Organizer: Michael Hintermüller
DFG-PP 6 : Non-smooth and Complementary-based Distributed Param-eter Systems: Simulation and Hierarchical Optimization
Thursday 14:00 - 16:00 Coudraystr. 11C, Room 101
SPP 1962: Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization
M. Hintermüller 14:00
Non-differentiable structures and partial differential operators arise naturally in numer-ous problems in the applied sciences, leading to non-smooth distributed parameter sys-tems. This non-smoothness may materialize in the formulation of the problem itself, in inequality constraints or complementarity systems, or as a result of competition and hierarchy. Some applications where such non-smoothness arises are quasivariational inequalities, generalized Nash equilibrium problems and mathematical programs with equilibrium constraints. This talk will describe the main goals of the program, present some of the model problems where non-smoothness is inherent, and outline its conse-quences on the resulting mathematical and numerical analysis. The talk will also touch on the crucial idea of transitioning from smoothing or simulation-based approaches to genuinely non-smooth and modern techniques to deal with the problems. Some recent advances in the field will also be presented.
Strong stationarity conditions for the optimal control of a Cahn-Hilliard-Navier-Stokes system
T. Keil, M. Hintermüller 14:20
This talk is concerned with the optimal control of two immiscible fluids with non-matched densities. For the mathematical formulation of the fluid phases, we use a coupled Cahn-Hilliard/Navier-Stokes system which has recently been introduced by Abels, Garcke and Grün in [1]:
∂tφ + v∇φ − div(m(φ)∇µ) = 0, (4.1)
−∆φ + ∂Ψ0(φ)− µ − κφ = 0, (4.2)
∂t(ρ(φ)v) + div(v⊗ ρ(φ)v) − div(2η(φ)ϵ(v)) + ∇p
+div(v⊗ J) − µ∇φ = 0, (4.3) v|∂Ω = ∂nφ|∂Ω= ∂nµ|∂Ω = 0, (4.4) divv = 0, (v, φ)|t=0 = (va, φa). (4.5) The free energy density Ψ0 associated with the underlying Ginzburg-Landau energy in the Cahn-Hilliard system is given by the double-obstacle potential. As a consequence, the system (4.1)-(4.2) becomes a variational inequality of fourth order.
We propose a suitable time discretization for the above system and verify the existence of solutions to the semi-discrete Cahn-Hilliard/Navier-Stokes system.
The optimal control problem is formulated by introducing an appropriate objective func-tional and a distributed control u which enters the Navier-Stokes equation (4.3) on the right-hand side.
We establish the existence of optimal controls and further investigate the sensitivity of the associated control-to-state operator verifying a Lipschitz estimate. A character-ization of the directional derivative of the constraint mapping is provided involving a system of variational inequalities and equations.
Finally, we present strong stationarity conditions for the optimal control problem which are derived via an variational approach pioneered by Mignot and Puel, cf. [2].
[1] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22, (2012).
[2] M. Hintermüller, B. S. Mordukhovich and T. M. Surowiec, Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints, Math. Program., 146, 555–582, (2014).
Efficient Methods for Optimization in Shape Spaces
K. Welker (Trier University), V. Schulz, M. Siebenborn (Trier University) 14:40
Shape optimization problems arise frequently in technological processes which are mod-elled in the form of partial differential equations (PDEs). In many practical circum-stances, the shape under investigation is parametrized by finitely many parameters, which on the one hand allows the application of standard optimization approaches, but on the other hand limits the space of reachable shapes unnecessarily. Shape calculus
presents a way out of this dilemma. Major effort in shape calculus has been devoted towards expressions in so-called Hadamard-forms, i.e., in forms of integrals over the surface of the shape under investigation. It is often a very tedious process to derive such surface expressions. Along the way, there appear volume formulations in the form of integrals over the entire domain as an intermediate step. In this talk, domain inte-gral formulations of shape derivatives are coupled with optimization strategies on shape spaces. Efficient shape algorithms reducing analytical effort and programming work are presented. In this context, a novel shape space is proposed.
The limiting normal cone in Lebesgue and Sobolev spaces F. Harder (TU Chemnitz), P. Mehlitz (TU Bergakademie Freiberg), G. Wachsmuth (TU Chemnitz)
15:00
The limiting normal cone, which is also denoted as Mordukhovich normal cone, is a cen-tral concept in modern variational analysis. It allows to derive optimality conditions for finite-dimensional nonsmooth optimization problems under very low regularity require-ments. In particular, it is applicable to mathematical programs with complementarity constraints (MPCCs), and one obtains optimality conditions of M-stationary type.
However, there literature concerning the limiting normal cone in infinite-dimensional spaces is scarce. In this talk, we are going to characterize the limiting normal cone in two important infinite-dimensional applications. First, we consider so-called decom-posable sets (i.e., sets with pointwise constraints) in Lebesgue spaces and we are able to give a precise characterization of their normal limiting cone. Second, we study the complementarity set in Sobolev spaces.
The analysis in both situations is delicate and this is mainly induced by the fact that the limiting normal cone is, in general, not closed.
DFG-PP 6 : Non-smooth and Complementary-based Distributed Param-eter Systems: Simulation and Hierarchical Optimization
Thursday 16:30 - 18:30 Coudraystr. 11C, Room 101
POD-based Set Oriented Multiobjective Optimal Control
D. Beermann (University of Konstanz), S. Peitz (Paderborn University), S. Volkwein (University of Konstanz), M. Dellnitz (Paderborn University)
16:30
In a wide range of applications one is interested in optimally controlling a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem (MOCP):
y∈Y,u∈Umin J (y, u) = min
y∈Y,u∈U
J1(y, u) ... Jk(y, u)
s. t. y = F (y, u),˙
where u is the control and y is the system state for which the dynamical behavior is described by F .
In the presence of multiple objectives, the set of optimal compromises, the so-called Pareto set, has to be approximated – in contrast to classical optimization problems, where one single optimal solution is computed. There exist various approaches to ad-dress MOCPs such as scalarization techniques, evolutionary algorithms or set oriented approaches. All of these have in common that a large number of function evaluations is typically required. Thus, the direct computation of the Pareto set can quickly be-come numerically infeasible. This is especially the situation when the dynamical system at hand is costly to solve, as is the case for problems described by (nonlinear) partial differential equations (PDEs).
Standard optimization methods for PDEs often make use of a discretization by finite elements or finite volumes which results in a high-dimensional system of ordinary differ-ential equations. In a multi query context such as optimization or parameter identifi-cation, this approach often exceeds the limits of today’s computing power and hence, a significant reduction of the computational cost is required. This can be achieved by ap-proximating the PDE by a reduced order model (ROM) of low dimension, where models obtained by Proper Orthogonal Decomposition and Galerkin projection [1] have proven to be well-suited for nonlinear problems.
In this presentation we extend previous results on PDE-constrained multiobjective op-timization using reduced order models. In [2], set oriented approaches were utilized to solve multiobjective optimal control problems governed by the Navier-Stokes equations using a pre-computed ROM. In [3] and [4], scalarization techniques were applied to semi-linear PDE-constrained problems and error bounds were derived. In order to so solve problems with a larger number of objectives and to address non-smooth problems in the future, we combine these results and extend a set oriented algorithm with uncertainties [5] to PDE-constrained problems using POD-based ROMs with error estimates for the cost function and the gradients.
[1] P. Holmes, J. L. Lumley, G. Berkooz. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge university press, 1998.
[2] S. Peitz, S. Ober-Blöbaum, M. Dellnitz. Multiobjective Optimal Control Methods for Fluid Flow Using Reduced Order Modeling. arXiv:1510.05819, 2015.
[3] L. Iapichino, S. Ulbrich, S. Volkwein. Multiobjective PDE-Constrained Optimiza-tion Using the Reduced-Basis Method. http://kops.uni-konstanz.de/handle/
123456789/25019, 2013.
[4] S. Banholzer, D. Beermann, S. Volkwein. POD-Based Bicriterial Optimal Control by the Reference Point Method. 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations, pp. 210–215, 2016.
[5] S. Peitz, M. Dellnitz. Gradient-Based Multiobjective Optimization with Uncertainties.
arXiv:1612.03815, 2016.
Towards MOR for multiphase flows with variable densities governed by dif-fuse interface models with nonsmooth energies
M. Hinze (Universität Hamburg), C. Gräßle (Universität Hamburg) 16:50
We develop, implement and analyse reduced order models for multiphase flows with variable densities governed by the Cahn Hilliard/Navier-Stokes system with a nonsmooth free energy. Special emphasis is taken on the treatment of adaptively obtained spatial snapshots, and on the resolution of the nonlinearity introduced through the nonsmooth free energy.
This is joint work with Carmen Gräßle, Fachbereich Mathematik, Universität Hamburg Optimal Control for Fracture Propagation Modeled by a Phase-Field Ap-proach
I. Neitzel (Rheinische Friedrich-Wilhelms-Universität Bonn), T. Wick (École Polytechnique), W. Wollner (TU Darmstadt)
17:10
We are concerned with an optimal control problem governed by a fracture model using a phase-field technique. To avoid the non-differentiability due to the irreversibility con-straint, the fracture model is relaxed using a penalization approach. Due to the removal of L∞ bounds on the phase-field, well posedness of the penalized fracture model needs to be analyzed. Existence of a solution to the penalized fracture model is shown and utilized to establish existence of at least one solution for the regularized optimal control problem.