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A New Coupled Complex Boundary Method for Bioluminescence Tomography

Part of: Numerical methods Partial differential equations, boundary value problems Numerical linear algebra Existence theories

Published online by Cambridge University Press:  15 January 2016

Rongfang Gong ,
Xiaoliang Cheng  and
Weimin Han
Rongfang Gong*
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Xiaoliang Cheng
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Weimin Han
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
*
*Corresponding author. Email addresses:grf_math@nuaa.edu.cn (R. Gong), xiaoliangcheng@zju.edu.cn (X. Cheng), weimin-han@uiowa.edu (W. Han)
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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.

Keywords

Bioluminescence tomographyTikhonov regularizationconvergence ratefinite element methodserror estimate

MSC classification

Secondary: 65N21: Inverse problems 65F22: Ill-posedness, regularization 49J40: Variational methods including variational inequalities 74S05: Finite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 226 - 250
Copyright
Copyright © Global-Science Press 2016 

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