Tikhonov Regularization for Identification Problems in Differential Equations

In this paper we investigate the method of Tikhonov regularization for solving nonlinear ill-posed inverse problems $$ F(x) = y, $$ (1) where instead of y noisy data y δ ∈ Y with ∥y — y δ ∥ ≤ δ are given, F: D(F) → Y is a nonlinear operator with domain D(F) ⊂ X and X, Y are Hubert spaces with corresponding inner products (•, •) and norms ∥ • ∥, respectively. Nonlinear ill-posed inverse problems arise in a number of applications and can be divided into explicit and implicit ill-posed inverse problems. A large class of explicit ill-posed inverse problems can be described by nonlinear integral equations of the first kind; implicit ill-posed inverse problems arise e.g. in problems connected with the identification of unknown coefficients q (which are in general functions) in distributed systems from certain observations y δ ∈ Y of the noise-free data y. Distributed systems are governed by differential equations, in general, which may be described by an operator equation of the form $$ T\left( {q,u} \right) = b, $$ (2) where T maps the couple (q, u) from the product space Q × U into the space of the right hand side of equation (2). This is of course formal and has to be made precise in each particular case.