An Introduction to Inverse Problems in Partial Differential Equations for Physicists, Scientists and Engineers

In an inverse problem where properties of a physical system are to be found indirectly from measured outputs, one seeks to define and then somehow invert an input/output mapping. When the unknown physical properties are characterized by a small number of constant parameters, the problem is referred to as a parameter identification problem and in such cases the input/output mapping is often simple enough that important properties of the map become transparent. In this first part of the tutorial, examples of such problems are presented to show that the input/output mapping is often a monotone map and to show how monotonicity can be exploited to invert the input/output mapping. Other examples illustrate how a parameterization that fails to exploit monotonicity can lead t incorrect or inferior results. These observations provide a basis for understanding more complex inverse prolems.