This chapter serves two purposes: it introduces several essential concepts of linear and nonlinear functional analysis that will be used in subsequent chapters and, as an illustration of them, studies the problem of unconstrained minimization of a convex functional. All the necessary notions of existence, uniqueness, and optimality conditions are presented and analyzed. Preconditioned gradient descent methods for strongly convex, locally Lipschitz smooth objectives in infinite dimensions are then presented and analyzed. A general framework to show linear convergence in this setting is then presented. The preconditioned steepest descent with exact and approximate line searches are then analyzed using the same framework. Finally, the application of Newton’s method to the Euler equations is discussed. The local convergence is shown, and how to achieve global convergence is briefly discussed.

In this section, we discuss fundamental methods, mostly based on gradient information, that yield descent, that is, the function value decreases at each iteration. We start with the most basic method, the steepest-descent method, analyzing its convergence under different convexity/nonconvexity assumptions on the objective function. We then discuss more general descent methods, based on descent directions other than the negative gradient, showing conditions on the search direction and the steplength that allow convergence results to be proved. We also discuss a method that also makes use of Hessian information, showing that it can find a point satisfying approximate second-order optimality conditions and finding an upper bound on the number of iterations required to do so. We then discuss mirror descent, a class of gradient methods based on more general distance metrics that are particularly useful in optimizing over the unit simplex – a problem that arises often in data science. We conclude by discussing the PL condition, a generalization of the strong convexity condition that allows linear convergence rates to be proved.