We consider finite element methods for a model linear-quadratic elliptic distributed optimal control problem with pointwise state constraints. For convex or smooth domains, this problem can be reformulated as a fourth order variational inequality that can be solved by many finite element methods originally designed for fourth order elliptic boundary value problems. We will present a general convergence analysis that can be applied to conforming, nonconforming and discontinuous Galerkin methods. We will also discuss the extension of this analysis to a P1 finite element method for the optimal control problem on nonconvex polyhedral domains.

We consider an optimal control problem of tracking type governed by a time-discrete regularized phase-field fracture or damage propagation model. The problem contains a regularization term that penalizes the violation of the irreversibility condition in the evolution of the fracture. We prove convergence of solutions of the regularized problem when taking the limit with respect to the penalty term, and obtain an estimate for the constraint violation in terms of the penalty Parameter.

An optimal control problem related to the Allen-Cahn equation is considered. In particular, a tracking type functional is minimized subject to the Allen-Cahn equation, by using controls of distributed type satisfying point-wise control constraints.Various regularity and continuity results of the control to state mapping are presented. In addition, first and second order necessary and sufficient conditions are discussed. A fully-discrete scheme based on the lowest order (in time) discontinuous Galerkin scheme combined with conforming finite element subspaces is considered. Provided that the temporal and spatial discretization parameters are suitably chosen, we present error estimates for the difference between locally optimal controls and their discrete approximations. Further estimates for the corresponding state and adjoint variables are also presented.

This talk introduces new reduced order model (ROM) approaches for the efficient solution of risk averse PDE constrained optimization problems.

Risk-averse optimization arises in science and engineering applications where one needs to determine inputs (controls or design parameters) to maximize the performance of a system specified by a quantity of interest (QoI). The system is modeled by PDEs and some system parameters such as material parameters are uncertain. Instead of maximizing the expected performance of the system, risk-averse optimization also includes a so-called risk measure for the deviation of the actual performance below the expected performance. One example of such risk measures is the Conditional Value at Risk (CVaR). Estimation of the risk-measure at a single control/design requires many evaluations of the PDE, each corresponding to a realization of the uncertain system parameters. More importantly, only realizations in tail of the probability distribution of the QoI enter the estimation. Unfortunately, this distribution is not known explicitly, but is determined by the probability distributions of the uncertain inputs and the PDE solution through which these uncertain inputs enter the performance objective. Optimization of risk-measures requires even more PDE solves.

This talk discussed the rigorous use of ROM for the efficient estimation of CVaR. Efficiency is gained by using ROMs to replace the expensive full order model in the CVaR estimation, and also using the structure of CVaR estimation to guide the ROM construction. Improvements gained by using ROMs are quantified and illustrated numerically. These CVaR estimation approaches are then used to optimize CVAR subject to PDE constraints with uncertain parameters. In this context it is important that key quantities that need to be approximated for CVAR objective function evaluations are
also the key quantities arising in gradient computations. Numerical results are given to illustrate the performance gains due to ROM.

We investigate the hard-thresholding method applied to optimal control problems with $L^0(\Omega)$ control
cost, which penalizes the measure of the support of the control. As the underlying measure space is non-atomic,
arguments of convergence proofs in $l^2$ or $\R^n$ cannot be applied.
Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous
along the iterates. That is, the function value in a weak limit point is less or equal than the lim-inf of the function
values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points,
which enables us to prove certain stationarity conditions for the limit points.
Numerical experiments are carried out, which show the performance of the method.
These indicates that the method is robust with respect to discretization.
In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of $L^1(\Omega)$-regularized problems.

We present an algorithm for the efficient solution of structured nonsmooth operator equations in Banach spaces. Here, the term structured indicates that we consider equations which are composed of a smooth and a semismooth mapping. Equations of this type occur, for instance, as optimality conditions of structured nonsmooth optimization problems. In particular, the algorithm can be applied to nonconvex PDE-constrained optimal control problems with sparsity.

The novel algorithm combines a semismooth Newton method with a quasi-Newton method. This hybrid approach retains the local superlinear convergence of both these methods under standard assumptions. Since it is known that, in general, quasi-Newton methods alone can not achieve superlinear convergence in semismooth settings, this is rather satisfying from a theoretical point of view. In addition, we observe in the infinite-dimensional setting that PDE-constrained optimal control problems are particularly well-suited for the application of the new method because of their inherent smoothing property.

The most striking feature of the new method, however, is its numerical performance. On nonsmooth PDE-constrained optimal control problems it is significantly faster than semismooth Newton methods, and these speedups persist when globalization techniques are added. Most notably, the hybrid approach can be embedded in a matrix-free limited-memory truncated trust-region framework to efficiently solve nonconvex and nonsmooth large-scale real-world optimization problems, as we will demonstrate by means of an example from magnetic resonance imaging. In this challenging environment it dramatically outperforms semismooth Newton methods, sometimes by a factor of fifty and more.

All of these topics are addressed in the talk.

Mean field game theory studies the behavior of a large number of interacting individuals
in a game theoretic setting. In this talk, we derive the mean field game PDE system from deterministic microscopic agent dynamics. The dynamics are given by a general ODE which defines a large class of differential games. We use the concept of Nash equilibria and apply dynamic programming to derive the mean field limit equation. We extensively study the scaling
behavior of our system and observe new reasonable configurations. We show that well known mean field game limit systems are a subclass of our model. We motivate the novel scales with an example of an agent-based financial market model, inspired by the econophysical Levy-Levy-Solomon model.

We report on recent progresses on models coupling a hysteresis operator to reaction-diffusion equations.
In a frist collaboration with M. Brokate, we prove weak-differentiability of the control-to-state mapping in a parabolic control problem with hysteresis. In a second paper with C. Münch, we present the derivation of hysteresis-reaction-diffusion systems
as singular fast-reaction limit of a suitable class of coupled ODE-PDE systems. Besides existence theory and numerical examples, we demonstrate in particular a hysteresis driven instability mechanism, where the shape of a generalised play operator may decide between large-time equilibration or large-time blow-up.

Dengue fever is a disease with about 390 Mio. cases per year (WHO, 2016). It is a vector borne disease which is transmitted by tiger mosquitoes.
In the year 2016 about 950 reported cases of dengue were identified in Germany, fortunately traced back to an infection outside of Germany.
Unfortunately tiger mosquitoes are already observed from self-preservative populations in 2018 in Freiburg im Breisgau, Sinsheim and Heidelberg.
Therefore the possibility of an outbreak of the disease in Germany is aware.
We consider a new dengue fever model with adapted parameters to Germany and present numerical simulations and optimal control strategies
in order to predict possible outbreak scenarios.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the unknown parameters of the model. We study the problem of sensitivity of the optimal solution via variational data assimilation with respect to observation errors. On the basis of relations between the error of the optimal solution and the errors of observational data through the Hessian of the cost functional, the algorithms are developed and justified for calculating the coefficients of sensitivity as the norms of the response operators occurring in the equations for errors. A numerical study of the sensitivity of the optimal solution on the example of the problem of variational data assimilation of sea surface temperature to restore the heat flows for the model of thermodynamics is presented. Numerical examples for data assimilation in the Baltic Sea dynamics model are given. This work was carried out within the Russian Science Foundation project 17-77-30001 (numerical experiments) and the project 18-01-00267 of the Russian Foundation for the Basic Research.

References
1. Gejadze I., Le Dimet F.-X., Shutyaev V. On analysis error covariances in variational data assimilation. SIAM J. Sci. Comput., 2008, v.30, no.4, pp. 1847–1874.
2. Gejadze I., Le Dimet F.-X., Shutyaev V. On optimal solution error covariances in variational data assimilation problems. J. Comp. Phys., 2010, v.229, pp. 2159–2178.
3. Gejadze I., Shutyaev V.P., Le Dimet F.-X. Analysis error covariance versus posterior covariance in variational data assimilation. Q. J. R. Meteorol. Soc., 2013, v.139, pp. 1826-1841.
4. Gejadze, I.Yu., Shutyaev, V. On gauss-verifiability of optimal solutions in variational data assimilation problems with nonlinear dynamics. J. Comp. Phys., 2015, v.280, pp.439-456.
5. Shutyaev, V., Gejadze, I., Vidard, A., and Le Dimet, F.-X. Optimal solution error quantification in variational data assimilation involving imperfect models. Int. J. Numer. Meth. Fluids, 2017, v.83, no.3, pp.276-290.
6. Gejadze, I., Oubanas, H., and Shutyaev, V. Implicit treatment of model error using inflated observation-error covariance. Q.J.R. Meteorol. Soc., 2017, v.143, pp.2496–2508.
7. Shutyaev, V. P., Le Dimet, F.-X., Parmuzin, E. I. Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation // Nonlin. Processes Geophys., 2018, v.25, pp.429-439.

The methods of data assimilation have become an important tool for analysis of complex physical phenomena in various fields of science and technology. These methods allow us to combine mathematical models, data resulting from observations and a priori information. The problems of variational data assimilation can be formulated as optimal control to find unknown model parameters such as initial and/or boundary conditions, right-hand sides in the model equations (forcing terms), distributed coefficients, based on minimization of the cost function related to observations. A necessary optimality condition reduces an optimal control problem to an optimality system which involves the model equations, the adjoint problem, and input data functions
In this work the variational data assimilation problems for the sea water area were formulated and studied [1]. We assume, that the unique function which is obtained by observation data processing is the function of Sea Surface Temperature (SST) and we permit that the function is known only on a part of considering area (for example, on a part of the sea). Numerical experiments on restoring the ocean heat flux and obtaining solution of the system (temperature, salinity, velocity, and sea surface level) of the primitive equation hydrodynamics model [2] with assimilation procedure were carried out with the use of the data error covariance matriх. The spatial resolution of the model grid with respect to the horizontal variables is 0.0625*0.03125 degree. The results of the numerical experiments are presented.
This study was supported by the Russian Foundation for Basic Research (project №19-01-00595).

[1] Agoshkov V.I., Parmuzin E.I., Zakharova N.B., Zalesny V.B., Shutyaev V.P., Gusev A.V. Variational assimilation of observation data in the mathematical model of the Baltic Sea dynamics, Russ. J. Numer. Anal. Math. Modelling, 2015, V. 30, No. 4, PP. 203–212.
[2] Zalesny V.B., Gusev A.V., Chernobay S.Yu., Aps R., Tamsalu R., Kujala P., Rytkönen J. The Baltic Sea circulation modelling and assessment of marine pollution, Russ. J. Numer. Analysis and Math. Modelling, 2014, V 29, No. 2, pp. 129–138.

Global feedback stabilizability results are presented for nonautonomous coupled parabolic-ode linear systems. Such systems arise, for example, when we linearize the monodomain equations around a given time-dependent trajectory. The monodomain equations are a model for the electrical activity in the heart.

The stabilizing feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. The location of the actuators plays an important role in the stabilizing performance of the controller. The optimal location will be discussed.

Subsequently, local feedback stabilizability results are presented for nonlinear coupled parabolic-ode systems of the FitzHugh-Nagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

The stochastic Landau-Lifshitz-Gilbert equation describes magnetization dynamics in ferromagnetic materials in a thermal bath. In this presentation I discuss the optimal control of a finite spin system governed by the stochastic Landau-Lifshitz-Gilbert equation in order to guide the configuration optimally into a target state. At the heart of our analysis is the problem statement with a Bellman PDE on the state manifold. We show the wellposedness of the Bellman formulation and establish the regularity of the value function and the optimal controls to prove the existence of a strong solution of the optimal control problem. Numerical experiments in a high-dimensional setting are presented. The talk is based on joint work with Ananta Majee, Andreas Prohl and Christian Schellnegger (Universitaet Tuebingen).

This talk is concerned with a shape optimization problem involving a viscous Eikonal equation as the
state equation. This problem is motivated by an inverse problem from cardiac electrophysiology. The
viscous Eikonal equation under consideration is a semilinear elliptic equation. Its solution models
the arrival time of a electromagnetic wave in the heart which is initiated at several activation sites.
The heart is modeled by a domain and the activation sites are given by small domains inside the
heart. The state equation is posed on the heart without the activation sites and thus has Dirichlet
boundary conditions on the surface of the activation sites. Given measurements on the surface of
the heart the locations of the activations sites are sought-after. This constitutes a shape optimization
problem. First the wellposeness of the state and adjoint state equation is discussed. Then shape
derivative of the involved cost functional is derived. Based on this derivative a perturbation field is
calculated which is used to translate the activation sites. The talk is concluded with two numerical
experiments in 2D and on a realistic geometry of a heart in 3D.

Fireshape is a shape optimization toolbox for the finite element library Firedrake. Fireshape can tackle optimization problems constrained by boundary value problems and offers the following features: decoupled discretization of control and state variables, automatic derivation of shape derivatives and adjoint equations, different metrics to define shape gradients, and numerous optimization algorithms. The shape optimization knowledge required to use this library is minimal.

Fireshape is available at https://github.com/fireshape/fireshape