joint work with Mohamad Hassan Farshbaf Shaker, Martin Gugat, Holger Heitsch
The talk addresses a PDE-constrained optimization problem, namely the optimal control of vibrating string under probabilistic constraints. The initial position of the string is supposed to be random (multiplicative noise on Fourier coefficients). The objective is to find a cost optimal control driving the final energy of the string to an amount smaller then epsilon with probability larger than p. The numerical solution of the resulting convex optimization problem relies on the ‘spheric-radial decomposition’ of Gaussian random vectors.
In this talk, we introduce and analyze a new class of optimal control problems constrained by elliptic equations with uncertain fractional exponents. We utilize risk measures to formulate the resulting optimization problem. We develop a functional analytic framework, study the existence of solution and rigorously derive the first-order optimality conditions. Additionally, we employ a sample-based approximation for the uncertain exponent and the finite element method to discretize in space.
joint work with Vesa Kaarnioja, Frances Kuo, Claudia Schillings, Ian Sloan
In this work we apply a quasi-Monte Carlo (QMC) method to an optimal control problem constrained by an elliptic PDE equipped with an uncertain diffusion coefficient. In particular the optimization problem is to minimize the expected value of a tracking type cost functional with an additional penalty on the control. It is shown that the discretization error of the solution is bounded by the discretization error of the adjoint state. For the convergence analysis the latter is decomposed into truncation error, finite element error and QMC error, which are then analysed separately.