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A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data

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  • Abney, Ray
  • Le, Thuy T.
  • Nguyen, Loc H.
  • Peters, Cam

Abstract

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.

Suggested Citation

  • Abney, Ray & Le, Thuy T. & Nguyen, Loc H. & Peters, Cam, 2025. "A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data," Applied Mathematics and Computation, Elsevier, vol. 494(C).
    • Handle: RePEc:eee:apmaco:v:494:y:2025:i:c:s009630032500013x
    DOI: 10.1016/j.amc.2025.129286
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