Chapter 4: Calculus of variations

Infinite-dimensional deterministic inverse problems are commonly formulated in terms of minimization of suitably defined functionals on function spaces. Understanding such problems requires tools from calculus of variations. This is the topic of this chapter. We begin our discussion in Section 4.1 by discussing a basic problem from calculus of variations—the shortest path problem. This motivates the subsequent discussion in Section 4.2, where we consider some classical theory and some well-known examples. To provide further insight regarding the basic tools discussed, we also consider a traditional application from analytical mechanics in Section 4.3. In Section 4.4, we consider a few examples that commonly occur in the discussions regarding inverse problems governed by PDEs. We conclude this chapter in Section 4.5, where we discuss notions of differentiability as well as the gradient and Hessian of functionals on Hilbert spaces. Our coverage here is brief. The reader is encouraged to consult books on calculus of variations and its applications for further details. See Section 4.6 for some references.