IOP PUBLISHING INVERSE PROBLEMS Inverse Problems 24 (2008) 034011 (22pp) doi:10.1088/0266-5611/24/3/034011 Adaptive finite element methods for the solution of inverse problems in optical tomography Wolfgang Bangerth 1 and Amit Joshi 2 1 Department of Mathematics, Texas A&M University, College Station TX 77843-3368, USA 2 Department of Radiology, Baylor College of Medicine, Houston TX 77030, USA E-mail: bangerth@math.tamu.edu and amitj@bcm.edu Received 16 July 2007, in final form 12 February 2008 Published 23 May 2008 Online at stacks.iop.org/IP/24/034011 Abstract Optical tomography attempts to determine a spatially variable coefficient in the interior of a body from measurements of light fluxes at the boundary. Like in many other applications in biomedical imaging, computing solutions in optical tomography is complicated by the fact that one wants to identify an unknown number of relatively small irregularities in this coefficient at unknown locations, for example corresponding to the presence of tumors. To recover them at the resolution needed in clinical practice, one has to use meshes that, if uniformly fine, would lead to intractably large problems with hundreds of millions of unknowns. Adaptive meshes are therefore an indispensable tool. In this paper, we will describe a framework for the adaptive finite element solution of optical tomography problems. It takes into account all steps starting from the formulation of the problem including constraints on the coefficient, outer Newton-type nonlinear and inner linear iterations, regularization, and in particular the interplay of these algorithms with discretizing the problem on a sequence of adaptively refined meshes. We will demonstrate the efficiency and accuracy of these algorithms on a set of numerical examples of clinical relevance related to locating lymph nodes in tumor diagnosis. (Some figures in this article are in colour only in the electronic version) 1. Introduction Fluorescence-enhanced optical tomography is a recent and highly active area in biomedical imaging research. It attempts to reconstruct interior body properties using light in the red and infrared range in which biological tissues are highly scattering but not strongly absorbing. It is developed as a tool for imaging up to depths of several centimeters, which includes in 0266-5611/08/034011+22$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1