Efficient reliable image reconstruction schemes for diffuse optical tomography

Abstract

This article deals with the design and analysis of reliable and efficient numerical methods for the solution of inverse problems in diffuse optical tomography. The proposed fully discrete algorithms are based on iterative regularization methods, derived and analysed on the continuous level, and their careful discretization by finite element methods. This guarantees convergence of the fully discrete algorithms under the same conditions as required on the continuous level, and allows to establish mesh-independent reliability and performance. The derivatives and adjoints can be shown to be exact on the discrete level, which implies that the Gauss–Newton systems used in the reconstructions are real symmetric and positive definite, and the conjugate gradient method can be used for their efficient solutions. We also present complexity estimates of the algorithms, and discuss a posteriori error estimators for assessing the discretization error. The efficiency and robustness of the proposed methods are demonstrated in numerical tests.