A generalized empirical interpolation method : application of reduced basis techniques to data assimilation 

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State of the art of the existing 3-D time-dependent neutron transport solvers

There exists quite a large amount of industrial codes solving the steady-state multigroup neutron transport equation for reactor core calculations. However, the extension of these codes for time-dependent computations seems to have been seldom implemented so far, mainly because of excessive computing times. For this reason, the existing time-dependent codes have used the diffusion approximation to the Boltzmann operator that has been presented in section 1.4. To the best of the author’s knowledge, only MINARET and TORT-TD [112] aim at solving the kinetic neutron transport equation (1.1) in three dimensional geometries. They are both multigroup SN solvers that use a Euler backward in time discretization. They also use the same numerical schemes (generalized Gauss-Seidel and source iteration). The main difference relies in the spatial discretization:
while TORT-TD works on cartesian grids, MINARET’s mesh is « partially » unstructured in the sense that it is built by an extrusion of an initial two-dimensional unstructured mesh.
In the present work, a special effort has been made in order to provide solutions to the long computational times of such solvers and some sequential and parallel acceleration techniques have been explored. These will be outlined in section 1.6 in which we have also outlined several other existing methods that seem interesting to keep in mind for future works. Concrete performances will be presented in chapter 2.

About acceleration techniques for a time-dependent multigroup neutron transport SN solver

We will focus here on the acceleration techniques for a solver built with:
• an Euler backward time discretization.
• a multigroup approximation for the energy.
• a discrete ordinates discretization for the directions.
This strategy is common in the field of neutronics as outlined in section 1.5 and it is also the one that has been followed in MINARET. For a given time step, we are led to the resolution of two embedded iterative algorithms (see algorithm 1.1):
1. a generalized Gauss-Seidel iterative algorithm for the resolution of the multigroup equations and.
2. the computation of a source iteration problem for each energy group.
Given this scheme, the resolution can first of all be accelerated by sequential methods. In section 1.6.1, we will explain two of these methods that have been implemented in MINARET: the Chebyshev extrapolation and the Diffusion Synthetic Acceleration. These methods can be combined with parallel techniques. Section 1.6.2 will outline the two parallel techniques explored in MINARET and also several other methods that seem promising tracks to be considered for future works.

Parallelization of the time domain: the parareal in time algorithm

The excellent scalability properties of the parallelization of the angular variable are unfortunately limited by the number of processors that can be assigned for this task. This comes from the fact that the number of SN angular directions is usually smaller than the classical number of available processors – because the number of directions is fixed in coherence with the accuracy of the rest of the variables. For this reason, if we have more processors at our disposal and wish additional speed-ups, the parallelization of other variables needs to be addressed. One can find in the literature techniques to parallelize the energy [34] and spatial variables [101]. However, regarding the parallelization of the time variable, the present work is (to the best of our knowledge) the first one that explores this type of parallelization in the neutron transport equation — preliminary works for neutron diffusion can be found in [14], [13] —. Several approaches have been proposed over the years to decompose the time direction when solving a partial differential equation (see, e.g., [96], [22], [44]). Of these, the parareal in time algorithm (as for « parallel in real time »), whose performances we explore in this work, was first proposed a decade ago by [72] and has received an increasing amount of attention in the last years.
During this time, the parareal method has been applied successfully to a number of applications (see, e.g., [7], [46], [110] among many others), demonstrating its versatility. Theoretical advances on this method include stability analysis ( [10], [115], [9], [31]), its coupling with spatial domain decomposition methods ( [86], [56]) and control problems ( [83], [85]).
To see how the method works and how it has been applied to the neutron transport equation, we start by noting that equation (1.1), can be written in the following compact form: 8< : @y @t + A(t; y) = 0 , t 2 [0, T].

Parallelization of the spatial domain

In the most internal loop of the numerical scheme given in algorithm 1.1, one has to solve a set of advection spatial problems of the form of (1.44) for D directions !d. Let us say that we solve this problem with a finite element approach like the one used in MINARET.
A first option to parallelize this equation is to use spatial domain decomposition methods. However, because of the advective nature of the PDE, the efficient techniques existing for elliptic problems (like the ones explored in [61] for the steady state neutron SPN equations) cannot be applied and the decomposition of the spatial domain in the transport equation remains still nowadays an open problem. We nevertheless mention the works of [50], [54] in this direction.
An alternative to this is to analyze the resolution of (1.44) from an algorithmic point of view. Since (1.44) is an advective equation, in the case of a resolution with finite elements, the problem is locally solved cell after cell by a sweeping technique. The order depends on the direction ! because a cell cannot be solved for a particular direction until its « upstream » neighbors have been  solved. In other words, a given cell can be computed provided that the incoming fluxes for this cell are known, i.e. the fluxes along cell faces for which !.n is negative. As an example, consider the situation of the simple mesh of figure 1.2(a): cell number 2 cannot be solved until we have not solved cell 1. This leads to the dependency graph of figure 1.2(b). Each vertex represents a cell and the arrows are the dependencies between cells. A vertex cannot be solved until its predecessors have not been computed.

Discretization and implementation in the MINARET solver

With the exception of some simple cases (see [107] for further references) where problem (2.1) can exactly be solved, the resolution of (2.1) needs to be numerically addressed and requires discretizations and approximations of the involved variables. The MINARET solver uses traditional discretization techniques and this section briefly explains them by putting special stress on the iterative numerical schemes that have been implemented.
We start by discretizing the energy variable and deriving the multigroup version of equation (2.1). The strategy is based on the division of the energy interval into G subintervals: [Emin,Emax] = [EG,EG−1] [ · · · [ [E1,E0]. For 1 g G, we denote by g the approximation of in the subinterval [Eg,Eg−1]. Further, let [0, T] = SN−1 n=0 [tn, tn+1] be a division of the full time interval and Tn+1 = tn+1 − tn. An Euler-backward scheme for the time variable is then applied. Let g,n(r,!) be the approximation of (t, r,!,E) at time t = tn and for E 2 [Eg,Eg−1]. Given { g,n(r,!)}Gg =1, the set of unknowns { g,n+1(r,!)}Gg =1 for the time tn+1 is the solution of the following set of coupled source problems: 8< : Find over R × S2 the angular flux g,n+1 (r,!) that is the solution of: Lg − Hg − ˜ Fg − ˜Qg g,n+1(r,!) = ˜ Sg,n(r,!), 8g 2 {1, . . . ,G}.

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Parallelization of the time variable

As has been outlined in the previous section, an efficient technique for the acceleration of the resolution of the time dependent neutron transport equation is the parallelization of the angular variable. Its performances seem to be only slightly degraded in weak scaling cases, which implies that arbitrary high Sn orders can be addressed in a reasonable time. The most usual case, however, is to fix the Sn angular accuracy in coherence with the accuracy fixed for other variables (like, e.g., the spatial variable in which the accuracy is given by the finite element polynomial approximation). For this reason, the number of allocated processors to efficiently accelerate a given calculation is upper bounded and, if we have more processors at our disposal and wish additional speed-ups, the parallelization of other variables needs to be addressed. In this context, it is interesting to consider the extra speed-up that can bring the parallelization of the temporal variable. In the present case, this task has been adressed by a domain decomposition technique: the parareal in time algorithm.
This section is organized as follows: after a brief recall of the basics of the parareal in time algorithm, an extension of the traditional theoretical speed-up formula will be proposed in order to properly take into account our particular case in which parareal is coupled with other iterative techniques at each time propagation. Finally, an analysis of the performances of the method for the resolution of transport transients with MINARET will be presented. The implemented results consider the parallelization of the time without coupling it with the parallelization of the angle. They are nevertheless representative enough of the accelerations that could be obtained in addition to the ones provided by the angular parallelization.

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 

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