Research

My research lies in numerical analysis and scientific computing, with interests in PDEs on manifolds and point clouds and wave localization in random media. My papers are grouped by topic below. You can also find them on Google Scholar.

Numerical methods for PDEs on manifolds and point clouds
Wave localization and random operators

Numerical methods for PDEs on manifolds and point clouds

We develop and analyze mesh-free methods for scalar and vector-valued PDEs on manifolds, with an emphasis on accuracy, stability, and boundary conditions.

RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions Peng Chen, Shixiao W. Jiang, Rongji Li, and Qile Yan. Engineering Analysis with Boundary Elements, 191, 106937, 2026. Journal arXiv
Generalized Moving Least-Squares Methods for Solving Vector-Valued PDEs on Unknown Manifolds Rongji Li, Qile Yan, and Shixiao W. Jiang. SIAM Journal on Scientific Computing, 48(3), A1101–A1132, 2026. Journal arXiv
Solving Forward and Inverse Partial Differential Equation Problems on Unknown Manifolds via Physics-Informed Neural Operators Anran Jiao, Qile Yan, John Harlim, and Lu Lu. SIAM Journal on Scientific Computing, 48(1), C136–C163, 2026. Journal arXiv
Generalized Finite Difference Method on Unknown Manifolds Shixiao W. Jiang, Rongji Li, Qile Yan, and John Harlim. Journal of Computational Physics, 502, 112812, 2024. Journal arXiv
Spectral Methods for Solving Elliptic PDEs on Unknown Manifolds Qile Yan, Shixiao W. Jiang, and John Harlim. Journal of Computational Physics, 486, 112132, 2023. Journal arXiv
Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds Qile Yan, Shixiao W. Jiang, and John Harlim. Journal of Scientific Computing, 94, Article 5, 2023. Journal arXiv

Wave localization and random operators

My ongoing research concerns spectral problems associated with wave propagation in disordered media. We are interested in random acoustic operators, localization landscapes, and the asymptotic behavior of eigenvalues and eigenfunctions on large domains.