Parallelization of the time variable

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Table of contents

Introduction (Version française) 
Motivations des présents travaux
Résumé des résultats par chapitres
Introduction (English version) 
Motivations of this work
Summary of the results by chapters
I Numerical models for time dependent neutron transport for safety studies 
1 Overview and modern challenges of neutronic calculations 
1.1 The time-dependent neutron transport equation
1.1.1 The equation
1.1.2 Boundary conditions
1.1.3 Existence theorems
1.2 The stationary case: resolution of a generalized eigenvalue problem
1.2.1 The equation
1.2.2 Existence and uniqueness of the stationary flux
1.3 Discretization of the time-dependent neutron transport equation
1.3.1 Discretization of the time variable
1.3.2 Discretization of the energy variable
1.3.3 Discretization of the angular variable
1.3.4 Spatial discretization
1.4 Approximations to the Boltzmann operator
1.4.1 The diffusion approximation
1.4.2 The Simplified PN
1.4.3 Quasi-static methods
1.5 State of the art of the existing 3-D time-dependent neutron transport solvers
1.6 About acceleration techniques for a time-dependent multigroup neutron transport SN solver
1.6.1 Sequential acceleration methods
1.6.2 Parallel methods
2 MINARET: Towards a parallel 3D time-dependent neutron transport solver 
2.1 Introduction
2.2 The time-dependent neutron transport equation
2.3 Discretization and implementation in the MINARET solver
2.4 Definition of the numerical test cases
2.5 Sequential acceleration techniques
2.6 Parallelization of the angular variable
2.7 Parallelization of the time variable
2.7.1 The parareal in time algorithm
2.7.2 Algorithmics and theoretical speed-up
2.7.3 Numerical application
2.7.4 A parareal in space and energy algorithm?
3 A coupled parareal reduced basis scheme 
3.1 Introduction
3.2 Convergence analysis of the parareal scheme with truncated internal iterations
3.3 An application to the kinetic neutron diffusion equation
3.3.1 The model
3.3.2 Some first results
II Numerical models for the real-time monitoring of physical processes 
4 A generalized empirical interpolation method : application of reduced basis techniques to data assimilation 
4.1 Introduction
4.2 Generalized Empirical Interpolation Method
4.2.1 Recall of the Empirical Interpolation Method
4.2.2 The generalization
4.2.3 Numerical results
4.2.4 The framework
4.2.5 The combined approach – numerical results
4.3 About noisy data
4.4 Conclusions
5 The generalized empirical interpolation method: stability theory on Hilbert spaces and an application to the Stokes equation 
5.1 The Generalized Empirical Interpolation Method
5.2 Further results in the case of a Hilbert space
5.2.1 Interpretation of GEIM as an oblique projection
5.2.2 Interpolation error
5.2.3 The Greedy algorithm aims at optimizing the Lebesgue constant
5.3 Practical implementation of the Greedy algorithm and the Lebesgue constant
5.4 A numerical study about the impact of the dictionary of linear functionals in the Lebesgue constant
5.4.1 Validation of the inf-sup formula
5.4.2 Impact of the dictionary of linear functionals
5.5 Application of GEIM to the real-time monitoring of a physical experiment
5.5.1 The general method
5.5.2 A numerical application involving the Stokes equation
5.6 Conclusion and perspectives
6 Convergence analysis of the Generalized Empirical Interpolation Method 
6.1 Introduction
6.2 The Generalized Empirical Interpolation Method
6.3 Convergence rates of GEIM in a Banach space
6.3.1 Preliminary notations and properties
6.3.2 Convergence rates for (n) in the case where ( n) is not constant
6.3.3 Convergence rates of the interpolation error
6.4 Convergence rates of GEIM in a Hilbert space
6.4.1 Preliminary notations and properties
6.4.2 Convergence rates for (n)
6.4.3 Convergence rates of the interpolation error
6.5 Conclusion
7 Improvement of cheap approximations by a post-processing/reduced basis rectification method 
7.1 Definition of the rectification operator
7.1.1 Definition of the rectification operator in the linear case
7.1.2 Definition of the rectification operator in the general case
7.2 A formula to derive the rectification map RM in practice
7.3 A numerical result
Conclusion and perspectives
Bibliography