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Commun. Comput. Phys., 19 (2016), pp. 226-250.
Published online: 2018-04
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In this paper, we introduce and study a new method for solving inverse source problems, through a working model 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 new finite 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.
}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.230115.150615a}, url = {http://global-sci.org/intro/article_detail/cicp/11087.html} }In this paper, we introduce and study a new method for solving inverse source problems, through a working model 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 new finite 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.