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Mathematical Problems in Engineering
Volume 2016, Article ID 3543571, 18 pages
Research Article

Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

1Laboratoire de Thermocinétique de Nantes (LTN), UMR CNRS 6607, Université de Nantes, rue C. Pauc, BP 50609, 44306 Nantes Cedex 3, France
2Chaire de Recherche Industrielle en Technologies de l'énergie et en Efficacité Énergétique (T3E), Ecole de Technologie Supérieure, 1100 rue Notre-Dame Ouest, Montréal, Canada H3C 1K3

Received 12 January 2016; Accepted 7 March 2016

Academic Editor: Maria Gandarias

Copyright © 2016 Y. Favennec et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, the -inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.