Research Interests

My research areas are applied mathematics and numerical analysis. More specifically, I am interested in

  • Discontinuous Galerkin Finite Element Methods
  • Variational Inequalities
  • Numerical Solutions of Stochastic Partial Differential Equations
  • Adaptive Numerical Methods for Partial Differential Equations
  • PDE-Constrained Optimization
  • Numerical Optimization

Grants

  • National Science Foundation DMS-2111004, PI, Novel Discontinuous Galerkin Methods for Deterministic and Stochastic Optimization Problems with Inequality Constraints, 09/2021-08/2024
  • CURM (The Center for Undergraduate Research in Mathematics) Minigrants, PI, Studying material transport in the vortical flow using Direct Numerical Simulations: Lagrangian and Eulerian approaches, 09/2021-05/2022
  • Simons Collaboration Grant, PI, 2021-2026 (withdraw in favor of the NSF grant)
  • New Faculty Grant, Office of Research and Engagement, UNC Greensboro, PI, 01/2018-06/2019

Research Projects

  • Variational Inequalities

    • Variational inequalities arise from many areas in science, engineering and finance that involve differential equations and optimization. They are fundamental tools for the modeling of applied problems that involve inequality constraints. My main focus in this area is to design, analyze and implement numerical algorithms for the displacement obstacle problem of Kirchhoff plates, which is an important class of variational inequalities. We developed a unified framework for the numerical analysis of various finite element methods including conforming, nonconforming, and discontinuous Galerkin methods.

    Quadratic C^0 interior penalty method on an L-shaped domain.
    Left: contact set. Right: adaptive mesh refinement
    Cubic C^0 interior penalty method on an L-shaped domain.
    Left: convergence history on uniform and adaptive meshes. Right: adaptive mesh refinement
  • PDE-Constrained Optimization and Optimal Control

    • The fields of optimization and optimal control subject to PDE-constraints, arising in a huge variety of industrial and technological applications, has received a significant attention in recent years. It is crucial to develop efficient and robust numerical algorithms for a comprehensive study of these problems. In this area, my focus is on finite element methods for an elliptic optimal control problem with pointwise state constraints. We studied quadratic and cubic C^0 interior penalty methods on two and three dimensions under different boundary conditions, by reformulating the original problem as a fourth order variational inequality. We proved the convergence of numerical solutions with rates in the H^2-like energy error and also introduced an adaptive algorithm.

    Discrete solutions (quadratic element) on uniform meshes
    Left: discrete optimal state. Right: discrete optimal control
    Left: discrete contact set for the state contraints. Right: adaptive mesh
  • Stochastic PDEs

    • Stochastic PDEs play a prominent role in understanding practical physical phenomenon since the physical environment is seldom deterministic. The uncertainty may arise from the thermal fluctuation, impurities of the materials and the intrinsic instabilities of the deterministic evolutions. My current interest in this area is to study numerical methods for the stochastic Allen-Cahn and Cahn-Hilliard equations. In particular, we studied finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noise and proved strong convergence with sharp rates.

    The sharp interface limit, as the diffuse interface thickness (ε) vanishes, of the stochastic Allen-Cahn equation is formally a stochastic mean curvature flow. The following figures illustrate the interplay of the geometric evolution parameter (ε) and gradient-type noise intensity (δ).

    Snapshots of the zero-level set of the average numerical solutions at different time points with ε = 0.01, and δ = 0.1, 1, 10. The zero-level set evolves faster and the shape undergoes more changes for larger δ.
    Snapshots of the zero-level set of the average numerical solutions at different time points with δ = 0.1, and ε = 0.01, 0.001, 0.02. This suggests the convergence of the zero-level set to the stochastic mean curvature flow as ε approaches 0.
  • Numerical Optimization

    • Optimization is an important tool that is widely used in science, engineering, economics, and industry. My interest is in developing fast numerical solvers arises from finite element approximations of variatinal inequalities, optimal control problems, stochastic PDEs and other applied problems. Recently, we designed a new homotopy-based approach for solving semidefinite programs without having to first find an interior point of the given program. We employ techniques from numerical algebraic geometry to handle various cases which can arise, such as adaptive precision path tracking to handle ill-conditioned areas, endgames to accurately approximate an optimizer, and projective space when the optimal value is not achieved.

    SDP

    Path of the homotopy with start point (○), endpoint (●), and feasible sets. The smaller (larger) ellipse is the feasibility set for the original (perturbed) semidefinite program. The start point (○) is not an interior point of the original feasible set.