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Table of Contents
Intro; Preface; References; Contents; About the Authors; Acronyms and Initialisms; Abstract; 1 Introduction; 1.1 Motivation; 1.2 Modelling Uncertainty in the Input Data. Illustrative Examples; 1.2.1 The Laplace-Poisson Equation; 1.2.2 The Heat Equation; 1.2.3 The Bernoulli-Euler Beam Equation; References; 2 Mathematical Preliminaires; 2.1 Basic Definitions and Notations; 2.2 Tensor Product of Hilbert Spaces; 2.3 Numerical Approximation of Random Fields; 2.3.1 Karhunen-Loève Expansion of a Random Field; 2.4 Notes and Related Software; References
3 Mathematical Analysis of Optimal Control Problems Under Uncertainty3.1 Variational Formulation of Random PDEs; 3.1.1 The Laplace-Poisson Equation Revisited I; 3.1.2 The Heat Equation Revisited I; 3.1.3 The Bernoulli-Euler Beam Equation Revisited I; 3.2 Existence of Optimal Controls Under Uncertainty; 3.2.1 Robust Optimal Control Problems; 3.2.2 Risk Averse Optimal Control Problems; 3.3 Differences Between Robust and Risk-Averse Optimal Control; 3.4 Notes; References; 4 Numerical Resolution of Robust Optimal Control Problems
4.1 Finite-Dimensional Noise Assumption: From Random PDEs to Deterministic PDEs with a Finite-Dimensional Parameter4.2 Gradient-Based Methods; 4.2.1 Computing Gradients of Functionals Measuring Robustness; 4.2.2 Numerical Approximation of Quantities of Interest in Robust Optimal Control Problems; 4.2.3 Numerical Experiments; 4.3 Benefits and Drawbacks of the Cost Functionals; 4.4 One-Shot Methods; 4.5 Notes and Related Software; References; 5 Numerical Resolution of Risk Averse Optimal Control Problems; 5.1 An Adaptive, Gradient-Based, Minimization Algorithm
5.2 Computing Gradients of Functionals Measuring Risk Aversion5.3 Numerical Approximation of Quantities of Interest in Risk Averse Optimal Control Problems; 5.3.1 An Anisotropic, Non-intrusive, Stochastic Galerkin Method; 5.3.2 Adaptive Algorithm to Select the Level of Approximation; 5.3.3 Choosing Monte Carlo Samples for Numerical Integration; 5.4 Numerical Experiments; 5.5 Notes and Related Software; References; 6 Structural Optimization Under Uncertainty; 6.1 Problem Formulation; 6.2 Existence of Optimal Shapes; 6.3 Numerical Approximation via the Level-Set Method
6.3.1 Computing Gradients of Shape Functionals; 6.3.2 Mise en Scène of the Level Set Method; 6.4 Numerical Simulation Results; 6.5 Notes and Related Software; References; 7 Miscellaneous Topics and Open Problems; 7.1 The Heat Equation Revisited II; 7.2 The Bernoulli-Euler Beam Equation Revisited II; 7.3 Concluding Remarks and Some Open Problems; References; Index
3 Mathematical Analysis of Optimal Control Problems Under Uncertainty3.1 Variational Formulation of Random PDEs; 3.1.1 The Laplace-Poisson Equation Revisited I; 3.1.2 The Heat Equation Revisited I; 3.1.3 The Bernoulli-Euler Beam Equation Revisited I; 3.2 Existence of Optimal Controls Under Uncertainty; 3.2.1 Robust Optimal Control Problems; 3.2.2 Risk Averse Optimal Control Problems; 3.3 Differences Between Robust and Risk-Averse Optimal Control; 3.4 Notes; References; 4 Numerical Resolution of Robust Optimal Control Problems
4.1 Finite-Dimensional Noise Assumption: From Random PDEs to Deterministic PDEs with a Finite-Dimensional Parameter4.2 Gradient-Based Methods; 4.2.1 Computing Gradients of Functionals Measuring Robustness; 4.2.2 Numerical Approximation of Quantities of Interest in Robust Optimal Control Problems; 4.2.3 Numerical Experiments; 4.3 Benefits and Drawbacks of the Cost Functionals; 4.4 One-Shot Methods; 4.5 Notes and Related Software; References; 5 Numerical Resolution of Risk Averse Optimal Control Problems; 5.1 An Adaptive, Gradient-Based, Minimization Algorithm
5.2 Computing Gradients of Functionals Measuring Risk Aversion5.3 Numerical Approximation of Quantities of Interest in Risk Averse Optimal Control Problems; 5.3.1 An Anisotropic, Non-intrusive, Stochastic Galerkin Method; 5.3.2 Adaptive Algorithm to Select the Level of Approximation; 5.3.3 Choosing Monte Carlo Samples for Numerical Integration; 5.4 Numerical Experiments; 5.5 Notes and Related Software; References; 6 Structural Optimization Under Uncertainty; 6.1 Problem Formulation; 6.2 Existence of Optimal Shapes; 6.3 Numerical Approximation via the Level-Set Method
6.3.1 Computing Gradients of Shape Functionals; 6.3.2 Mise en Scène of the Level Set Method; 6.4 Numerical Simulation Results; 6.5 Notes and Related Software; References; 7 Miscellaneous Topics and Open Problems; 7.1 The Heat Equation Revisited II; 7.2 The Bernoulli-Euler Beam Equation Revisited II; 7.3 Concluding Remarks and Some Open Problems; References; Index