# Research

My research is driven by the challenges of high dimensional analysis and computation in applied mathematics, statistics, and data science. A common goal across these fields is the rigorous and efficient estimation of quantities of interest to scientists and engineers. The complexity of high dimensionality can manifest in various ways: in physical problems where a wide range of scales necessitates numerous degrees of freedom; in data science where the feature space is high dimensional and datasets are enormous; and in the integration of physical models and data towards data assimilation and scientific machine learning. To address these challenges, two fundamental questions that motivate me are:

- How to leverage low complexity structures in physical and data science problems to achieve provably efficient numerical estimation?
- How to dynamically improve our uncertainty estimations, particularly in the efficient modeling and sampling of probability distributions?

In my research, I draw insights from multiscale analysis and randomization to identify low complexity structures and automate/accelerate scientific computing, in areas with heterogeneous physics PDEs and large-scale data matrices. I have recently been working on analyzing and designing statistically and numerically desirable dynamics of measures to navigate the space of high-dimensional probability distributions, with applications in Bayesian inverse problems and generative modeling for imaging, climate forecasting, and material science.

Some specific topics are as follows:

**Multiscale analysis, randomized algorithms**: numerical homogenization, randomized numerical linear algebra**Gaussian processes, kernel methods**: scalable algorithms, uncertainty quantification, learning theory**Stochastic modeling, dynamical sampling**: Monte Carlo, gradient flows, optimal transport, diffusion models

### Publications

##### Preprints

**Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias**

Yifan Chen, Xiaoou Cheng, Jonathan Niles-Weed, Jonathan Weare

Submitted. [arXiv]**Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities**

José A. Carrillo, Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Dongyi Wei

Submitted. [arXiv]**Efficient, Multimodal, and Derivative-Free Bayesian Inference with Fisher-Rao Gradient Flows**

Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart

Submitted. [arXiv] [Code] [Slide]**Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors**

Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, Katherine L. Bouman

Submitted. [arXiv]**Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation**

Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart

Submitted. [arXiv] [Code]**Sequential-in-time Training of Nonlinear Parametrizations for Solving Time-dependent Partial Differential Equations**

Huan Zhang, Yifan Chen, Eric Vanden-Eijnden, Benjamin Peherstorfer

Submitted. [arXiv]**Sampling via Gradient Flows in the Space of Probability Measures**

Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart

Submitted. [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. [arXiv] [Code]**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. [arXiv] [Code]

##### Conference Publications

**Probabilistic Forecasting with Stochastic Interpolants and Follmer Processes**

Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden

*ICML*, 2024. [Proceedings] [arXiv] [Slide] [Code] [Poster]**Inpainting crystal structure generations with score-based denoising**

Xinzhe Dai, Peichen Zhong, Bowen Deng, Yifan Chen, Gerbrand Ceder

*ICML Workshop AI4Science*, 2024. [Proceedings]

##### Journal Publications

**Provable Probabilistic Imaging using Score-Based Generative Priors**

Yu Sun, Zihui Wu, Yifan Chen, Berthy T. Feng, Katherine L. Bouman

*IEEE Transactions on Computational Imaging*, 2024. [Journal] [arXiv] [Code]**Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes**

Yifan Chen, Houman Owhadi, Florian Schaefer

*Mathematics of Computation*, 2024. [Journal] [arXiv] [Code] [Slide]**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+]**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

- Reviewer for NeurIPS 2024
- Reviewer for Journal of Functional Analysis
- Reviewer for Mathematics of Computation
- Reviewer for Journal of Computational Physics
- Reviewer for SIAM on Uncertainty Quantification
- Reviewer for SIAM on Multiscale Modeling and Simulation
- Reviewer for SIAM on Numerical Analysis
- Reviewer for SIAM on Control and Optimization
- Reviewer for IMA Journal of Numerical Analysis
- Reviewer for Linear Algebra and Its Applications
- Reviewer for Nature Machine Intelligence
- Reviewer for European Journal of Applied Mathematics
- Reviewer for Foundations of Data Science
- Reviewer for Research in the Mathematical Sciences
- Reviewer for Computational Methods in Applied Mathematics
- Reviewer for International Journal of Computer Mathematics
- Reviewer for 4th International Conference, GSI 2019