Titles and slides of talks
- G. Barles (Université de Tours)
Recent progress on Hamilton-Jacobi Equations and Deterministic Control Problems with Discontinuities
The theory of viscosity solutions has provided a very satisfactory framework for dealing with a wide variety of deterministic control problems by the dynamic programming approach. But this theory is only really effective when the dynamics and costs are continuous, even if some more general cases can be treated. Recently, much effort has been made to understand more systematically situations where discontinuities appear. After recalling these different types of problems (networks, regional or stratified problems), our aim is to describe the main results for regional control problems and some examples of results for standard or not standard stratified problems.
- S. Bianchini (SISSA, Trieste)
A decomposition of vector fields in $\mathbb R^{d+1}$
- G. Bouchitté (Université de Toulon)
Convex relaxation for a class of free boundary problems
- C. Canuto (Politecnico di Torino)
Adaptivity in high-order numerical methods for PDEs
- G. Dal Maso (SISSA, Trieste)
Gamma-convergence and stochastic homogenisation of free-discontinuity functionals
We study Gamma-convergence of sequences of free-discontinuity functionals. The main results are: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form, and homogenisation formulas without periodicity assumptions. These results are then applied to stochastic homogenisation of free-discontinuity functionals.
- A. Ern (ENPC and INRIA, Paris)
Polynomial-degree-robust liftings in H1 and H(div)
We study liftings of piecewise polynomial data prescribed on faces and elements of a patch of simplices sharing a vertex in the H1 and H(div) settings. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken H1 and H(div) spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One important application of these results is in a posteriori error analysis. This is joint work with M. Vohralik (INRIA).
- G. Leugering (Universität Erlangen-Nürnberg)
Exact boundary controllability of networks of quasi-linear strings and springs coupled with point-masses
Abstract.
- S. Müller (Universität Bonn)
Data driven problems in elasticity
Abstract.
- E. Rocca (Università di Pavia)
Diffuse interface models of tumor growth: optimal control and other issues
We will present some results on optimal control problems for a diffuse interface model of tumor growth. The state equations couple a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we will also allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions. Other issues, like the existence of solutions to more complex models, including the velocity field, for example, will be discussed.
- B. Schweizer (T. U. Dortmund)
Description of waves in discrete and in heterogeneous media with dispersive effective equations
Dispersion occurs in media in which waves with different wave-length travel with different speed. A linear wave equation with constant coefficients does not show dispersion. A linear wave equation with periodic coefficients and a small periodicity can be replaced, in the homogenization limit, by a linear wave equation with constant coefficients, we hence do not expect dispersive effects. On the other hand, numerical experiments show that solutions have a dispersive behavior, at least after long time. We discuss this effect and derive dispersive effective equations. We furthermore investigate the wave equation in a discrete spring-mass model. The discrete character of the model introduces small-scale oscillations, which result in a dispersive long time behavior. We derive the dispersive effective wave equations for the discrete model. We present joint work, obtained with A. Lamacz, T. Dohnal, and F. Theil.
- E. Trélat (Université Pierre et Marie Curie)
Optimal shape and location of sensors or actuators in PDE models
We consider the problem of optimizing the shape and the location of sensors or actuators for systems whose evolution is driven by a linear PDE model. This problem is frequently encountered in applications where one wants for instance to maximize the quality of the reconstruction of solutions by using only partial observations. For example, we model and solve the following informal question: What is the optimal shape and location of a thermometer? We stress that we want to optimize not only the placement but also the shape of the observation domain, over the class of all possible measurable subsets of the domain having a prescribed measure. We model this optimal design problem as the one of maximizing a functional that we call the randomized observability constant, which reflects what happens for random initial data, and which is of a spectral nature. Solving this problem is then strongly dependent on the PDE model under consideration. For parabolic equations, we prove the existence and uniqueness of a best domain, regular enough, and whose algorithmic construction depends in general on a finite number of modes. In contrast, for wave or Schrodinger equations, relaxation may occur, and our analysis reveals intimate relations with quantum chaos, more precisely with quantum ergodicity properties of the eigenfunctions. These works are in collaboration with Y. Privat and E. Zuazua.
- A. Walther (Universität Paderborn)
Adjoint-based optimization of a complete design chain in CFD
An optimization in CFD that covers the complete design chain including a CAD tool to describe the object to be optimized is still a severe challenge. In this talk we present the algorithmic differentiation of the CAD kernel within OpenCASCADE Technology using the AD tool ADOL-C. This will be coupled with a correspondingly extended flow solver for optimization purposes in CFD. First numerical results for the optimization of a U-bend pipe and of the TU Berlin stator are shown including also a verification of the computed derivatives.