Research

Overview

My overall research goal is to develop novel, fast, and robust numerical methods to efficiently solve scientific problems. I find this field particularly fascinating because recent development in technology has provided computers with greater computational power which has increased the potential to implement large-scale problems with high-performance and even greater parallelism. Therefore, there is a need to develop novel numerical linear algebra and numerical analysis tools to fully exploit this advanced computing potential. I believe that, in the near future, a combination of mathematics and scientific computing will provide even more powerful means to explore and simulate large-scale scientific problems with applications in physics, chemistry, and biology.

Keywords: Numerical Analysis; Scientific Computing; Numerical Linear Algebra; Differential Equations; Preconditioning; Multigrid; Approximations; Modeling of Biological Systems; Mathematics Education.

Below, I briefly describe several projects that I have been working on with my collaborators: Relja Vulanovic, Niall Madden, Scott MacLachlan, Martin Stynes, Vinh Mai, and Zakia Hammouch. Please also visit my publication page for more details.

Numerical Analysis for Singularly Perturbed Differential Equations

The interest in singular perturbation differential equations is motivated by their numerous applications including Navier-Stokes equations with large Reynolds numbers and other convection-diffusion equations modeling water-pollution problems, oil-extraction processes, flows in chemical reactors, convective heat-transport problems with large Péclet numbers, and semiconductor devices. Finding numerical solutions to the singularly perturbed problems is a great challenge. The difficulties in applying classical numerical schemes stem from the fact that, typically, derivatives of the exact solution u of order p have magnitude O($\varepsilon^{-p}$), where p is a positive integer and $\varepsilon$ is a small positive perturbation parameter. Classical techniques do not have the property of being parameter robust (also known as “uniformly convergent” and “$\varepsilon$-uniform” in the literature) because they rely on certain derivatives being bounded, which is not the case as $\varepsilon \to 0$. That is, methods that work well when $\varepsilon = O(1)$, may fail to give meaningful solutions when $\varepsilon$ is small, unless one makes unreasonable assumptions such as the discretization parameter, N, being $O(\varepsilon^{-1})$. Our goal is to develop and analyze novel parameter-uniform computational algorithms to solve singularly perturbed differential equations.

Scientific Computing

Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the study of the numerical approximation of solutions to differential equations. Consequently, much is known about how to accurately and stably discretize such equations on a priori adapted meshes, in order to properly resolve the layer structure present in their continuum solutions (see the aforementioned project). However, despite being a key step in the numerical simulation process, much less is known about the efficient and accurate solution of the linear systems of equations corresponding to these discretizations. Our objective in this direction is to develop parameter-robust and efficient preconditioning strategies that are tuned to the matrix structure induced by using layer-adapted meshes for singularly perturbed differential equations. In some particular settings (e.g., for convection-diffusion problems [MacLachlan, Madden, Nhan, SIMAX Vol 43. (2), 561--583, 2022]), our numerical results show the efficiency of the resulting preconditioners, with time-to-solution of less than one second for representative problems on 1024x1024 meshes and up to 40x speedups over standard sparse direct solvers.

Modeling and Computational Biology

Enzymes are biological catalysts naturally present in living organisms, and they are capable of accelerating biochemical reactions in the metabolism process. Cells use many regulatory mechanisms to regulate the concentrations of cellular metabolites at physiological levels. Enzymatic inhibition is one of the key regulatory mechanisms naturally occurring in cellular metabolism, especially the enzymatic non-competitive inhibition by product. This inhibition process helps the cell regulate enzymatic activities. Our goal is to derive effective mathematical models describing the enzymatic competitive inhibition by product. Our models usually consist of a coupled system of nonlinear ordinary differential equations for the species of interest. Our analysis provides qualitative and quantitative insights to gain a better understanding of how the enzymes are regulated in cells.

Math Education and Pedagogy

Nowadays, there is a wide range of math courses that are typicallydesignated to serve either full-time/part-time working learners and/or returning students who want to regain their confidence in math and to seek basic mathematical skills for solving problems in their daily life. Therefore, the main challenge for instructors who teach this class is how to create class activities in order to keep the students motivated and engaged in the learning process after their long day at work, and even to keep them coming to the class regularly. Along with traditional whiteboard lectures, the instructor should also be willing to design other creative, interactive, and innovative activities to achieve a high attendance rate for a 16-week-long semester. Our goal in this pedagogical direction is to use a data-driven approach to efficiently and actively enrich learning outcomes in mathematics classrooms.