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An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems

Wenrui Hao, Sun Lee, Xiangxiong Zhang

TL;DR

The paper develops a GPU-accelerated quasi-Newton framework for quasi-linear elliptic PDEs by approximating the Jacobian with $A_h + \beta I$, exploiting tensor-product and Kronecker structures to reduce inversion costs. It provides a convergence analysis showing local convergence under an optimally chosen $\beta_n = \tfrac{\lambda_{\min} + \lambda_{\max}}{2}$ of the nonlinear Jacobian, and demonstrates significant computational gains via diagonalization and GPU-based tensor operations. Numerical experiments in 1D, 2D, and 3D—including SEM in multiple dimensions and FDM—validate robustness, accuracy, and substantial speedups over classical Newton methods, especially at scale. The work broadens the feasibility of large-scale simulations on modern hardware and lays groundwork for extending to other nonlinear PDEs with adaptive regularization strategies.

Abstract

In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method's robustness and computational gains with tensor-product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equation and system solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.

An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems

TL;DR

The paper develops a GPU-accelerated quasi-Newton framework for quasi-linear elliptic PDEs by approximating the Jacobian with , exploiting tensor-product and Kronecker structures to reduce inversion costs. It provides a convergence analysis showing local convergence under an optimally chosen of the nonlinear Jacobian, and demonstrates significant computational gains via diagonalization and GPU-based tensor operations. Numerical experiments in 1D, 2D, and 3D—including SEM in multiple dimensions and FDM—validate robustness, accuracy, and substantial speedups over classical Newton methods, especially at scale. The work broadens the feasibility of large-scale simulations on modern hardware and lays groundwork for extending to other nonlinear PDEs with adaptive regularization strategies.

Abstract

In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method's robustness and computational gains with tensor-product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equation and system solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.

Paper Structure

This paper contains 25 sections, 1 theorem, 121 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Model Problem
  3. Numerical Discretization
  4. Spectral Element Method in 2D
  5. Spectral Element Method in 3D
  6. Finite Difference Method
  7. The Quasi-Newton Method
  8. Implementation with tensor product
  9. Diagonalization of Matrices
  10. Extend to nonlinear systems of differential equations
  11. Eigenvalue Problem
  12. Convergence Analysis
  13. Numerical Results
  14. One-Dimensional Examples
  15. Example 1
  16. ...and 10 more sections

Key Result

Theorem 1

Let $F^h(U) = A_h \textcolor{black}{\operatorname{vec}(U)} + N_h(\hbox{\boldmath$\mathbf{x}$}, U)$, where $A_h$ is a symmetric positive definite (SPD) matrix, and the Jacobian of $N_h = \operatorname{vec}([f(\hbox{\boldmath$\mathbf{x}$},U)])$ with respect to $U$ denotes $(N_h)_U(\hbox{\boldmath$\mat will converge locally to $U^\ast$with $\|U^\ast-U_0\|_{L^2}\ll1$ when where $\lambda_{\max}((N_h)_

Figures (6)

  • Figure 1: Examples of square domain $Q^2$ elements. Rectangles represent the meshes, and dots are the Gauss points within the rectangles. From left to right, the meshes have sizes $1 \times 1$, $2 \times 2$, and $4 \times 4$, each with $3 \times 3$ Gauss points inside. The corresponding values are $N_x = N_y = 3, 5, 9$, respectively, and the degrees of freedom are $N_x \times N_y$.
  • Figure 2: Numerical solutions of Eq. (\ref{['ex2']}) with $N=1000$ grid points. The unstable solution is plotted with dashed lines, while the stable solution is represented with solid lines.
  • Figure 3: Numerical solutions of Eq. (\ref{['example3']}) with $N=1000$ grid points. The unstable solution is plotted with dashed lines, while the stable solution is represented with solid lines.
  • Figure 4: Four solutions of the Eq. \ref{['2dex']} on $200 \times 200$ mesh.
  • Figure 5: Eight solutions of the Gray-Scott model, showing only $A(x,y)$ from the 2D system of Eq. \ref{['2dgrayeq']} with a degree of freedom $1600 \times 1600$.
  • ...and 1 more figures

Theorems & Definitions (6)

  • remark thmcounterremark
  • Theorem 1
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark