A New Coupled Complex Boundary Method for Bioluminescence Tomography. (15th January 2016)

• Record Type: Journal Article
• Authors: Gong, Rongfang; Cheng, Xiaoliang; Han, Weimin
• Is Part Of: Communications in computational physics. Volume 19:Number 1(2016:Jan.)
• Page Start: 226
• Page End: 250
• Publication Date: 2016-01-15
• Subjects: 65N21, -- 65F22, -- 49J40, -- 74S05; Bioluminescence tomography, -- Tikhonov regularization, -- convergence rate, -- finite element methods, -- error estimate; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.230115.150615a
• Languages: English
• ISSNs: 1815-2406

Abstract

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.