A novel coupled complex boundary method for solving inverse source problems

Abstract

In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional boundary conditions. Unlike the existing methods found in the literature, which usually use some of the boundary conditions to form a boundary value problem for the elliptic partial differential equation and the remaining boundary conditions in the objective functional for optimization to determine the source term, the novel method that we propose here has coupled complex boundary conditions. We use a complex elliptic partial differential equation with a Robin boundary condition coupling the Dirichlet and Neumann boundary data, and optimize with respect to the imaginary part of the solution in the domain to determine the source term. Then, on the basis of the complex boundary value problem, Tikhonov regularization is used to obtain a stable approximate source function and the finite element method is used for discretization. Theoretical analysis is given for both the continuous model and the discrete model. Several numerical examples are provided to show the usefulness of the proposed coupled complex boundary method.

A new class of accelerated regularization methods, with application to bioluminescence tomography

“…The evolution equation (4) with an appropriate numerical discretization scheme for the artificial time variable yields a concrete iterative method. This has motivated us to develop some novel iterative regularization methods based on the continuous method (4). The goal of this section is to realize this idea.…”

A modified coupled complex boundary method for an inverse chromatography problem

“…Also, let V be the complex version of V, see [2] for details. Denote H (S) as the standard real Sobolev space, where S = {C ∈ V : ≤ C ≤ C max } and C max is given.…”

“…Recently, a CCBM has been proposed to solve an inverse source problem [2] and the inverse conductivity problem [13], which transfers the data fitting from the boundary ∂Q ≡ {L} × T to the domain Q which leads to a more robust reconstruction process. In our previous research [36] for the inverse chromatography problem, numerical results indicate that a domain fitting formulation performs better than the corresponding boundary fitting formulation.…”

“…In this study, we consider another domain fitting functional by means of a coupled complex boundary method (CCBM). In contrast to the Kohn-Vogelius-type formulation, this method was introduced in [2] for solving an inverse source problem and has its own merits. Compared to the linear inverse source problem, the nonlinear inverse chromatography problem is far more complicated and the recovered adsorption isotherm is a function of the solution of the chromatographic equation.…”

“…In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem.…”