Fréchet Derivatives for Some Bilinear Inverse Problems

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 an unknown parameter of the inverse problem. For the numerical reconstruction of p, the heuristically derived Fréchet derivative R' of the mapping R:p→ "measurementfunctional of u" is often 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 to more general problems than are established methods for similar inverse problems.