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Optical tomography: forward and inverse problems. (English) Zbl 1188.35197
The present paper is a very interesting survey on mathematical and computational optical tomography with a very rich list of references.
The paper starts with a general description of direct and inverse problems involving electromagnetic waves, the radiative transport equation (RTE) for inhomogeneously absorbing media (IAM) as well as the derivation of the diffusion equation.
Roughly speaking, optical tomography essentially consists in recovering unknown coefficients in both diffusion and RTE equations. Concerning diffuse optical tomography the authors first highlight the method of linear transforms and linearization techniques to solve inverse problems – via regularization – at first in simple geometries. Then such techniques are used to solve inverse problems related to general bounded domains \(\Omega\) by suitably extending the available information on \(\partial \Omega\) to \(\partial B\), \(B\) being any ball compactly containing \(\Omega\).
Finally, the authors deal with linearized inverse problems for RTE related to IAM.
The last part of the paper is devoted to numerical methods and a discussion on shape reconstruction techniques.
The survey ends with a few open problems.

MSC:
35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A70 Biological applications of optics and electromagnetic theory
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
82C70 Transport processes in time-dependent statistical mechanics
78M25 Numerical methods in optics (MSC2010)
Software:
L-BFGS
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