A New Coupled Complex BoundaryMethod for Bioluminescence Tomography

In this paper,we introduce and study a new method for solving inverse source problems,through aworkingmodel that arises in bioluminescence tomography(BLT).In the BLT problem,one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal’s body surface.The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions.In conventional Tikhonov regularization,it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions.The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition,followed by the Tikhonov regularization.By properly adjusting the parameter in the Robin boundary condition,we achieve two important properties for our new method:first,the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy;second,the convergence order of the regularized solutions reaches one with respect to the noise level.Then,the finite element method is used to compute numerical solutions and a newfinite element error estimate is derived for discrete solutions.These results improve related results found in the existing literature.Several numerical examples are provided to illustrate the theoretical results.