A Caltech Library Service

On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems


Chen, Yifan (2023) On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/83p4-c644.


This thesis focuses on numerical methods for scientific computing and scientific machine learning, specifically on solving partial differential equations and inverse problems. The design of numerical algorithms usually encompasses a spectrum that ranges from specialization to generality. Classical approaches, such as finite element methods, and contemporary scientific machine learning approaches, like neural nets, can be viewed as lying at relatively opposite ends of this spectrum. Throughout this thesis, we tackle mathematical challenges associated with both ends by advancing rigorous multiscale and statistical numerical methods.

Regarding the multiscale numerical methods, we present an exponentially convergent multiscale finite element method for solving high-frequency Helmholtz's equation with rough coefficients. To achieve this, we first identify the local low-complexity structure of Helmholtz's equations when the resolution is smaller than the wavelength. Then, we construct local basis functions by solving local spectral problems and couple them globally through non-overlapped domain decomposition and Galerkin's method. This results in a numerical method that achieves nearly exponentially convergent accuracy regarding the number of local basis functions, even when the solution is highly non-smooth. We also analyze the role of a subsampled lengthscale in variational multiscale methods, characterizing the tradeoff between accuracy and efficiency in the numerical upscaling of heterogeneous PDEs and scattered data approximation.

As for the statistical numerical methods, we discuss using Gaussian processes and kernel methods to solve nonlinear PDEs and inverse problems. This framework incorporates the flavor of scientific machine learning automation and extends classical meshless solvers. It transforms general PDE problems into quadratic optimization with nonlinear constraints. We present the theoretical underpinning of the methodology. For the scalability of the method, we develop state-of-the-art algorithms to handle dense kernel matrices in both low and high-dimensional scientific problems. For adaptivity, we analyze the convergence and consistency of hierarchical learning algorithms that adaptively select kernel functions. Additionally, we note that statistical numerical methods offer natural uncertainty quantification within the Bayesian framework. In this regard, our further work contributes to some new understanding of efficient statistical sampling techniques based on gradient flows.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Scientific computing, scientific machine learning, multiscale methods, Helmholtz's equations, Gaussian processes, kernel methods, fast algorithms, randomized numerical linear algebra, hierarchical learning, statistical sampling, gradient flows
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey and Co. Prize in Applied Mathematics, 2023.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Hou, Thomas Y. (advisor)
  • Owhadi, Houman (advisor)
  • Stuart, Andrew M. (advisor)
Thesis Committee:
  • Schroeder, Peter (chair)
  • Hou, Thomas Y.
  • Owhadi, Houman
  • Stuart, Andrew M.
Defense Date:17 May 2023
Non-Caltech Author Email:yifanc96 (AT)
Funding AgencyGrant Number
Kortchak scholarshipUNSPECIFIED
NSF GrantsDMS-1912654
Air Force Office of Scientific ResearchFA9550-20-1-0358
NSF GrantsDMS-2205590
Record Number:CaltechTHESIS:05292023-175108484
Persistent URL:
Related URLs:
URLURL TypeDescription adapted to Chapter 3 adapted to Chapter 2 adapted to Chapter 4 adapted to Chapter 5 adapted to Chapter 6 partly adapted to Chapter 7 partly adapted to Chapter 7 related to this thesis, partly summarized in Chapter 1 related to this thesis, partly summarized in Chapter 1 related to this thesis, partly summarized in Chapter 1 related to this thesis, partly summarized in Chapter 1
Chen, Yifan0000-0001-5494-4435
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15224
Deposited By: Yifan Chen
Deposited On:01 Jun 2023 16:19
Last Modified:16 Jun 2023 16:30

Thesis Files

[img] PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page