Stochastic Gradient Descent and its Application for Parametrized Boundary Value Problems under Uncertainties

Abstract

In this thesis we want to give a theoretical and practical introduction to stochastic gradient descent (SGD) methods. In the theoretical part, we prove two fundamental convergence results that hold under certain assumptions, like a strongly convex objective function. The first result covers the convergence behaviour of SGD running with a fixed step size sequence and is expanded to the second result, which deals with SGD running with a diminishing step size sequence. For both cases, we provide an upper bound for the expected optimality gap. At the expense of a concrete convergence rate, we then generalize both results to non-convex objective functions. The practical part of this thesis deals with the application of SGD as a convincing and stable optimizer for parametrized boundary value problems under uncertainties. Firstly, we discretize an ordinary differential equation (ODE) Dirichlet problem using finite differences (FD) and improve the results by using preconditioning techniques and a weighted norm. Secondly, we generalize the results to an elliptic partial differential equation (PDE) Dirichlet problem and aim for a weak solution using a finite element (FE) discretization. For both problems, the SGD algorithm convinces with stable results and provides convergence in expectation.