Session: PDE-Constrained Optimization and Optimal Control
Chair: Henrik Wyschka
Cluster: nan
Talk 1: Numerical Solution of p-Laplace Problems for Shape Optimization
Speaker: Henrik Wyschka
Abstract: 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 efficient solution of a p-Laplace problem in each descent step. Therefore, we extend an algorithm based on an interior-point method without resorting to homotopy techniques for high orders p. Further, we discuss modifications for 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. Building upon this work, we also aim to reduce the number of outer iterations and thus calls of the algorithm by proposing a trust-region method. Due to the structure of the algorithm for finite p, we are able to introduce a constraint on the gradient of the solution naturally. Consequently, the obtained deformation fields also fulfill a trust-radius in terms of the Lipschitz topology.
Talk 2: A Variational and Adjoint Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws
Speaker: Jannik Breitkopf
Abstract: 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 interesting, 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. Finally, we investigate the adjoint problem as an efficient way to compute the gradient of the objective functional. The adjoint problem is a linear system of transport equations with a discontinuous coefficient and possibly discontinuous terminal data. In general, problems of this kind do not admit unique solutions. We derive interior boundary conditions to characterize the correct reversible solution. This is a joint work with Stefan Ulbrich.
Talk 3: A Variational Calculus for Optimal Control of Networks of Scalar Conservation or Balance Laws
Speaker: Marcel Steinhardt
Abstract: Networks of scalar conservation or balance laws provide models for vehicular traffic flow, supply chains or transmission of data. These networks usually consist of initial boundary value problems (IBVPs) of scalar conservation or balance laws on every edge coupled by node conditions. For the optimal control of solutions 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 results such as continuous Fréchet differentiability for functionals of tracking-type and an adjoint based gradient representation on the network. Joint work with Stefan Ulbrich
