Reduced order models for large scale simulations of urban air pollution
Urban air pollution emerges as a major global challenge nowadays because of its negative consequences on ecosystems, health, and climate change. In particular, urban traffic appears to be responsible for the dispersion of pollutants such as benzene particulate matter and carbon monoxide. The situation is aggravated by projections of future urbanization that will result in a substantial expansion of urban areas, which will also increase pollution.
The problem proves to affect communities of people living in areas of higher urbanization. Suffice it to say that research conducted by the World Health Organization estimated that 9 out of 10 people breathe polluted air [1].
The implications on people’s health are varied, the most important of which include a higher rate of incidence of childhood asthma and coronary and carotid arteriosclerosis [2].
These consequences translate into an estimated life cost of 1.8 million deaths in 2019, and an economic cost of $5 trillion annually [3]. Given the importance of this problem, air quality monitoring and management appear as one of the United Nations Sustainable Growth Goals.
The importance of a monitoring framework
Research on the epidemiological effects of traffic has shown that adverse effects due to exposure to traffic pollutants on human health are not uniformly distributed in urban areas, but are correlated with distance from sources and volume of emissions.
The above comment implies the importance of monitoring the concentration of pollutants for the development of policies to mitigate the effects of pollution itself.
In particular, combining traditional measurement tools with numerical modeling techniques allows us to achieve an important analytical framework capable of extracting insights of various kinds from the collected statistics, such as the relationship between urban traffic and the effectiveness of prevention strategies.
This concept is well explicated by a recent European Commission directive requiring air quality to be measured by combining the use of monitoring stations with appropriate mathematical modeling tools.
The necessity for Reduced Order Models
Monitoring urban air pollution requires studying the evolution of the concentration of pollutants in the atmosphere.
The mathematical model is that of atmospheric dispersion, which is a system of partial differential equations (PDEs) known as advection-diffusion-reaction, and coupled through a term modeling chemical production [4].
The transport-diffusion equation is a linear partial differential equation, which takes the form:
In the previous equation, we are interested in finding the unknown function c(x,t), which represents the concentration of the pollutant (which can be for example NO2) at the spatial point x at time t.
Each term in the equation has a different meaning.
Specifically, the term “f” on the right hand side is used as a proxy for the source emission, meaning how much pollutant is released into the atmosphere, in the unit of time.
The term “-ν∆c” models the change in concentration due to the diffusion effect, that is, when a substance moves due to its different concentration in different areas; this phenomenon is well observed by throwing some ink in a glass of water.
The term “∇ · (uc)” models the convective transport effect, that is, the transport of the pollutant due to the motion of the fluid in which it is immersed. In fact, in order to obtain such a field, it is necessary to use computational fluid dynamics (CFD) techniques, which in the case of the urban scale require an excessive computational cost. It is therefore necessary to use high-performance computing facilities to obtain results within a reasonable time [5].
In particular, the problem is parametric, that is, it is to be solved for different values of certain quantities that describe the problem itself. For example, it is our intention to be able to find a solution to the problem by varying the source term “f” and the velocity field “u” which in turn depends on the value taken on the boundary of the area we are interested in modeling.
However, if we were to repeatedly solve the equations for different parameter values, as in our case, do we have to recalculate the solution each time from scratch?
The answer is no if we employ the Reduced Basis method [6], which involves processing information from a set of already known solutions to obtain a compressed representation to be used for the new instances of the problem.
Test case: the campus of Bologna university
We addressed the problem described above by creating a framework capable of solving the pollutant transport problem in real-time, parameterized with respect to pollutant emission and wind distribution at the domain boundary.
The model was obtained through an interdisciplinary collaboration between the mathLab group of SISSA International School for Advanced Studies (Italy) and Széchenyi István University of Gyor (Hungary) [7].
We are considering two test cases on different scales. A small mesh that models the university campus of Bologna, and a bigger mesh for the entire city of Gyor for which we dispose of real-world traffic counts obtained by a network of cameras and loop detectors.
To propose a reduced-order model, we used a well-known technique in the field of reduced-order modeling, known as Proper Orthogonal Decomposition [8], which can be used to compress information in different settings by extracting a low-dimensional representation of a given data set. In particular, we created an innovative framework in which our model is trained on a subset of the time series and then used for testing on future state prediction of the source term.
Another novelty in our model lies in the use of a neural network for velocity reconstruction from boundary conditions. This network was trained using empirical data prescribing boundary conditions (see Figure below) for an entire calendar year.
The result of our work is a model that can quickly provide the solution for any distribution of emission and weather conditions. For example, the results shown in the figure below represent the simulation for a given day and at a given time. It is observed that the ROM solution (commonly referred to as online) is virtually indistinguishable from that which would have been obtained by means of traditional numerical methods (referred to as offline). Moreover, to obtain the ROM solution the time taken is significantly less, in fact, our method is 1500 times faster than its classical counterpart.
References
[1] T. N. Sophie Gumy, K. Dushaj, and A. Pruss-Ustun. WHO Global Urban Ambient Air Pollution Database (update 2016). Department of Public Health, Environmental and Social Determinant of Health World Health Organization, 2016.
[2] C. A. Pope Iii, R. T. Burnett, M. J. Thun, E. E. Calle, D. Krewski, K. Ito, and G. D. Thurston. Lung cancer, cardiopulmonary mortality, and long-term exposure to fine particulate air pollution. Jama, 2002.
[3] V. A. Southerland, M. Brauer, A. Mohegh, M. S. Hammer, A. van Donkelaar, R. V. Martin, J. S. Apte, and S. C. Anenberg. Global urban temporal trends in fine particulate matter (pm2·5) and attributable health burdens: estimates from global datasets. The Lancet Planetary Health, 2022.
[4] J. Seinfeld and S. Pandis. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. Wiley, 2016.
[5] L. Kornyei, Z. Horvath, A. Ruopp, A. Kovacs, and B. Liszkai. Multi-scale modelling of urban air pollution with coupled weather forecast and traffic simulation on hpc architecture. Association for Computing Machinery, 2021.
[6] J. S. Hesthaven, G. Rozza, and B. Stamm. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer International Publishing, 2015.
[8] A. Chatterjee. An introduction to the proper orthogonal decomposition. Current Science, 2000.
