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Numerische und
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In many inverse problems a functional of u is given by measurements where u solves a partial differential equation of the type L(p)u+Su=Q. Here, Q is a known source term and L(p), S are operators with p as unknown parameter of the inverse problem. For the numerical reconstruction of p often the heuristically derived Frechet derivative R' of the mapping R:p -> 'measurement functional of u' is used. We show for three problems --- a transport problem in optical tomography, an elliptic equation governing near infrared tomography, and the wave equation in moving media --- that R' is the derivative in the strict sense. Our method is applicable in more general problems than established methods for similar inverse problems.
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