, \gamma)$	$(\varepsilon_1, \, \varepsilon_2)$	$RelErr_q$	$RelErr_\varphi$	Figure
0.001	(0.008, 0.0001)	(0.0001, 0.002)	2.8775e-02	2.9445e-02	Fig. 3 (A)–(D)
0.005	(0.01, 0.0001)	(0.0001, 0.002)	3.4949e-02	3.0530e-02	Fig. 3 (E)–(H)
0.01	(0.01, 0.0001)	(0.0001, 0.005)	4.3614e-02	2.3588e-02	Fig. 3 (I)–(L)
0.05	(0.03, 0.0001)	(0.0001, 0.005)	4.4124e-02	2.8089e-02	Fig. 3 (M)–(P)

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Table 2. Numerical results of Example 2 for different noise levels

$ \epsilon$	$(\beta, \, \gamma)$	$(\varepsilon_1, \, \varepsilon_2)$	$RelErr_q$	$RelErr_\varphi$	Figure
0.001	(0.008, 0.0001)	(0.0001, 0.002)	2.0766e-02	4.5529e-02	Fig. 5 (A)–(D)
0.005	(0.01, 0.0001)	(0.0001, 0.002)	2.6773e-02	4.5755e-02	Fig. 5 (E)–(H)
0.01	(0.01, 0.0001)	(0.0001, 0.005)	3.5652e-02	4.6567e-02	Fig. 5 (I)–(L)
0.05	(0.03, 0.0001)	(0.0001, 0.005)	4.9123e-02	5.0302e-02	Fig. 5 (M)–(P)

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