Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

TL;DR

This paper introduces a normalized Wolfe-Powell-type local minimax method (NWP-LMM) for efficiently finding multiple unstable solutions of nonlinear elliptic PDEs, extending traditional methods with convergence guarantees and improved performance.

Contribution

The paper develops a new NWP-LMM framework with rigorous convergence analysis, incorporating general descent directions and conjugate gradient methods for better efficiency.

Findings

NWP-LMM guarantees global convergence for general descent directions.

Combining NWP-LMM with CG directions accelerates convergence.

Numerical results show improved robustness and effectiveness over existing methods.

Abstract

The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.