2024/06/14 by Yifan Chen, Houman Owhadi, Florian Schäfer · 11 citations
Computer Science · Decision Sciences · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Cholesky decomposition #Computer science #Discrete mathematics #Eigenvalues and eigenvectors #Gaussian Processes and Bayesian Inference #Incomplete Cholesky factorization #Kernel (algebra) #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Pointwise #Probabilistic and Robust Engineering Design
paper · doi:10.1090/mcom/3992
published in Mathematics of Computation (American Mathematical Society (AMS))
crossref issued 2024/06/14 · crossref published 2024/06/14 · crossref published-online 2024/06/14 · openalex publication_date 2024/06/14 · crossref created 2024/06/14 · openalex created_date 2025/10/10 · crossref deposited 2026/03/16 · openalex updated_date 2026/08/01 · crossref indexed 2026/08/08
In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives , a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ ε -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ( N / ϵ ) ) O(Nlog d(N/ε )) in space and O ( N log 2 d ( N / ϵ ) ) O(Nlog 2d(N/ε )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.