Research

1. Data Assimilation

I have been working on developing continuous data assimilation algorithms. By extending the existing Azouani–Olson–Titi (AOT) algorithm through the introduction of dynamic feedback control, we can control the convergence rate, which in turn helps estimate the amount of observational data needed in time to successfully recover the initial conditions for a numerical forecast model.

2D shallow water model. Solution profiles over the time period [0, 3]. Approximating solutions (lower row: h = water depth, p = x-momentum) start from a constant state and recover most of the reference solutions (upper row) using sparse observed data (red dots) as time increases, capturing shocks and fine-scale details.

More recently, we have developed an Interpolated Discrepancy Data Assimilation (IDDA) method for PDEs with sparse observations, which improves on the standard AOT algorithm by constructing a smooth interpolant of the discrepancy between the observed and simulated states before applying feedback control.

2. Adaptive Moving Mesh Methods

One major difficulty in developing numerical methods for hyperbolic systems is that these systems admit non-smooth solutions such as shocks and contact discontinuities. To achieve high resolution while improving computational efficiency, I have developed new adaptive moving mesh (AMM) central-upwind schemes for hyperbolic systems.

Shallow water equations. AMM central-upwind scheme for the Saint-Venant system. The scheme is well-balanced and positivity preserving; discontinuities in the solution are sharply captured by the adaptive mesh.

Two-species chemotaxis system. A 2-D AMM upwind scheme for the Patlak–Keller–Segel chemotaxis system with two non-competing species of different chemotactic sensitivity. The solution develops multi-spiky structures that are resolved with high resolution by the AMM scheme.

The AMM framework has been applied to Saint-Venant shallow water systems, two-species chemotaxis models, and general hyperbolic systems of conservation laws. The adaptive redistribution of mesh points concentrates resolution near solution features, significantly reducing computational cost compared to fixed uniform grids.

3. Shallow Water Equations

Shallow water models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. Solving the shallow water equations numerically is a challenging task for several reasons, including the preservation of physically relevant steady-state solutions (well-balancing) and the positivity preservation of water depth.

Shallow Water Equations with Friction. Beyond the standard numerical difficulties, the bottom friction term becomes extremely stiff when water depth is very small. I introduced a semi-implicit-explicit version of the central-upwind scheme that is second-order accurate, well-balanced, and positivity preserving. The stiff friction term is treated semi-implicitly, so no extra time-step restriction is imposed.

Laboratory experiment of Cea, Garrido, and Puertas (2010) modelling rainfall in an urban area. The designed scheme achieves remarkable agreement with experimental results.

Additional contributions in this area include well-balanced central-upwind schemes for shallow water equations with wet/dry fronts via flux globalization, operator-splitting schemes for moving bottom topography, moving-water equilibria preserving schemes, and a three-layer approximation of two-layer shallow water systems.

4. Network-Based Epidemiology

Example bipartite network used to model heterosexual transmission of sexually transmitted infections.

Many natural and social dynamics, including the spread of infectious diseases, can be described using networks. In such models, each node represents an individual and each edge represents a possible transmission route. For sexually transmitted diseases such as Chlamydia, which are transmitted only through heterosexual partnerships, bipartite network models are particularly appropriate.

When generating synthetic networks for disease modeling, it is essential to preserve the relevant structure of real-world contact networks, including the degree distribution and the joint-degree distribution. I designed a new algorithm to construct bipartite networks that exactly match both a prescribed degree distribution and a prescribed joint-degree distribution; the resulting networks have been used to model transmissions of sexually transmitted diseases.

More recent work extends to spatial models for Wolbachia invasion dynamics. Wolbachia is a bacterial endosymbiont that, when introduced into mosquito populations, can suppress the transmission of dengue, Zika, and chikungunya. I have developed reaction-diffusion PDE models and multistage spatial models that describe Wolbachia spread and inform optimal release strategies for mosquito-borne disease control.