Many practical problems require the optimization of systems (e.g., differential equations) with uncertain inputs such as noisy problem data, unknown operating conditions, and unverifiable modeling assumptions. In this talk, we formulate these problems as infinite-dimensional, risk-averse stochastic programs for which we minimize a quantification of risk associated with the system outputs. For many popular risk models, the resulting risk-averse objective function is not differentiable, significantly complicating the numerical solution of the optimization problem. Unfortunately, traditional methods for nonsmooth optimization converge slowly (e.g., sublinearly) and consequently are often intractable for problems in which the objective function and any derivative information is expensive to evaluate. To address this challenge, we introduce a novel trust-region algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the primal-dual risk minimization algorithm and employs smooth approximate risk measures at each iteration. In addition, this algorithm permits and rigorously controls inexact objective function value and derivative (when available) computations, enabling the use of inexpensive approximations (e.g., adaptive discretizations). We discuss convergence of the algorithm under mild assumptions and demonstrate its efficiency on various examples from PDE-constrained optimization.

In this talk we discuss the introduction of a control variable into the coefficients of an optimal control problem governed by an obstacle problem. As is known, the associated solution operator of the obstacle problem is not Gateaux differentiable, therefore we utilize a regularization approach to compute first order limiting optimality conditions for this problem.

The novelty and focus of this talk is the introduction of a matrix valued control into the coefficients of the problem. To handle the resulting multiplicative coupling of the control and the gradient of the state, we will utilize H-convergence arguments in a bootstrapping approach and prove strong $L^p$ convergence for this coefficient control variable.

In this talk, we discuss the quasi-optimal a priori error estimates for an optimal control problem constrained by an
elliptic obstacle problem where the finite element discretization is carried out using the symmetric interior
penalty discontinuous Galerkin method. The main proofs are based on the improved $L^2$-error estimates
for the obstacle problem, the discrete maximum principle, and a well-known quadratic growth property. The standard (restrictive) assumptions on mesh are not assumed here.

Shape optimization constrained to partial differential equations is a vibrant field of research with high relevance for industrial-grade applications. Recent developments suggest that using a p-harmonic approach to determine descent directions is superior to classical Hilbert space methods. This applies in particular to the representation of kinks and corners in occurring shapes. However, the approach requires the solution of a vector-valued p-Laplace problem with a boundary force for each descent direction. We present an algorithm to solve these problems for finite p efficiently and discuss extensions to the limit setting. A key challenge in this regard is that the Lipschitz deformations obtained as solutions in limit setting are in general non-unique. Thus, we focus on solutions which are in a sense limits to solutions for finite p and aim to preserve mesh quality throughout the optimization.