Non-Intrusive Reduced Basis Methods Applied to Outdoor Pollutant Transport Models
Résumé
The challenges of understanding the full impact of air pollution on human health require incorporating the knowledge and research of a multitude of scientific fields. In collaborative work between two disciplines, applied mathematics and epidemiology exposure assessment, we are working to develop numerical methods for advanced models of air pollution to better estimate individual exposure, enabling current epidemiologic studies to evaluate the association of these exposures to various health problems. This study aims to investigate the use of reduced basis methods [1] to diminish the cost of resolution of parameter-dependent PDEs present in air pollution models developed for California. Numerical modeling has met growing success over the last decades in CFD modeling, leading to resolution of larger and larger nonlinear problems. Computation times for these problems in 3D can take tens of hours, making reduction techniques particularly appealing. The idea of reduced basis methods is to compute a cheap and accurate approximation of the solution by using approximation spaces made of suitable sample of solutions of the parameterized PDE associated to the problem. One of the keys of this technique is the decomposition of the computational work into an offline and online stage. The reduced basis functions, as well as all expensive parameter-independent terms, are computed once during the offline stage and stored, whereas inexpensive parameter-dependent quantities are evaluated during the online stage, for each new value of the parameters. Parameter-independent matrices and vectors are built only once and saved during the offline stage. This part of the offline stage requires accessing and modifying the assembly routines of the corresponding computational code, leading to an intrusive procedure. In the case of air pollution modeling, many sophisticated models have been developed and are in use; modifying the computational code would be impractical and risky. In addition, the correct inputs to these models are often unknown. For this reason, we propose an alternative, less intrusive method using data assimilation for measured pollution concentrations, introduced in [2-3]. California has been the study area for multiple air quality measurement campaign and epidemiology studies examining the impact of air pollution on children's health for over a decade [4, 5]. As air pollution concentrations are known to be highly heterogeneous, sophisticated physically-based models are of great interest and enable to better estimate individual exposure. The goal of this work is to extend the application of reduced basis methods to air pollution modeling, using physically-based air quality models and pollutant concentration data provided by the epidemiology exposure assessment team at University of California-Berkeley to rapidly estimate the concentration of one or several pollutants around an area of interest at a fine urban scale. References: [1] Prud'homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday, Y., Patera, A. T., & Turinici, G. (2002). 'Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods'. Journal of Fluids Engineering, 124(1), 70-80
[2] R. Chakir, P. Joly, Y. Maday and P. Parnaudeau, 'A Non-intrusive reduced basis method: application to computational fluid dynamics', 2nd ECCOMAS Young Investigators Conference (YIC 2013), Sep 2013, Bordeaux, France, . [3} Y. Maday, A.T Patera, J.D. Penn and M. Yano, 'A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics', Int. J. Numer. Meth. Engng (2014). [4] Noth, E.M., Hammond, S.K., Biging, G.S., Tager, I.B., 'A spatial-temporal regression model to predict daily outdoor residential PAH concentrations in an epidemiologic study in Fresno, CA', Atmospheric Environment 45, 2394-2403 (2011). [5] Wilson, J.G., Kingham, S., Pearce, J., Sturman, A.P., 'A review of intraurban variations in particulate air pollution: Implications for epidemiological research', Atmospheric Environment 39, 6444-6462 (2005)