Méthodes des bases réduites pour la modélisation de la ville

Université Paris-Est, école doctorale Mathématiques et Sciences et Technologies de l'Information et de la Communication

• IFSTTAR : Frédéric Bourquin, Rachida Chakir

Reduced Basis Method, Model Order Reduction, Data Assimilation, Air Quality Modeling

The principal objective of this thesis is the development of low-cost numerical tools for spatial mapping of pollutant concentrations from field observations and advanced deterministic models. With increased pollutant emissions and exposure due to mass urbanization and development worldwide, air quality measurement campaigns and epidemiology studies of the association between air pollution and adverse health effects have become increasingly common. However, as air pollution concentrations are highly variable spatially and temporally, the sensitivity and accuracy of these epidemiology studies is often deteriorated by exposure misclassification due to poor estimates of individual exposures. Data assimilation methods incorporate available measurement data and mathematical model to provide improved approximations of the concentration. These methods, when based on an advanced deterministic air quality models (AQMs), could provide spatially-rich small-scale approximations and can enable better estimates of effects and exposures. However, these methods can be computationally expensive. They require repeated solution of the model, which could it self be costly. In this work we investigate a combined reduced basis (RB) data assimilation method for use with advanced AQMs on urban scales. We want to diminish the cost of resolution, using RB arguments, and incorporate measurement data to improve the quality of the solution. We extend the Parameterized-Background Data-Weak (PBDW) method to physically based AQMs. This method can rapidly estimate “online” pollutant concentrations at urban scale, using available AQMs in a non-intrusive and computationally efficient manner, reducing computation times by factors up to hundreds. We apply this method in case studies representing urban residential pollution ofPM2.5, and we study the stability of the method depending on the placement of air quality sensors. Results from the PBDW are compared to the Generalized Empirical Interpolation Method (GEIM) and a standard inverse problem, the adjoint method, in order to measure efficiency of the method. This comparison shows possible improvement in precision and great improvement in computation cost with respect to classical methods. We find that the PBDW method shows promise for the real-time reconstruction of a pollution field in large-scale problems, pro-viding state estimation with approximation error generally under10%when applied to an imperfect model.

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Our study [1] covered different scenarios of pollution in an idealized urban geometry representing a residential area. We studied the stability and accuracy of the PBDW method for varying sensor locations and norms. We began with a 2D computational domain of dimensions 75m x 120m, with traffic pollution sources. The pollutant is transported by a decoupled turbulent velocity field, computed using the CFD software Code\_Saturne [2]. Two Update spaces were built, one using sensors placed randomly, and another where sensor placement was selected by the double-greedy algorithm from a grid of proposed locations. The stability of the PBDW system is evaluated and the PBDW method was compared to the GEIM interpolation method and to an adjoint-type method.

The robustness of these methodological adaptations to the PBDW method was studied in an extension towards a real-world scenario over Fresno, CA [3]. The objective is to show the feasibility of these methods in real applications. We considered an urban domain of dimensions 800m x 800m in a residential neighborhood (figure 1.a), and a velocity field corresponding to real conditions on 2001/4/1 (figure 1.b). This velocity field propagated the pollution from two large streets for this simple test case using a dimensionless advection-diffusion model.

The adimensionalization procedure and the selection of the norm are important in the case of a large computational domain over an urban neighborhood. The PBDW method was examined for an imperfect model with simulated observation data from a shifted model (advection-diffusion-reaction) on concentration fields after a preliminary study on sensor placement.

[1] J.K. Hammond, et al. ``PBDW: A Non-Intrusive Reduced Basis Data Assimilation Method and Its Application to an Urban Dispersion Modeling Framework.'' Applied Mathematical Modelling, vol. 76, Dec. 2019, pp. 1-25. Crossref, doi:10.1016/j.apm.2019.05.012.

[2] F. Archambeau, et al, Code Saturne: a Finite Volume Code for the Computation of Turbulent Incompressible flows Int, J. Finite Volumes.

[3] J.K. Hammond, R. chakir, « A non-intrusive reduced order data assimilation method applied to the monitoring of urban flow, CSMA 2019- 14 ème Colloque National en Calcul des Structures, May 2019, Giens, France (hal-02186298)