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 unknown parameter of the inverse problem. For the numerical reconstruction of p often the heuristically derived Fréchet 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. AMS subject classifications. 35R30, 35R25, 65M32, 65N21, 92C55, 92F05, 35L20, 35J25