Space-time parallel methods

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Table of contents

List of Figures
List of Tables
Résumé de la Thèse
Introduction Générale
Contributions de cette thèse
État de l’art
General Introduction
Contributions of this thesis
1 State of the art
1.1 Data assimilation (DA)
1.1.1 Sequential methods
1.1.2 Variational methods
1.2 Space-time parallel methods
1.2.1 The Parareal algorithm
1.2.2 Space-time parallel methods and DA
1.3 Bathymetry estimation
1.3.1 Wave modeling
1.4 Blade element momentum (BEM) theory
2 Time-parallelization of sequential data assimilation problems
2.1 The Luenberger observer
2.2 Time-parallelization setting
2.2.1 Framework
2.2.2 The Diamond strategy
2.3 Parallelization
2.3.1 The Parareal algorithm
2.3.2 Combination with Luenberger observer
2.3.3 Complexity analysis
2.4 Numerical experiments
2.4.1 Diagonalized system
2.4.2 Evolution of k`
2.4.3 Observed efficiency
2.5 Perspectives
3 Bathymetry optimization
3.1 Derivation of the wave model
3.1.1 From Navier-Stokes system to Saint-Venant equations
3.1.2 Helmholtz formulation
3.2 Description of the optimization problem
3.2.1 Weak formulation
3.2.2 Continuous optimization problem
3.2.3 Continuity of the control-to-state mapping
3.2.4 Existence of optimal solution
3.3 Boundedness/Continuity of solution to Helmholtz problem
3.3.1 C0-bound for the general Helmholtz problem
3.3.2 C0-bounds for the total and scattered waves
3.4 Discrete optimization problem
3.4.1 Convergence of the Finite element approximation
3.4.2 Convergence of the discrete optimal solution
3.5 Numerical experiments
3.5.1 Numerical methods
3.5.2 Example 1: a wave damping problem
3.5.3 Example 2: an inverse problem
3.6 Perspectives
4 Mathematical analysis of the Blade element momentum theory
4.1 The Blade element momentum theory
4.1.1 Variables
4.1.2 Glauert’s modeling
4.1.3 Simplified model
4.1.4 Corrected model
4.2 Analysis of Glauert’s model and existence of solution
4.2.1 Simplified model
4.2.2 Corrected model
4.2.3 Multiple solutions
4.3 Solution algorithms
4.3.1 Usual algorithm
4.3.2 Alternative algorithms
4.4 Optimization
4.4.1 Simplified model and usual design procedure
4.4.2 Asymptotical analysis of the corrected model
4.5 Numerical experiments
4.5.1 A practical example
4.5.2 Solution algorithms
4.5.3 Optimization
4.6 Perspectives
Appendix 4.A Convergence in the simplified case
References