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

TL;DR

This paper presents a GPU-optimized quasi-Newton method with tensor product implementation for efficiently solving large-scale quasi-linear elliptic equations, demonstrating improved computational performance and convergence properties.

Contribution

The paper introduces a novel quasi-Newton method that approximates Jacobians using tensor products, reducing computational costs for large PDE systems on GPU hardware.

Findings

Demonstrates local convergence under optimal regularization parameters

Achieves significant computational gains in 2D and 3D numerical experiments

Validates robustness and efficiency of the proposed method

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.