Tikhonov Regularization for Identification Problems in Differential Equations

Abstract

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.

Thanks are due to the organizers of the International Workshop on Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology in Karlsruhe, April 10–12 1995 – and especially to Johannes Gottlieb – for the opportunity to participate and to present this material.