Year: 2016
Communications in Computational Physics, Vol. 19 (2016), Iss. 1 : pp. 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.230115.150615a
Communications in Computational Physics, Vol. 19 (2016), Iss. 1 : pp. 226–250
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
-
Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurement
Gong, Rongfang | Liu, Xinran | Shen, Jun | Huang, Qin | Sun, Chunlong | Zhang, YeInverse Problems, Vol. 40 (2024), Iss. 7 P.075002
https://doi.org/10.1088/1361-6420/ad49cb [Citations: 0] -
Solving a nonlinear inverse Robin problem through a linear Cauchy problem
Gong, R. F. | Yu, P. J. | Jin, Q. N. | Cheng, X. L. | Han, W.Applicable Analysis, Vol. 99 (2020), Iss. 12 P.2093
https://doi.org/10.1080/00036811.2018.1553037 [Citations: 1] -
A homotopy method for bioluminescence tomography
Gong, R. F. | Cheng, X. L. | Han, W.Inverse Problems in Science and Engineering, Vol. 26 (2018), Iss. 3 P.398
https://doi.org/10.1080/17415977.2017.1310854 [Citations: 11] -
A new class of accelerated regularization methods, with application to bioluminescence tomography
Gong, Rongfang | Hofmann, Bernd | Zhang, YeInverse Problems, Vol. 36 (2020), Iss. 5 P.055013
https://doi.org/10.1088/1361-6420/ab730b [Citations: 17]