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Volume 19, Issue 1
A New Coupled Complex Boundary Method for Bioluminescence Tomography

Rongfang Gong, Xiaoliang Cheng & Weimin Han

Commun. Comput. Phys., 19 (2016), pp. 226-250.

Published online: 2018-04

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  • Abstract

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.

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@Article{CiCP-19-226, author = {Rongfang Gong, Xiaoliang Cheng and Weimin Han}, title = {A New Coupled Complex Boundary Method for Bioluminescence Tomography}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {1}, pages = {226--250}, abstract = {

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} }
TY - JOUR T1 - A New Coupled Complex Boundary Method for Bioluminescence Tomography AU - Rongfang Gong, Xiaoliang Cheng & Weimin Han JO - Communications in Computational Physics VL - 1 SP - 226 EP - 250 PY - 2018 DA - 2018/04 SN - 19 DO - http://doi.org/10.4208/cicp.230115.150615a UR - https://global-sci.org/intro/article_detail/cicp/11087.html KW - AB -

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.

Rongfang Gong, Xiaoliang Cheng and Weimin Han. (2018). A New Coupled Complex Boundary Method for Bioluminescence Tomography. Communications in Computational Physics. 19 (1). 226-250. doi:10.4208/cicp.230115.150615a
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