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Simultaneous identification of initial value and source strength in a transmission problem for a parabolic equation

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Abstract

We are concerned with an inverse problem of simultaneously recovering the initial value and source strength in a transmission problem for a parabolic equation from extra measurements. First, we establish a conditional stability of the inverse problem by combining the Carleman estimate with the logarithmic convexity theory. Second, we construct a regularized minimizing functional and transform the inverse problem into an optimization problem. The existence and stability of solutions to the optimization problem are analyzed rigorously. Then, we introduce a surrogate functional and propose an iterative thresholding algorithm for solving the optimization problem. The algorithm is cheap and easy to be implemented numerically. Finally, we present several numerical examples for the two-dimensional (2D) and three-dimensional (3D) cases to show the validity of the proposed algorithm.

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Funding

The first author is supported by the National Natural Science Foundation of China (No. 12071072) and China Scholarship Council (No. 202106090240). The second author is supported by the National Natural Science Foundation of China (No. 11871240, 12271197) and self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (No. CCNU20TS003). The third author is supported by the National Natural Science Foundation of China (Nos. 12071072, 11671082, 11971104).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Shuli Chen & Haibing Wang

  2. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Daijun Jiang

  3. Nanjing Center for Applied Mathematics, Nanjing, 211135, People’s Republic of China

    Haibing Wang

Authors
  1. Shuli Chen
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  2. Daijun Jiang
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  3. Haibing Wang
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Correspondence to Haibing Wang.

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Communicated by: Bangti Jin

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Chen, S., Jiang, D. & Wang, H. Simultaneous identification of initial value and source strength in a transmission problem for a parabolic equation. Adv Comput Math 48, 77 (2022). https://doi.org/10.1007/s10444-022-09983-x

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  • DOI: https://doi.org/10.1007/s10444-022-09983-x

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