Research
My research lies in the broad area of applied and computational mathematics, with a focus on partial differential equation(PDE)-constrained optimization, inverse problems and numerical analysis of PDEs. In my research I am passionate about understanding the interplay of data and PDE models. I use efficient tools in data science for specific problems in physics and engineering with a focus on kinetic and wave type equations. In this regard, I have two main lines of research.
1. Given sufficient data, how can we improve PDE models?
I am deeply interested in developing mathematical strategies for estimating unknown or partially known parameters in PDE models. Specifically, we investigate the reconstruction of heat conductance coefficients using the phonon transport equation, which is a kinetic equation. We also study the associated diffusion limit of the phonon transport equation.
Reconstructing the thermal phonon transmission coefficient at solid interfaces in the phonon transport equation (SIAM J. Appl. Math., 82(1), 194–220, Jan 2022) (arXiv)
with Irene Gamba and Qin LiSecond-order diffusion limit for the phonon transport equation-asymptotics and numerics (Partial Differ. Equ. Appl. 3, 38 (2022)) (arXiv)
with Qin Li and Weiran Sun
2. How do PDEs help in tuning data?
I am also interested in understanding how PDEs aid in improvising data collection. Specifically, we use the paraxial wave equation (PWE) to find optical beams that have a certain desired property after propagating through atmospheric or oceanic turbulence.
Estimating the time-evolving refractivity of a turbulent medium using optical beam measurements: a data assimilation approach (submitted) (arXiv)
with Qin Li and Samuel StechmannScintillation minimization vs intensity maximization in optimal beams (Optics Letters 48 (15), 3865-3868 (2023)) (arXiv)
with Qin Li and Samuel StechmannComputation of optimal beams in weak turbulence (Opt. Continuum 1, 1867-1887 (2022)) (arXiv)
with Qin Li and Samuel StechmannOptimal beams and minimizing scintillation (Propagation Through and Characterization of Atmospheric and Oceanic Phenomena(2022)) (Proceedings)
with Qin Li and Samuel Stechmann
other academic projects
Compressing a high fidelity PDE solution and regularizing with a coarse solver (Graduate internship)
Mentor: Prof. Cory Hauck, Computer Science and Mathematics Division, Oak Ridge National LaboratoryModelling of Electron-Solid Interaction for Electron Probe Micro-Analysis (Undergraduate internship)
Mentor: Prof. Manuel Torrilhon, Center for Computational Engineering Sciences, RWTH Aachen, Germany