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Abstract This paper proposes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier–Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works.
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This content will become publicly available on June 1, 2026
A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data
We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the ``reduced dimensional method.'' Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples.
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- PAR ID:
- 10580587
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Applied Mathematics and Computation
- Volume:
- 494
- Issue:
- C
- ISSN:
- 0096-3003
- Page Range / eLocation ID:
- 129286
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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