Invited Session Fri.3.MA 415

Friday, 15:15 - 16:45 h, Room: MA 415

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

PDE constrained optimization with uncertain data

 

Chair: Volker H. Schulz

 

 

Friday, 15:15 - 15:40 h, Room: MA 415, Talk 1

Hanne Tiesler
Stochastic collocation for optimal control problems with stochastic PDE constraints

Coauthors: Robert Mike Kirby, Tobias Preusser, Dongbin Xiu

 

Abstract:
The use of stochastic collocation schemes for the solution of optimal control problems, constrained by stochastic partial differential equations (SPDE), is presented. The constraining SPDE depends on random data and accordingly, the randomness will propagate to the states of the system, whereas the control is assumed to be deterministic. There exist different efficient numerical schemes for the solution of SPDEs, one of them is the stochastic collocation method, which is based on the generalized polynomial chaos. For the minimization of the constrained optimization problems we combine the stochastic collocation method with a gradient descent method as well as a sequential quadratic program (SQP). In the presented work, different optimization problems are considered, i.e., we define different objective functions of tracking type to show different application possibilities. The functions involve several higher order moments of the random states as well as classical regularization of the control. The developed methods are compared to the widely used Monte Carlo method. Numerical results illustrate the performance of the new optimization approach with stochastic collocation.

 

 

Friday, 15:45 - 16:10 h, Room: MA 415, Talk 2

Claudia Schillings
On the influence of robustness measures on shape optimization with stochastic uncertainties

Coauthor: Volker Schulz

 

Abstract:
The unavoidable presence of uncertainties poses several difficulties to the numerical treatment of optimization tasks. In this talk, we discuss a general framework attacking the additional computational complexity of the treatment of uncertainties within optimization problems. Appropriate measure of robustness and a proper treatment of constraints to reformulate the underlying deterministic problem are investigated. In order to solve the resulting robust optimization problems, we propose efficient discretization techniques of the probability space as well as algorithmic approaches based on multiple-setpoint ideas in combination with one-shot methods. Finally, numerical results considering optimal aerodynamic design under shape uncertainties will be presented.

 

 

Friday, 16:15 - 16:40 h, Room: MA 415, Talk 3

Matthias Heinkenschloss
A trust-region based adaptive stochastic collocation method for PDE constrained optimization with uncertain coefficients

 

Abstract:
Many optimization problems in engineering and science are governed by partial differential equations (PDEs) with uncertain parameters. Although such problems can be formulated as optimization problems in Banach spaces and derivative based optimization methods can in principle be applied, the numerical solution of these problems is more challenging than the solution of deterministic PDE constrained optimization problems. The difficulty is that the PDE solution is a random field and the numerical solution of the PDE requires a discretization of the PDE in space/time as well as in the random variables. As a consequence, these optimization problems are substantially larger than the already large deterministic PDE constrained optimization problems.
In this talk we discuss the numerical solution of such optimization problems using stochastic collocation methods. We explore the structure of this method in gradient and Hessian computations. We use a trust-region framework to adapt the collocation points based on the progress of the algorithms and structure of the problem. Convergence results are presented. Numerical results demonstrate significant savings of our adaptive approach.

 

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