Optimization problems are relevant in many areas of technical, industrial, and economic applications. At the same time, they pose challenging mathematical research problems in numerical analysis optimization. Harald Held considers an elastic body subjected to uncertain internal and external forces. Since simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings, he uses techniques from level set-based shape optimization and two-stage stochastic programming. Taking advantage of the PDE's linearity, he is able to compute solutions for an arbitrary number of scenarios without significantly increasing the computational effort. The author applies a gradient method using the shape derivative and the topological gradient to minimize, e.g., the compliance. The stochastic programming perspective also allows incorporating risk measures into the model which might be more appropriate objective in many practical applications.
Author(s): Harald Held
Edition: 2009
Language: English
Pages: 140
Cover......Page 1
Stochastic Programming......Page 3
Shape Optimization
under Uncertainty
from a Stochastic
Programming Point
of View......Page 4
ISBN 3834809098......Page 5
Foreword......Page 6
Acknowledgments......Page 7
Abstract......Page 8
Contents......Page 9
Symbol Index......Page 10
1 Introduction......Page 11
1.1 The Elasticity PDE......Page 14
1.1.1 Variational Formulation......Page 16
1.2 Shape Optimization Problems......Page 23
1.3 Two-Stage Stochastic Programming......Page 27
1.3.1 Expected Value......Page 30
1.3.2 Risk Measures......Page 32
2 Solution of the Elasticity PDE......Page 35
2.1.1 Construction for the Neumann Boundary......Page 38
2.1.1.1 Implementational Remarks......Page 43
2.1.2 Construction for the Dirichlet Boundary......Page 47
2.1.2.1 Implementational Remarks......Page 50
2.1.2.2 Simple 1D Example......Page 53
2.1.3 Mixed Boundary Conditions......Page 54
2.1.4 Computation of the System Matrix and the Right-Hand Side Vector......Page 57
3 Stochastic Programming Perspective......Page 58
3.1 Stochastic Shape Optimization Problem......Page 59
3.1.1 Two-Stage Stochastic Shape Optimization Problem......Page 60
3.1.2 Dual Problem and Saddle Point Formulation......Page 63
3.2 Reformulation and Solution Plan for the Expectation-Based Model......Page 70
3.3 Expected Excess......Page 79
3.3.1 Barrier Method......Page 80
3.3.2 Smooth Approximation......Page 81
3.4 Excess Probability......Page 82
4 Solving Shape Optimization Problems......Page 85
4.1 Level Set Formulation......Page 86
4.1.1 Computation of the Mean Curvature......Page 88
4.2 Shape Derivative......Page 89
4.3 Topological Derivative......Page 97
4.4 Steepest Descent Algorithm......Page 102
4.4.1 Regularized Descent Direction......Page 106
5 Numerical Results......Page 109
5.1 Deterministic and Expectation-Based Results......Page 110
5.1.1 VSS and EVPI......Page 120
5.2 Risk Aversion......Page 122
A.1 Notation......Page 128
A.2 Important Facts and Theorems......Page 131
References......Page 133
