Kernel and Lengthscale Selection on the Performance of the Sparse Cholesky Factorization Algorithm Open Access

Abstract

In the realms of science and engineering, many challenges arise that demand the repeated solving of intricate systems of partial differential equations (PDEs) across various parameter values. Such scenarios are common in fields like molecular dynam- ics, micro-mechanics, and turbulent flows. Machine learning methods have shown promising capabilities in automating scientific computations, especially in PDE solving. Among these methods, Gaussian Process (GP) and kernel methods are notable for their interpretable and theoretically grounded function representation. In our study, we initially focused on the Sparse Cholesky-accelerated Gauss-Newton Algorithm [6]. However, we identified a need for further exploration regarding the choice of kernel and its parameters. Utilizing provided code, we conducted experiments to investigate the impact of different kernels and their lengthscales on accuracy, identifying optimal lengthscales. Moreover, we explored nonlinear elliptic PDEs, testing various solutions and observing limitations in achieving low relative error as high frequency terms became more significant, leading to non-convergence of some solutions. We also adjusted algorithm parameters and saw some accuracy improvements, although some questions still remain unanswered.

Table of Contents

1 Introduction 1

1.1 Linear PDE................................ 2 1.2 Non-Linear PDE ............................. 2 1.2.1 Semi-linear equations....................... 3 1.2.2 Quasi-linear equations ...................... 3 1.2.3 Fully nonlinear equations..................... 4 1.3 Classical numerical methods for solving non-linear PDEs . . . . . . . 4 1.3.1 Finite-difference methods..................... 4 1.3.2 Finite-element methods...................... 5 1.4 Machine learning methods ........................ 6 1.4.1 Neural Networks ......................... 7 1.4.2 GP and kernel methods ..................... 7 1.5 Contributions ............................... 8

2 Overall Process to solve non-linear PDE 10

2.1 Solving nonlinear PDEs via GPs..................... 10 2.2 Sparse Cholesky factorization algorithm. . . . . . . . . . . . . . . . . 12 2.2.1 Ordering of the measurement .................. 12 2.2.2 Select non-zero entries of Uρ ................... 15 2.2.3 Optimize Uρ............................ 15 2.3 Second order optimization methods ................... 16

3 Our Purpose 18

3.1 Kernel and Lengthscale.......................... 18 3.2 Truth function of Nonlinear Elliptic PDEs . . . . . . . . . . . . . . . 19 3.3 GN Steps and other parameters ..................... 19

4 Experiments 21

4.1 Kernel and Lengthscale influence on accuracy . . . . . . . . . . . . . 22 4.1.1 Nonlinear elliptic PDEs ..................... 22 4.1.2 Viscous Burgers’ equation .................... 23 4.1.3 Monge-Ampère equation in two-dimensional space . . . . . . . 25 4.2 Change truth function in Nonlinear elliptic PDEs . . . . . . . . . . . 26 4.2.1 Truncate the low-frequency terms................ 26 4.2.2 Change the degree of k...................... 28 4.2.3 Representative truth functions.................. 29 4.3 Improvements............................... 30 4.3.1 Increase Gauss-Newton steps................... 30 4.3.2 Increase small (Algorithm 2) KNN value . . . . . . . . . . . . 31 4.3.3 Increase big KNN value ..................... 32 4.3.4 Increase small ρ value ...................... 32 4.3.5 Increase big ρ value........................ 33 4.4 Summary ................................. 33 4.4.1 Kernel and Lengthscale...................... 33 4.4.2 Truth function of Nonlinear Elliptic PDEs . . . . . . . . . . . 34 4.4.3 GN Steps and other parameters................. 34

5 Conclusion 42

Bibliography 44

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Numerical Analysis PDE
Committee Chair / Thesis Advisor	Yuanzhe Xi, Emory University Tianshi Xu, Emory University
Committee Members	Alessandro Veneziani, Emory University Julianne Chung, Emory University

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