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

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Abstract

The local minimax method (LMM) proposed by Li and Zhou (2001) and Li and Zhou (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 Li and Zhou (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.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12171148 and 11771138) and the Construct Program of the Key Discipline in Hunan Province. Wei Liu was supported by National Natural Science Foundation of China (Grant Nos. 12101252 and 11971007). Wenfan Yi was supported by National Natural Science Foundation of China (Grant No. 11901185), National Key R&D Program of China (Grant No. 2021YFA1001300) and the Fundamental Research Funds for the Central Universities (Grant No. 531118010207).

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  1. Wei Liu

    Present address: Current address: South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510631, China

Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), Hunan Normal University, Changsha, 410081, China

    Wei Liu & Ziqing Xie

  2. School of Mathematics, Hunan University, Changsha, 410082, China

    Wenfan Yi

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  1. Wei Liu
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  2. Ziqing Xie
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  3. Wenfan Yi
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Correspondence to Wenfan Yi.

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Liu, W., Xie, Z. & Yi, W. Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs. Sci. China Math. 66, 2361–2384 (2023). https://doi.org/10.1007/s11425-021-2093-1

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  • DOI: https://doi.org/10.1007/s11425-021-2093-1

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