Classical Definition of Inverse Problems

Abstract

It is known that various mathematical models in groundwater modeling can be represented commonly by operator equations relating state variables and parameters. In forward problems we solve for state variables when parameters are given, while in inverse problems we solve parameters when state variables are measured. Thus, the general theory of inverse problems should be based on the solution of operator equations. In this section, we will introduce accurate and approximate solutions of operator equations and discuss their well-posedness. Introducing the classical definition of inverse problems not only makes these concepts more general and clear, but also helps to understand their origin and developments in depth. If the reader is not familiar with the terminologies used in this section, such as mapping, vector space, norm and etc., it is suggested to read Appendix A of the book first. In this section, we assume the model to be free of structure error. Therefore, it is unnecessary to differentiate between a “real system” and a “model”.