The applicability of stochastic programming models and methods to PDE constrained stochastic optimization problem is discussed. The problem concerning mathematical model involves a PDE-type constaint and an uncertain parameter related to the exernal load. A computational scheme for this type of problems is proposed, including discretization methods for random elements (scenario based two-stage stochastic programming) and the PDE constraint (finite difference method). Several deterministic reformulations are presented and compared using numerical and graphical results. and Obsahuje seznam literatury
The purpose of the paper is to discuss the applicability of stochastic programming models and methods to civil engineering design problems. In cooperation with experts in civil engineering, the problem concerning an optimal design of beam dimensions has been chosen. The corresponding mathematical model involves an ODE-type constraint, uncertain parameter related to the material characteristics and multiple criteria. As a result, a multi-criteria stochastic nonlinear optimization model is obtained. It has been shown that two-stage stochastic programming offers a promising approach to solving similar problems. A computational scheme for this type of problems is proposed, including discretization methods for random elements and ODE constraint. An approximation is derived to implement the mathematical model and solve it in GAMS. The solution quality is determined by an interval estimate of the optimality gap computed by a Monte Carlo bounding technique. The parametric analysis of a multi-criteria model results in efficient frontier computation. Furthermore, a progressive hedging algorithm is implemented and tested for the selected problem in view of the future possibilities of parallel computing of large engineering problems. Finally, two discretization methods are compared by using GAMS and ANSYS.