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Numerical computations for electromagnetic scattering
This project covers the acoustic and electromagnetic equations, especially with regards to scattering problems. Scattering problems involving layered media arise in many engineering applications in electromagnetics, optics, and acoustics. Over the years, robust and accurate simulation capability has received increased attention as a cost effective tool for predictive measurement and analysis of modern physical systems. We study quasi-periodic solutions of the scalar Helmholtz equation in two or three dimensions in the context of layered media scattering problems. Multiple layered media with periodic surface interfaces is considered. A high-order spectral element method is presented for solving layered media scattering problems featuring an operator that transparently enforces the far-field boundary condition.
Deep Neural Networks: Safe AI, Adversarial Examples, and Uncertainty
Machine learning models (or deep neural networks) have been used in a variety of applications including autonomous robots, vehicles, and drones. When deploying AI systems to the physical world, the reliability of algorithms is crucial for safety. Guaranteeing such safety includes specification, robustness, and assurance. Given a concrete purpose of the system (specification), the AI system should be robust to perturbations and attacks (adversarial examples). Further, the uncertainty of predictions by models helps monitor and control the AI system's activity. In this line of thought, we study uncertainty of models (e.g., Bayesian Neural Networks) and adversarial examples from both attacker and defender perspectives. This topic may fall in the intersection of AI and security.
Numerical computations for the humid atmosphere
The inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation are considered in the presence of topography. Because of its high accuracy, the hydrostatic equation is well accepted as a fundamental equation of the atmosphere, and is considered as a starting point for studying the extremely complicated atmospheric phenomena, and for predicting the weather and possible climate changes. The main focus has been to develop effective simulations such as numerical weather predictions using primitive equations of the atmosphere with saturation, which introduces a change of phase from the physical point of view. One purpose of numerical simulations is to study the rain shadow effect; a region in the leeward of mountains that receives less rainfall than the region windward of the mountains (see video below).
Numerical study of the second-order water wave
Attention is given to the study of the propagation of long-crested wave motions with an incompressible perfect fluid in a uniform horizontal channel. When the fluid motion is irrotational, inviscid and uniform in the cross channel direction, the two-dimensional version of the Euler equations provide a good model of waves on the surface of water. However, in many practical applications, full water wave model appears to be more complex than is necessary, and consequently further approximations are often made in the shallow water regime. In this project, we numerically study the unidirectional fifth-order (second-order correct) KdV-BBM, and examine behavior of solutions arising from long-crested waves propagation. The video below depicts the fission processes of the KdV-BBM equation and second-order correct equations; for more details, see  of my publication list.
Numerical method for singularly perturbed problems
Attention is given to the singularly perturbed boundary value problems with small parameter, which are suitable linearizations of the Navier-Stokes or Boussinesq equations. The numerical computations of the convection-dominated equations raise substantial difficulties since, without care, the computed solutions are highly oscillatory when the diffusive term is small because the singularity at the boundary propagates inside the domain along the characteristics. The computational difficulties can be overcome with the use of a correction function usually called "corrector", which resolves the discrepancy between the zero boundary conditions and those of the outer solutions.