A two-stage stochastic program with elliptic pde constraints

A two-stage stochastic program with elliptic pde constraints

Článek recenzovaný mimo WoS a Scopus

The purpose of the paper is to introduce an optimization model for a problem involving random elements and constraints formed by an elliptic partial differential equation and related conditions. The discussed problem may serve as a prototype for various applications in mechanical and civil engineering. The problem con- cerning an optimal design of a membrane under a random load has been chosen. The corresponding mathematical model involves a partial differential equation (PDE) type constraint, specifically, the elliptic PDE is considered. It has been shown that the two-stage stochastic programming related scheme offers a promising approach to study similar problems. The formally correct description of the model serves as the initial step towards a computable model. A computational scheme for this type of problems is proposed, including discretization methods to tackle randomness and continuity involved in the formal description. By means of the derived approximation, the mathematical model is detailed, implemented, and solved in GAMS. Then the solution quality is tested by Monte Carlo technique. Finally, the graphical and numerical results are presented and interpreted.

The purpose of the paper is to introduce an optimization model for a problem involving random elements and constraints formed by an elliptic partial differential equation and related conditions. The discussed problem may serve as a prototype for various applications in mechanical and civil engineering. The problem con- cerning an optimal design of a membrane under a random load has been chosen. The corresponding mathematical model involves a partial differential equation (PDE) type constraint, specifically, the elliptic PDE is considered. It has been shown that the two-stage stochastic programming related scheme offers a promising approach to study similar problems. The formally correct description of the model serves as the initial step towards a computable model. A computational scheme for this type of problems is proposed, including discretization methods to tackle randomness and continuity involved in the formal description. By means of the derived approximation, the mathematical model is detailed, implemented, and solved in GAMS. Then the solution quality is tested by Monte Carlo technique. Finally, the graphical and numerical results are presented and interpreted.

Finite difference method; GAMS; Membrane design; Monte Carlo; Partial differential equation constraint; Scenarios; Two-stage stochastic programming

Finite difference method; GAMS; Membrane design; Monte Carlo; Partial differential equation constraint; Scenarios; Two-stage stochastic programming

POPELA, P.; ČAJÁNEK, M.

2016

22.06.2010

978-80-214-4120-0

Conference Proceedings Mendel 2010

1803-3814

Mendel Journal series

2010

1

Česká republika

447

452

6