Nonlinear Inverse Problems: Examples and Difficulties

Abstract

AbstractWe present in this chapter the nonlinear least-squares (NLS) approach to parameter estimation and inverse problems, and analyze the difficulties associated with their theoretical and numerical resolution.We begin in Sect. 1.1 with a simple finite dimensional example of nonlinear parameter estimation problem: the estimation of four parameters in the Knott–Zoeppritz equations. This example will reveal the structure of inverse problem, and will be used to set up the terminology.Then we define in Sect. 1.2 an abstract framework for NLS problems, which contains the structure underlying the example of Sect. 1.1. Next we review in Sect. 1.3 the difficulties associated with the resolution of NLS problems, and hint at possible remedies and their location in the book.Finally, Sects. 1.4–1.6 describe infinite dimensional parameter estimation problems of increasing difficulty, where the unknown is the source or diffusion coefficient function of an elliptic equation, to which the analysis developed in Chaps. 2–5 will be applied.Examples of time marching problems are given in Sects. 2.8 and 2.9 of Chap. 2 KeywordsInverse ProblemReflection CoefficientElliptic EquationObservation OperatorParameter Estimation ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.