Sampling-based approximations

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Table of contents

A Context and theoretical tools 
Introduction en français
–1 Contexte et objectifs
–2 Plan et principales contributions
Introduction 
–1 Context and main objectives
–2 State of the art
–2.1 Uncertainty Quantification
–2.2 Optimization
–2.3 Reliability-Based Design Optimization (RBDO)
–2.4 Robust Design Optimization (RDO)
–3 Industrial context and contributions
–3.1 Context
–3.2 Contributions
–4 Outline of this work
I Introduction to kernel methods 
I–1 An intuitive introduction to Gaussian processes
I–1.1 The basics: Linear Regression
I–1.2 Adding complexity: Feature Regression
I–1.3 Adding generality: Non-parametric regression
I–1.4 A probabilistic view: Bayesian Regression
I–1.5 Gaussian processes
I–2 Reproducing Kernel Hilbert Spaces
I–2.1 RKHS theory fundamentals
I–2.2 Kernel Mean Embeddings
B Methods and algorithms
II A general approach for constrained multi-objective optimization under uncertainty 
II–1 Formulation of the problem
II–1.1 Multi-objective optimization
II–1.2 Uncertainty-based optimization
II–1.3 Robustness and reliability measures
II–2 A probabilistic setting
II–2.1 Probabilistic Pareto dominance
II–2.2 Pareto-Optimal Probability
II–3 The SAMATA algorithm
II–3.1 Surrogate-Assisting strategy
II–3.2 Measure Approximation with Tunable Accuracy
III Uniform approximations: Bounding Boxes 
III–1 Bounding-Box context
III–1.1 Definition of a Bounding-Box
III–1.2 Boxed Pareto dominance
III–1.3 SABBa general flowchart
III–2 Theoretical considerations
III–2.1 Pareto-optimal sets
III–2.2 Bounding-Box approach
III–2.3 Convergence analysis
III–2.4 Surrogate-Assisting model
III–2.5 Algorithm
III–3 Noisy optimization with tunable accuracy
III–3.1 Numerical ingredients
III–3.2 Applications
III–4 Optimization under uncertainty
III–4.1 Measures computation and refinement
III–4.2 POP computation
III–4.3 Quality indicator
III–4.4 Uncertainty-based SABBa algorithm
III–4.5 Numerical tests
IV Sampling-based measure approximations 
IV–1 Sampling-based setting
IV–2 Coupled-space realizations for joint samplings
IV–2.1 Formulation and challenge
IV–2.2 Inducing points strategy
IV–2.3 Implementation in SAMATA
IV–3 KDE-based random field surrogate
IV–3.1 Kernel Density Estimation
IV–3.2 Surrogate-model construction
IV–4 Application to parametric uncertainties
IV–4.1 Cost assessment
IV–4.2 Benefit of sampling-based approximations
IV–5 Application to non-parametric uncertainties
IV–5.1 Adjustment of SAMATA
IV–5.2 Analytical application
C Applications
V Engineering applications 
V–1 Two bar truss structure
V–2 Optimization of a Thermal Protection System for atmospheric reentry
V–3 ORC turbine blade optimization
V–3.1 Physical application
V–3.2 Problem setting
V–3.3 Numerical ingredients
V–3.4 Mono-objective results
V–3.5 Bi-objective results
D Perspectives
VI Conclusions 
VI–1 Main achievements
VI–1.1 Tunable accuracy formulation
VI–1.2 Bounding-Box approximations
VI–1.3 Sampling-based approximations
VI–1.4 Properties of the proposed framework
VI–2 Limitations
VI–3 Perspectives
VI–3.1 Improvements within the SAMATA framework
VI–3.2 Ideas
Bibliography