Research Interests

• High order numerical methods: Discontinuous Galerkin finite element methods, Spectral methods, etc.

• Numerical methods for kinetic equations

• Post-processing techniques

• Numerical methods for fractional differential equations

• Machine learning, Neural network

• Numerical methods for uncertainty quantification

• Multiscale Computational Methods

• Mathematical applications across diverse disciplines

Research focus

My research is deeply rooted in scientific computing and numerical analysis, with a core mission to elevate the precision, efficiency, and versatility of numerical methods across a diverse spectrum of mathematical models. Within this realm, my work encompasses the creation, scrutiny, and implementation of exceptionally efficient numerical algorithms, with wide-ranging applications spanning fluid dynamics, kinetic theory, finance, and engineering mechanics.

My research pursuits converge into four focal areas: (Click each item for project details.)

• Precision Enhancement for Nonsmooth Problems: A primary focus lies in enhancing the accuracy of numerical simulations, especially for functions featuring singularities. At the core of this effort lies the development and analysis of innovative reconstruction techniques aimed at significantly improving precision and potentially restoring spectral accuracy from poorly behaving numerical simulation data. Moreover, I am actively investigating the integration of neural networks to automate singularity detection from spectral data, which could lead to additional improvements in simulation accuracy.

  • Project 1: Postprocessing techniques

    To be added. slides "Recovering exponential accuracy in spectral methods involving piecewise smooth functions"
  • Project 2: Solution-enriched numerical methods

    To be added.
  • Project 3: Neural network-base singularity detectors

    To be added.
• Kinetic Simulations: Within the domain of kinetic models, my research is committed to maximizing the efficiency and accuracy of numerical methods. This involves exploring multiscale algorithms and hybrid methods through their design and analysis, strategically applying stochastic Galerkin methods, and integrating deep neural networks to enhance the efficiency and accuracy of kinetic system simulations.

  • Project 1: Numerical methods for simulating linear kinetic models - design and analysis

    To be added. slides "Multiscale Convergence Properties for Spectral Approximations of a Model Kinetic Equation"
  • Project 2: Numerical methods for kinetic models with uncertainties

    To be added.
  • Project 3: Fast solvers

    To be added.
• High-Order Numerical Methods: My expertise lies in high order numerical methods, with a particular focus on crafting and analyzing high order structure-preserving discontinuous Galerkin (DG) finite element methods customized for time-dependent partial differential equations (PDEs). These methods ensure not only high accuracy and efficiency but also alignment with fundamental physical principles. Additionally, I employ advanced techniques like Weighted Essentially Non-Oscillatory (WENO) methods, crucial for guaranteeing precise solutions in time-dependent hyperbolic problems.

  • Project 1: High-order numerical methods for various PDEs and their applications

    To be added.
  • Project 2: Structure-preserving high-order numerical schemes for time-dependent PDEs

    To be added. slides "Third-order Maximum-Principle-Satisfying Direct DG methods for convection-diffusion equations on unstructured triangular mesh"

    Nonlinear porous medium equation

    Incompressible Navier–Stokes equation - Vortex patch problem
• Cross-Disciplinary Collaborations: My research philosophy extends beyond conventional mathematical boundaries. Collaborating with experts across diverse fields enables me to apply mathematical expertise to real-world challenges.

  • Project 1: Application in Finance

    Teaming up with experts in accounting and finance, I've employed data processing techniques to investigate the influence of CEOs on the readability of financial reports. This exploration has revealed invaluable insights into corporate governance.
  • Project 2: Application in Civil Engineering

    In collaboration with colleagues in civil engineering, we're leveraging cutting-edge machine learning techniques to uncover the intricate relationships between geometric descriptors and material behavior. This effort aims to deepen our understanding of material microstructure and fracture properties, with significant implications for materials science.