A New Coupled Complex Boundary Method for Bioluminescence Tomography (Communications in Computational Physics, 2018)
In this paper, we introduce and study a new method for solving inversesource problems, through a working model that arises in bioluminescence tomography(BLT). In the BLT problem, one constructs quantitatively the bioluminescence sourcedistribution inside a small animal from optical signals detected on the animal's bodysurface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularizationis used to obtain stable approximate solutions. In conventional Tikhonovregularization, it is crucial to choose a proper regularization parameter for trade offbetween the accuracy and stability of approximate solutions. The new method isbased on a combination of the boundary condition and the boundary measurementin a parameter-dependent single complex Robin boundary condition, followed by theTikhonov regularization. By properly adjusting the parameter in the Robin boundarycondition, we achieve two important properties for our new method: first, the regularizedsolutions are uniformly stable with respect to the regularization parameterso that the regularization parameter can be chosen based solely on the considerationof the solution accuracy; second, the convergence order of the regularized solutionsreaches one with respect to the noise level. Then, the finite element method is used tocompute numerical solutions and a new finite element error estimate is derived for discretesolutions. These results improve related results found in the existing literature.Several numerical examples are provided to illustrate the theoretical results.