Publication detail

A two-stage stochastic program with elliptic pde constraints

POPELA, P. ČAJÁNEK, M.

Original Title

A two-stage stochastic program with elliptic pde constraints

English Title

A two-stage stochastic program with elliptic pde constraints

Type

journal article - other

Language

en

Original Abstract

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.

English abstract

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.

Keywords

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

Released

22.06.2010

ISBN

978-80-214-4120-0

Book

Conference Proceedings Mendel 2010

Pages from

447

Pages to

452

Pages count

6

BibTex


@article{BUT124323,
  author="Pavel {Popela} and Michal {Čajánek}",
  title="A two-stage stochastic program with elliptic pde constraints",
  annote="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.

",
  booktitle="Conference Proceedings Mendel 2010",
  chapter="124323",
  howpublished="print",
  number="1",
  volume="2010",
  year="2010",
  month="june",
  pages="447--452",
  type="journal article - other"
}