Chapter 10: Discretization and optimization

Consider a variational inverse problem of the form \begin{align*}&\min_{m} \; \hat{J}(m) := J(m,u(m)), \\& \text{where}\\& c(u,m) = 0.\end{align*} (10.1)Here, \(J\) and \(c(u,m) = 0\) are abstract notations for the cost functional and the governing PDEs, respectively. As discussed in Section 5.4, there are two approaches to solving such problems: discretize-then-optimize (DTO) and optimize-then-discretize (OTD). In the DTO approach we first discretize the cost functional \(\hat {J}\) and consider a finite-dimensional optimization problem. On the other hand, in the OTD approach, we first derive the necessary optimality conditions for minimization of \(\hat {J}\), after which discretization is performed; see Figure 10.1.