A deterministic optimal design problem for the heat equation

PDE Seminar
Wednesday, March 30, 2016 - 2:05pm for 1 hour (actually 50 minutes)
Skiles 270
Alden Waters – CNRS Ecole Normale Superieure
Wilfrid Gangbo
In everyday language, this talk addresses the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using wave packet decompositions from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated by solving a sequence of finite-dimensional optimization problems. This talk will also address future applications to inverse problems.