Nonlinear Cauchy/Robin inverse problems solved by an optimal splitting-linearizing method

doi:10.1016/j.ijheatmasstransfer.2023.124329

2023-10-01

PRE

Chein‐Shan Liu

Chih‐Wen Chang *

DOI：10.1016/j.ijheatmasstransfer.2023.124329

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In the paper we develop a novel optimal splitting-linearizing method (OSLM) to iteratively solve a non-linear inverse Cauchy problem in a simply-connected domain. The nonlinear term in the nonlinear ellip-tic equation is decomposed at two sides through a splitting parameter, which is then linearized around the value at the previous iteration step. The multiple-scale Pascal-polynomial method together with the OSLM is employed to solve the Cauchy problem, of which the optimal value of the splitting parameter is achieved by minimizing a theoretic merit function. Then, we solve the Cauchy/Robin inverse problem of a nonlinear elliptic equation in a doubly-connected domain for recovering the unknown Cauchy data and Robin transfer coefficient on an inner boundary. Two-parameter bases are derived to automatically sat-isfy the prescribed Cauchy boundary conditions on the outer boundary. When the solution is convergent after solving a sequence of linear systems, one can retrieve the Cauchy data very accurately. Simultane-ously, the unknown Robin transfer coefficient is recovered from a given convective boundary condition on the inner boundary by either a division method or a linear system method. To overcome the ill-posed property of Cauchy/Robin problems, the optimal splitting parameter and a scaling factor play the role as regularization parameters. These methods assembled are new techniques to solve the Cauchy/Robin inverse problems. Although a few overspecified data are merely given on the outer boundary, the novel method is quite accurate, robust against large noise, and is convergent very fast to find the entire so-lution, the Cauchy data and the Robin transfer coefficient. We assess the convergence by the computed order of convergence (COC) of the proposed iterative algorithms.& COPY; 2023 Elsevier Ltd. All rights reserved.

Keyword:

Nonlinear elliptic equation

Cauchy

robin inverse problem

Two-parameter bases

Optimal splitting-linearizing method

Regularization parameters