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

Abstract

In the paper we develop a novel optimal splitting-linearizing method (OSLM) to iteratively solve a nonlinear inverse Cauchy problem in a simply-connected domain. The nonlinear term in the nonlinear elliptic 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 satisfy 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. Simultaneously, 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 solution, the Cauchy data and the Robin transfer coefficient. We assess the convergence by the computed order of convergence (COC) of the proposed iterative algorithms.