Adaptive mesh refinement strategy for a non conservative transport problem

Aymard, Benjamin; Clément, Frédérique; Postel, Marie

doi:10.1051/m2an/2014014

Cited by 2 publications

(4 citation statements)

References 15 publications

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“…On the one hand, existence and uniqueness of the solution in our case is proven for bounded initial conditions, and hence as well the existence of zeroth and first order moments [30]. On the other hand, the same asymptotic behavior as in the smooth velocity case [26], tending towards a monokinetic distribution, is observed in numerical simulations [2,4]. Nevertheless, we will show that if we relax the hypotheses made in [26] on the velocity field, the reduced model can have only measure solutions in some cases, and is therefore of little use for practical and numerical purposes, while in other cases the monokinetic behavior [26] is preserved.…”

Section: Introductionsupporting

confidence: 77%

“…We know from Theorem 3.1 that any solution (ζ,ρ) defines a particular solution of the 2D model using the mono kinetic ansatz (2.22). Furthermore, Theorem 3.3 ensures that for long time, the support of the 2D solution should remain bounded in the y direction by the solutions ζ i (t, x), corresponding to some s = ζ 0 i , i = 0, 1 which can be deduced from the values of t s and ζ s i for i = 0, 1 defined in the proof of Theorem 3.3 4 . In this section we illustrate that in practice the comparison between the 2D and 1D model is quite robust quantitatively and satisfies the following properties …”

Section: Numerical Comparison Of the Reduced 1d Model With The 2d Modelmentioning

confidence: 97%

“…The consequences of the velocity discontinuities on the well-posedness of the model have been tackled in [30], where the existence and uniqueness of the solution is obtained, while a specific numerical treatment has been designed in [2] and [4]. Several strategies have already been explored to reduce the computational costs (parallel computing [1], adaptive mesh refinement [4]). …”

Section: Multiscale 2d Modelmentioning

confidence: 99%

“…Postponing the theoretical study to future work, we check that for realistic parameter values, the zeroth and the first order moments are numerically comparable. We refer the reader to [2,4] for the description and validation of the 2D numerical method used to perform the simulations. The density for both set-ups is computed using a second order Finite Volume scheme on multiresolution-driven adaptive meshes with the parameter values provided in Table 5.1.…”

Section: Numerical Simulations In a Non Affine Casementioning

confidence: 99%

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Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

Clément¹,

Coquel²,

Postel³

et al. 2017

Multiscale Model. Simul.

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Abstract. We consider a class of kinetic models for which a moment equation has a natural interpretation. We show that, depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We detail the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained by computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. For arbitrary initial conditions, we compare 1D and 2D model solutions in asymptotically large time. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy. The numerical robustness is studied in the scalar case, and a full scale vector case is presented.

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Section: Introductionsupporting

confidence: 77%

Section: Numerical Comparison Of the Reduced 1d Model With The 2d Modelmentioning

confidence: 97%

Section: Multiscale 2d Modelmentioning

confidence: 99%

Section: Numerical Simulations In a Non Affine Casementioning

confidence: 99%

See 2 more Smart Citations

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

Clément¹,

Coquel²,

Postel³

et al. 2017

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

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Cell-Kinetics Based Calibration of a Multiscale Model of Structured Cell Populations in Ovarian Follicles

Aymard¹,

Cleźment²,

Monniaux³

et al. 2016

SIAM J. Appl. Math.

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Abstract. In this paper, we present a strategy for tuning the parameters of a multiscale model of structured cell populations in which physiological mechanisms are embedded into the cell scale. This strategy allows one to cope with the technical difficulties raised by such models, that arise from their anchorage in cell biology concepts: localized mitosis, progression within and out of the cell cycle driven by time-and possibly unknown-dependent, and nonsmooth velocity coefficients. We compute different mesoscopic and macroscopic quantities from the microscopic unknowns (cell densities) and relate them to experimental cell kinetic indexes. We study the expression of reaching times corresponding to characteristic cellular transitions in a particle-like reduction of the original model. We make use of this framework to obtain an appropriate initial guess for the parameters and then perform a sequence of optimization steps subject to quantitative specifications. We finally illustrate realistic simulations of the cell populations in cohorts of interacting ovarian follicles.Key words. transport equations, parameter calibration, structured cell populations, cell kinetics AMS subject classifications. 92B05,35Q92Introduction. In this paper, we deal with the question of the numerical calibration of an existing multiscale model of cell-structured populations in the physiological context of ovulation. This model was formulated as a system of weakly coupled, non conservative transport equations with controlled velocities and sink terms, where the unknowns are the cell densities in each follicle [9, 8]. A number of theoretical studies have established the well-posedness of the model [19], examined optimal control problems related to the ovulatory trajectories in the framework of hybrid optimal control theory [6], and studied the reachability of final states corresponding to either ovulatory or atretic cases in the framework of backwards reachable sets [8]. Implementation of the model in an efficient and reliable computing environment has involved the design of a finite-volume scheme dealing with the discontinuous coefficients [3], embedding this scheme within a dedicated adaptive mesh based on a multi-resolution approach [4], and implementing it on parallel architecture [2]. This has left the question of model calibration to biological specifications to be resolved. We have to face a generic, yet unsolved issue in parameter fitting for physiologicallyoriented multiscale mathematical models: although mechanistic knowledge in molecular and cell biology is available on the lower scales, quantitative experimental data are rather available on the higher scales. In our case, the question is how to infer the parameters entering the microscopic functions (on the level of the follicular cells) from mesoscopic (on the level of the individual follicles, i.e. the number of follicular cells) or macroscopic (on the level of the populations of follicles) information. In addition, even on the macroscopic level, data remain rather scarce and ...

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Adaptive mesh refinement strategy for a non conservative transport problem

Cited by 2 publications

References 15 publications

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

Cell-Kinetics Based Calibration of a Multiscale Model of Structured Cell Populations in Ovarian Follicles

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