Nonlinear Inverse Problems

In the previous chapters, we considered linear problems which we wrote as \(Kx=y\), where K was a linear and (often) compact operator between Hilbert spaces. Needless to say that most problems in applications are nonlinear. For example, even in the case of a linear differential equation of the form \(-u^{\prime \prime }+cu=f\) for the function u the dependence of u on the parameter function c is nonlinear; that is, the mapping \(c\mapsto u\) is nonlinear. In Chapters 5, 6, and 7 we will study particular nonlinear problems to determine parameters of an ordinary or partial differential equation from the knowledge of the solution.