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Below are the problems to be presented at the 2023 Graduate Student Mathematical Modeling Camp.
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Hangjie Ji, North Carolina State University
Solvent evaporation from the liquid-air interface naturally occurs in porous media saturated with particle-laden solutions. Such drying dynamics play a crucial role in environmental processes and engineering applications such as water resource management and industrial drying processes. For example, after a membrane filter (a thin layer of porous media) is employed to remove undesired particles from a fluid, one needs to dry the filter to reuse it. During the drying process, some particles get deposited on the internal structure of the porous media. Meanwhile, the evaporation rate from the media is significantly influenced by solvent properties, thermal effects, particle deposition, and media porosity. Understanding the interplay between the moving liquid/air interface and particle deposition is crucial in predicting the media structure after the filter is dried. Furthermore, this problem is multiscale in nature, requiring a pore-scale microscopic model to describe the evaporation from individual pores and a continuum model to account for the macroscopic evolution of membrane internal morphology during the drying process.
This project will start with an idealized pore-scale model to describe the drying process in porous media. The developed model should account for factors such as evaporation, particle deposition, and the pore geometry. Based on this model, we will explore the effects of pore geometry, temperature field, and initial particle concentration on the drying dynamics. The students will also be encouraged to extend their findings to a multiple-pore structure or a macroscopic continuum model. This session will prepare participants to better understand the context of filtration and drying in porous media and gain familiarity with modeling techniques (nondimensionalization of models and asymptotic analysis), basic ODE/PDE analysis, and numerical simulation.
Read the final report
Peter Kramer, Rensselaer Polytechnic Institute
Predictions from structured dynamical models can be combined with emerging data in a process broadly referred to as ``filtering,'' when the reliabilities of each can be at least roughly characterized. As a guiding example, we can think about models of disease spread (say of the SIR or SEIR type) which are used alongside observed case counts as the disease outbreak is occurring. Discrepancies between the model and data can be due to missing or inadequately represented features of the model and/or errors in the observation process, and the filtering process attempts to reconcile these inconsistencies. Kalman filtering gives an efficient and rational solution when the dynamics are linear and both model and observational error can be treated as Gaussian noise uncorrelated in time. In more general settings, filtering approaches can be formulated mathematically in terms of Bayesian inference, and the question becomes how to implement these Bayesian updates in a computationally efficient manner. Particle filters, which here refer to a computational statistical methodology rather than a porous medium, are a popular class of approaches, in which the uncertainty in the dynamical state is characterized by a distribution of particles representing different possible states with certain probabilistic weights. These particles are to be given rules for how to update their state values and weights as new observations become available. These choices need to both be consistent with the statistical representation of uncertainty and with concerns of computational fidelity and cost.
The aim of this working group will be to gain familiarity with some particle filtering approaches by computational experiments on various kinetic models including models of disease spread. Students in this session will also be encouraged to apply the particle filters to dynamical models of their own interest. One outcome will be to see how standard particle filtering approaches that should work in principle can run into practical computational problems. This session will thereby prepare participants to better understand the background and context for particle flow.
Manuchehr Aminian, Cal Poly Pomona
On December 30, 2021, there was a devastating in the greater Boulder area in Colorado, as the result of a fire carried by intense (70-100mph) winds propagating what would normally be a localized grass fire, jumping into urban areas and destroying over one thousand homes and various other property. This was surprising to many - and was understood to be the result of a unique combination of weather and climate circumstances; see the following for a longer explanation:
High Winds and Marshall Fire on December 30th, 2021
In the wake of this event, our (fake) insurance company wants to re-evaluate overall fire risk for a variety of geographic regions which accounts for potential events as the result of similar weather patterns to the Marshall fire, through some combination of mathematical modeling and data analysis/data-driven methods.
Broadly, we have two main tasks for participants:
1. Identify existing mathematical models which are "expressive" enough to both capture the weather effects seen during the Marshall Fire, and also capture "nominal" weather. If they do not exist, construct one. Ideally, the relative frequency of these "intense" events can reflect reality to some degree. While we prefer a smaller, more useful model, an existing kitchen-sink, black-box weather simulation code which participants can successfully explain is acceptable.
2. A ranking, scoring, or flagging of a geographic regions throughout the US based on their relative risk for events such as the Marshall Fire. Any approach here is valid, as long as it is well-explained. Aside from "plugging in" the model in part 1, purely data-driven methods (e.g. data analysis; feature extraction/selection, outlier detection, anomaly detection, etc) are also welcome, but (again) explainability is important.
Read the final report