joint work with Sven Leyffer, Lars Ruthotto, Bart van Bloemen Waanders
We aim to find the number and location of sources from discrete measurements of concentration on computational domain. We formulate a forward PDE model and then invert it for the unknown sources by reformulating the latter as a MINLP, resulting in mixed-integer PDE constrained optimization (MIPDECO). For MINLP solvers, solving such MINLPs is computationally prohibitive for fine meshes and 3D cases. We propose a trust-region improvement heuristic that uses continuous relaxation of MIPDECO problem with different rounding schemes.
joint work with Stefan Ulbrich
We present a multilevel optimization algorithm for PDE-constrained problems. The basis of the algorithm is an adaptive SQP method, using adaptive grid refinement to reduce the computational cost. This method is extended by introducing a level build with reduced order models (ROM) for state and adjoint equation. For this algorithm we can show convergence, when suitable a posteriori error estimators on the discretization and the ROM are available. The algorithm is applied to a benchmark problem of fluid-structure interaction, which is a problem of interest in many engineering applications.
joint work with Stefan Ulbrich
We study optimal control problems governed by balance laws with initial and boundary conditions and pointwise state constraints. The boundary data switch between smooth functions at certain points, where we consider the smooth parts and the switching points as control. State constraints are challenging in this context since solutions of nonlinear hyperbolic balance laws may develop shocks after finite time which prohibits the use of standard methods. We derive optimality conditions and prove convergence of the Moreau-Yosida regularization that is used to treat the state constraints.