# Design and performance analysis of a multiscale scheme for controlled population dynamics

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## Investigation of numerical noise appearance

In this section we investigate the circumstances under which numerical noise can appear. For this we consider a simplied 1D problem with piecewise constant speeds. We compute the approximated solution with a simple rst order FV scheme. This allows us to perform one time step of the algorithm by hand and completely describe the interaction of the doubling condition with a two-level MR algorithm. We illustrate this example by numerical simulations performed with a method coupling the 2D FV scheme with a mean value MR adaptive strategy.

### Explication of the numerical noise appearance

In this paragraph we describe the mechanism for numerical noise appearance due to the coupling of Multiresolution with doubling uxes. Suppose that jdn K􀀀ij < « , for i 0 and jdn K+1j « . Due to the preceding rule, the mesh D0;K = D1;2K [ D1;2K+1 is subdivided, to account for possible displacement of the singularity (see Algorithm 1). The mesh D0;K􀀀1 is kept at the coarser level, but the value of the solution on its right subdivision D1;2K􀀀1, which enters the FV scheme to advance the solution on D1;2K, is available using the prediction operator (4.6) in the quadratic case.

#### Formalization and generalization of the doubling strategy

The result (4.17) also applies to Haar reconstruction (4.12). This means that the appearance of the spurious noise is due not only to the discontinuities in the solution induced by the localized doubling, but also to the enhancement of the thresholding error. In particular, even if we modied the prediction scheme in the vicinity of interfaces in order to account for the doubling of the solution, this would not stop the increase of noise.
To formalize this remark, we generalize the MR scheme introduced in Paragraph 2.2 to an arbitrary number of levels L. We introduce the linear prediction operator P`􀀀1 ` such that formula (4.6) can be expressed generically as ^n` = P`􀀀1 ` n `􀀀1.

Numerical validation of the adaptive thresholds strategies

In this paragraph we perform numerical tests to validate the adaptive strategies (4.19) and (4.20). We then study the relationship between the two thresholds and « .

Eect of the adaptive doubling strategy on the solution

To justify the doubling strategy designed to tackle the spurious noise introduced by the MR we show the inuence of applying (4.20) instead of (4.2), independently of the MR, and even of any numerical scheme, by computing the exact solution of the 1D problem (4.1). As shown in [7], the exact solution for a single interface located at x = xs with condition (4.2) is 1(t; x) = 8>>< >>: 0(x 􀀀 g0t) for x < xs.

Link between the MR and the doubling thresholds

In this paragraph we study the relation between the MR threshold parameter  » and the threshold in the adaptive doubling interface (4.20). Still for 1D test case (4.1) with the piecewise constant initial condition, we run extensive tests for dierent values of both thresholds. We also study the inuence of the number of levels in the MR hierarchy, the precision of the FV algorithm, and the eect of normalizing the MR threshold by the overall mass of the solution, as it increases with time (see Eq. (4.19)).
Figure 4.10 displays the L1 norm of error at nal time t = 20, between the adaptive solution and the solution computed on the uniform nest grid. The adaptive solution is computed with dierent values of the MR threshold « , xed once for all times, and for values of the threshold varying between « =100 and 10″. For two, three and four levels in the MR hierarchy we display the results in two dierent fashions : The standard error visualization is shown in the left panels, where the interesting feature is the linear trend of the error with  » which is preserved by the adaptive doubling strategy. However normalizing by  » both the error on the vertical axis and on the horizontal axis highlights the correlation between  » and since the minimum of the error is observed for a similar ratio range for each set of curves.
Figure 4.10 displays results computed with the rst order upwind FV scheme (4.7- 4.8). The value of which minimizes the error lies between 0:2″ and 2″ and increases with both  » and the number of levels. We also made the same test for four levels and the third order FV scheme. The results are displayed in the top panels of Figure 4.11 ; the value of minimizing the error is shifted from about  » in the rst order case to about 2″.

READ  CAPE TOWN AND HER NEIGHBOURHOODS

1 Introduction
1.1 Historique et Motivations
1.2 Modèle multiéchelles
1.2.1 Inconnues du problème
1.2.2 Système de lois de conservations
1.2.3 Contrôle non local
1.3 Apports de la thèse
1.3.1 Numerical simulation of the selection process of the ovarian follicles
1.3.2 A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics
1.3.3 Adaptive mesh renement strategy for a non conservative transport problem
1.3.4 Calibration
1.3.5 Schéma numérique multiéchelles et analyse de performances par propagation d’incertitudes
2 Numerical simulation of the selection process of the ovarian follicles
2.1 Biological and biomathematical background
2.1.1 Biological background
2.1.2 Biomathematical model
2.2 Numerical Method
2.2.1 Discretization
2.2.2 Macroscopic scale : piecewise constant approximation of the hormonal control
2.2.3 Mesoscopic scale : nite volume scheme
2.2.4 Macro/Meso scale Parallelization
2.2.5 Parallel algorithm and improvement to order 2 in time
2.3 Numerical simulations
2.3.1 Initial condition
2.3.2 Sets of default parameters
2.3.3 Time evolution of the cell density for one follicle
2.3.4 Competition between ten follicles
2.3.5 Convergence rate test
2.3.6 HPC test
3 A numerical method for transport equations with discontinuous ux functions : application to mathematical modeling of cell dynamics
3.1 Introduction
3.2 Toy models
3.3 Numerical scheme
3.3.1 Discretization
3.3.2 Treatment of discontinuous coecients
3.4 Numerical validation
3.4.1 1D test cases
3.4.2 2D test cases
3.5 Application to cell dynamics
3.5.1 Numerical simulation
3.6 Appendix : Exact solutions
3.6.1 Exact solutions of the 1D problem with piecewise constant speeds
3.6.2 Exact solution of the 2D problems with piecewise constant speeds
4 Adaptive mesh renement strategy for a non conservative transport problem
4.1 Introduction
4.2 Investigation of numerical noise appearance
4.2.1 Description of the simplied set-up
4.2.2 Finite volume discretization on a two-level MR hierarchy
4.2.3 Explication of the numerical noise appearance
4.2.5 Formalization and generalization of the doubling strategy
4.3 Numerical validation of the adaptive thresholds strategies
4.3.1 Eect of the adaptive doubling strategy on the solution
4.3.2 Time adaptation of the MR threshold « t
4.3.3 Link between the MR and the doubling thresholds
4.4 Application to a biological model
4.4.1 Model and numerical method
4.4.2 Numerical performances
4.4.3 Comparison of the numerical performances for both models of mitosis
4.5 Appendix : Description of the model for follicular development
4.6 Appendix : Example of spurious numerical noise appearance
4.7 Numerical noise appearance
5 Calibration
5.1 Degrés de liberté
5.1.1 Conditions initiales
5.1.2 Fonctions de vitesses
5.1.3 Variables de contrôle
5.2 Spécications biologiques
5.2.1 Mesures disponibles
5.2.2 Trajectoires folliculaires
5.3 Aides à la calibration
5.3.1 Contraintes a priori
5.3.2 Visualisation à une échelle microscopique
5.3.3 Diagnostic a posteriori
5.4 Méthode de calibration pour un follicule
5.4.1 Détermination de la maturité asymptotique
5.4.2 Détermination des paramètres aectant la dynamique
5.4.3 Exemple de situation de référence à 1 follicule
5.5 Méthode de calibration pour une cohorte de follicules
5.5.1 Réglage de la pression sélective
5.5.2 Abaques multi follicules
5.5.3 Exemple de situation de référence à 10 follicules
6 Design and performance analysis of a multiscale scheme for controlled population dynamics
6.1 Multiscale model
6.1.1 Example : selection process of ovarian follicles
6.2 Multiscale numerical method
6.2.1 Microscopic level : adaptive nite volume scheme
6.2.2 Mesoscopic and macroscopic scale : Numerical moments and control
6.2.3 Parallel computing
6.3 Performance analysis
6.3.1 Method
6.3.3 Mean weak scalability test
7 Conclusion
A CodeFollicleMR : Notice utilisateur (getting started)
A.1 Simulation avec paramètres par défaut
A.2 Visualisation
A.3 Simulation avec plusieurs follicules, en choisissant un jeu de paramètre204
A.4 Simulation en créant son propre jeu de paramètres
A.5 Interface Utilisateur
B dualMR : Notice utilisateur (getting started)
B.1 Comment utiliser dualMR
B.2 Visualiser les sorties macroscopiques et mésoscopiques
B.3 Visualiser les sorties microscopiques
C CodeFollicleMR : Notice programmeur
C.1 Structures de données
C.2 Organisation du code source
C.3 Entrées/sorties
Bibliographie

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