PT Unknown
AU Angerhausen, S
TI Stochastic Primal-Dual Proximal Splitting Method for Risk-Averse Optimal Control of PDEs
PD 04
PY 2022
DI 10.17185/duepublico/78165
LA en
AB In this thesis we consider a non-convex optimization problem that is constrained by a partial differential equation (PDE) with uncertain coefficients. The random field PDE solution is taken into account in the objective function by means of the Conditional Value-at-Risk (CVaR), which is a well-known risk measure. A particularly useful feature of CVaR comes to light when it is used in the context of a proximal point method, since the proximal operator of its Fenchel conjugate is just the metric projection onto the so-called bounded probability simplex. Consequently, we propose a stochastic primal-dual proximal splitting method which is adapted from the wellknown Chambolle-Pock method and solves the aforementioned problem. The stochasticity or randomness of the algorithm arises from what we call component-wise gradient freezing or CGF. It is motivated by randomized coordinate descent methods and requires that only a subset of the coordinates of an occurring gradient is recalculated in each iteration. We provide an abstract proof of almost sure weak convergence of the algorithm and specify the results for the case of scalar and deterministic step sizes. Furthermore, we present an algorithm for computing the aforementioned simplex projection and prove its convergence. The reduction of iteration costs due to CGF in terms of saved PDE solutions is presented by means of two numerical examples which are implemented in the Julia programming language.
ER