Research

Many computational and inference tasks in applied mathematics, statistics, and machine learning exhibit high heterogeneity, dimensionality, or necessitate specialized labor and effort. Analytical and algorithmic innovations are needed to achieve enhanced accuracy, efficiency, and general-purpose automation. My interest lies in exploring multiscale, randomized, and dynamical approaches to tackle these challenges, particularly in the realms of partial differential equations (PDEs) and data science.

Multiscale analysis and randomization prove instrumental in identifying low-complexity structures within these heterogeneous and high-dimensional problems, while dynamical approaches often decompose the task into smaller, simpler, and sequential subproblems, allowing us to tackle it gradually. The interesting questions would be how to embed multiscale and randomization ideas to get low complexity, robust, and automated algorithms in various scientific computing settings, and how to design dynamics that converge fast and can be numerically approximated, or learned, effciently. Specific domain knowledge and abstract mathematical formulation and analysis play important roles in answering these questions.

Some specific topics that I have worked on:

  • Multiscale, randomized algorithms: numerical homogenization, randomized numerical linear algebra
  • Gaussian processes, kernel methods: scalable algorithms, uncertainty quantification, learning theory
  • Dynamical, stochastic sampling: Monte Carlo, gradient flows, optimal transport, generative modeling

Publications

Preprints

  • Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations
    Huan Zhang, Yifan Chen, Eric Vanden-Eijnden, Benjamin Peherstorfer
    Submitted, 2024. [arXiv]

  • Probabilistic Forecasting with Stochastic Interpolants and Follmer Processes
    Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden
    Submitted, 2024. [arXiv]

  • Provable Probabilistic Imaging using Score-Based Generative Priors
    Yu Sun, Zihui Wu, Yifan Chen, Berthy T. Feng, Katherine L. Bouman
    Submitted, 2023. [arXiv]

  • Sampling via Gradient Flows in the Space of Probability Measures
    Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart
    Submitted, 2023. [arXiv] [Code] [Slide]

  • Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs
    Pau Batlle, Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart
    Submitted, May, 2023. [arXiv] [Code]

  • Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes
    Yifan Chen, Houman Owhadi, Florian Schaefer
    Submitted, April, 2023. [arXiv] [Code] [Slide]

  • Randomly Pivoted Cholesky: Practical Approximation of A Kernel Matrix with Few Entry Evaluations
    Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber
    Submitted, July, 2022. [arXiv] [Code]

Journal Publications

  • Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    SIAM Multiscale Modeling and Simulation, 2023. [Journal] [arXiv] [Slide] [Slide+]

  • Exponentially Convergent Multiscale Finite Element Method
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    Communications on Applied Mathematics and Computation, 2023. [Journal] [arXiv] [Slide] [Slide+]

  • Multiscale Elliptic PDEs Upscaling and Function Approximation via Subsampled Data
    Yifan Chen, Thomas Y. Hou
    SIAM Multiscale Modeling and Simulation, 2022. [Journal] [arXiv] [Code]

  • Solving and Learning Nonlinear PDEs with Gaussian Processes
    Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart
    Jounal of Computational Physics, 2021. [Journal] [arXiv] [Code] [Slide] [Slide+] [Slide++]

  • Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    SIAM Multiscale Modeling and Simulation, 2021. [Journal] [arXiv] [Slide] [Slide+]

  • Consistency of Empirical Bayes and Kernel Flow For Hierarchical Parameter Estimation
    Yifan Chen, Houman Owhadi, Andrew M. Stuart
    Mathematics of Computation, 2021. [Journal] [arXiv] [Code] [Slide] [Longer Slide] [Short Video] [Longer Video]

  • Function Approximation via The Subsampled Poincare Inequality
    Yifan Chen, Thomas Y. Hou
    Discrete & Continuous Dynamical Systems - A, 2020. [Journal] [arXiv]

  • Optimal Transport Natural Gradient for Statistical Manifolds with Continuous Sample Space
    Yifan Chen, Wuchen Li
    Information Geometry, 2020. [Journal] [arXiv] [Code] [Slide]

  • Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers
    Yifan Chen, Yuejiao Sun, Wotao Yin
    Mathematical Programming, 2019. [Journal] [arXiv] [Slide]

  • The Quadratic Wasserstein Metric for Earthquake Location
    Jing Chen, Yifan Chen, Hao Wu, Dinghui Yang
    Journal of Computational Physics, 2018. [Journal] [arXiv] [Slide]

Thesis

  • On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems
    Yifan Chen
    Ph.D. Thesis, The W.P. Carey and Co. Prize in Applied Mathematics, 2023 [CaltechTHESIS] [Slide]

Notes

  • Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance
    Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart
    [arXiv]

Professional Experience

Referee Service