We propose a non-intrusive reduced-order modeling method based on the notion of space-timeparameter proper orthogonal decomposition for approximating the solution of non-linear parametrized time-dependent partial differential equations. A two-level proper orthogonal decomposition method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced-order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced-order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection-reaction-diffusion problem. We demonstrate that our approach leads to reduced-order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time.

SUMMARYWe aim to evaluate environmental and genetic effects on the expansion/proliferation of committed single cells during embryonic development, using melanoblasts as a paradigm to model this phenomenon. Melanoblasts are a specific type of cell that display extensive cellular proliferation during development. However, the events controlling melanoblast expansion are still poorly understood due to insufficient knowledge concerning their number and distribution in the various skin compartments. We show that melanoblast expansion is tightly controlled both spatially and temporally, with little variation between embryos. We established a mathematical model reflecting the main cellular mechanisms involved in melanoblast expansion, including proliferation and migration from the dermis to epidermis. In association with biological information, the model allows the calculation of doubling times for melanoblasts, revealing that dermal and epidermal melanoblasts have short but different doubling times. Moreover, the number of trunk founder melanoblasts at E8.5 was estimated to be 16, a population impossible to count by classical biological approaches. We also assessed the importance of the genetic background by studying gain-and lossof-function b-catenin mutants in the melanocyte lineage. We found that any alteration of b-catenin activity, whether positive or negative, reduced both dermal and epidermal melanoblast proliferation. Finally, we determined that the pool of dermal melanoblasts remains constant in wild-type and mutant embryos during development, implying that specific control mechanisms associated with cell division ensure half of the cells at each cell division to migrate from the dermis to the epidermis. Modeling melanoblast expansion revealed novel links between cell division, cell localization within the embryo and appropriate feedback control through b-catenin.

SUMMARYThis paper presents a methodology for constructing low-order surrogate models of finite element/finite volume discrete solutions of parameterized steady-state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high-dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high-dimensional parameter spaces. For numerical experiments and validation, several non-linear steady-state convection-diffusion-reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two-dimensional spatial case with two parameters, it is shown that a 7×7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13×6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design.

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134et al. (4) and Hall and Barrow (5), analyses about the effects of adverse weather on traffic indicators have seen an expansion over the last years. For example, Rakha et al. (6) report a maximum reduction in the range of 6% to 9% in free-flow speed and 8% to 14% in speed-at-capacity if the rain intensity is 1.6 cm/h. In a previous study (7 ), we noticed an average decrease of 15.5% of the capacity during rainy conditions and a drop of 9% in free-flow speed. This trend was confirmed by Cools et al. (8) and Unrau and Andrey (9). In spite of these results, the impact of rain on traffic still needs to be addressed: there is no consensus on the main findings. The main reasons for that are twofold: (i) a lack of standardized methodology dealing with the quantification of the rain effects and (ii) a lack of comprehensive data, which often prevents separating the study according to the intensity of rainfall.The originality of the proposed study resides in a multilevel approach: from individual data provided by double loop sensors, a microscopic analysis is carried out, enabling observation of individual drivers' behavior under adverse weather conditions. Next, the same data could lead to a mesoscopic study, that is, an observation of the rain influence in terms of platoons. Finally, the study can be extended to a macroscopic point of view with the assessment of the rain impact on the fundamental diagram through the use of traffic models. Indeed, the changes in the relationships between speed, flow, and density must capture the weather effects. Such results could lead to a parameterization of the fundamental diagram according to the intensity of rain. In this paper, a systematic methodology is proposed. Next, this methodology is implemented through an empirical analysis on a French interurban motorway. The promising results could enable integration of the new findings about the rain effects into a decision support system, allowing road managers to deal online with both traffic and weather data. Figure 1 describes a systematic methodology that enables an analysis of the rain impact on traffic by tackling the problem at three different levels: micro-, meso-, and macroscopic. Such an approach requires the use of individual traffic data collected by loop detectors. The following information is recorded for each vehicle: date, hour, METHODOLOGY Multilevel Assessment of the Impact of Rain on Drivers' Behavior Standardized Methodology and Empirical AnalysisRomain Billot, Nour-Eddin El Faouzi, and Florian De Vuyst For all road managers, inclement weather events are a source of uncertainty that can affect traffic operations and safety. Regarding safety, various studies reveal significant effects of adverse weather conditions on the frequency and severity of crashes. Regarding mobility, because of a lack of data, there are few comprehensive studies, although the quantification of the effects of adverse weather on traffic represents the first step toward the development of weather-responsive traffic management strategies. Th...

In this paper, we propose a symbolic control synthesis method for nonlinear sampled switched systems whose vector fields are one-sided Lipschitz. The main idea is to use an approximate model obtained from the forward Euler method to build a guaranteed control. The benefit of this method is that the error introduced by symbolic modeling is bounded by choosing suitable time and space discretizations. The method is implemented in the interpreted language Octave. Several examples of the literature are performed and the results are compared with results obtained with a previous method based on the Runge-Kutta integration method.

In 2002, Després and Lagoutière [Després and Lagoutière (2002)] proposed a low-diffusive advection scheme for pure transport equation problems, which is particularly accurate for step-shaped solutions, and thus suited for interface tracking procedure by a color function. This has been extended by Kokh and Lagoutière [Kokh and Lagoutière (2010)] in the context of compressible multifluid flows using a five-equation model. In this paper, we explore a simplified variant approach for gas-liquid three-equation models. The Eulerian numerical scheme has two ingredients: a robust remapped Lagrange solver for the solution of the volume-averaged equations, and a low diffusive compressive scheme for the advection of the gas mass fraction. Numerical experiments show the performance of the computational approach on various flow reference problems: dam break, sloshing of a tank filled with water, water-water impact and finally a case of Rayleigh-Taylor instability. One of the advantage of the present interface capturing solver is its natural implementation on parallel processors or computers.

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