In this talk, we discuss the solution of high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value-at-Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. Our approach is based on tensor train approximations of random fields discretized using stochastic collocation. To avoid the non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, an unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush-Kuhn-Tucker (KKT) system in the full space formulation. We also present numerical experiments to demonstrate that the proposed method enables accurate CVaR optimization constrained by PDEs with random coefficients.

In this talk, we analyze optimal control problems for quasilinear strictly hyperbolic systems of conservation laws where the control is the initial state of the system. The problem is of interest, for example, in the context of fluid mechanics or traffic flow modelling. Similar problems for scalar conservation laws have already been studied. However, the case of hyperbolic systems is more involved due to the coupling of the characteristic fields.

We begin our analysis by considering the Generalized Riemann Problem, which has a piecewise smooth initial state with exactly one discontinuity. This is a natural choice since it is well known that solutions to hyperbolic conservation laws generally develop discontinuities even for smooth data. For piecewise $C^1$ initial data we obtain the existence, uniqueness and stability of an entropy solution by a careful fixed point argument built on the associated Riemann Problem with piecewise constant initial states. The construction yields insights into the structure and regularity of the solution and provides a foundation to derive differentiability results of the control-to-state mapping.

The entropy solution is piecewise $C^1$. Its smooth parts are separated by $C^2$ curves which are either shock curves or boundaries of rarefaction waves. In a subsequent step, we show that these curves depend differentiably on the initial state. This allows the transformation to a fixed space-time domain as a reference space. In this reference space, we can show that the transformed solution depends differentiably on the initial state in the topology of continuous functions. For this, a detailed knowledge of the structure of the solution and the behaviour of the shock curves is crucial. As an immediate consequence, the differentiability of tracking type functionals for the optimal control problem follows.

Networks of scalar conservation or balance laws provide models for vehicular traffic flow, supply chains or transmission of data. Such networks usually consist of initial boundary value problems (IBVPs) of scalar conservation or balance laws on every edge coupled by node conditions. For their optimal control a variational calculus is desirable that implies differentiability of objective functionals w.r.t. controls. In the last decade research on IBVPs successfully introduced a variational calculus which implies differentiability of objective functionals of tracking type and also yields an adjoint based gradient representation for the functional. This talk presents recent progress in an extension of these results to networks of scalar conservation or balance laws. Regarding node conditions we introduce a framework for their representation compatible with the known approach on single edges which allows us to extend the desired results from IBVPs. Therefore, in the optimal control on networks of scalar conservation or balance laws we obtain continuous Fréchet differentiability for functionals of tracking-type w.r.t. controls and an adjoint based gradient representation on the network.

Industrial manufacturing processes are often described by systems of differential equations with respective material laws. Since these material laws depend on material-specific parameters, which are frequently unknown or can only be obtained through a range of expensive experiments, there is a need for numerical parameter estimation strategies. In this context, we consider a fiber spinning process modeled by a boundary value problem (BVP) of ordinary differential equations where some material parameters need to be estimated on the basis of few experimental data. We have developed a numerical method involving collocation and continuation methods to solve BVPs arising in a generalized Newtonian setting. Furthermore we are able to calculate gradients of the BVP solutions with respect to the material parameters and apply nonlinear optimization techniques to optimize the material parameters of interest with respect to given measurement data. In this talk we aim to replace the computationally demanding BVPs through surrogate models, leading to fast online optimization. Furthermore we investigate the training process of these surrogates and transfer learning approaches for different fiber spinning configurations.