A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.