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Discretization and index-robust error analysis for constrained high-index saddle dynamics on the high-dimensional sphere

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Abstract

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on the high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12225102, 12050002 and 12288101) and the National Key Research and Development Program of China (Grant No. 2021YFF1200500).

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research, Center for Machine Learning Research, Center for Quantitative Biology, Peking University, Beijing, 100871, China

    Lei Zhang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Pingwen Zhang

  3. School of Mathematical Sciences, Laboratory of Mathematics and Applied Mathematics, Peking University, Beijing, 100871, China

    Pingwen Zhang

  4. School of Mathematics, Shandong University, Jinan, 250100, China

    Xiangcheng Zheng

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  1. Lei Zhang
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  2. Pingwen Zhang
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  3. Xiangcheng Zheng
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Correspondence to Xiangcheng Zheng.

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Zhang, L., Zhang, P. & Zheng, X. Discretization and index-robust error analysis for constrained high-index saddle dynamics on the high-dimensional sphere. Sci. China Math. 66, 2347–2360 (2023). https://doi.org/10.1007/s11425-022-2149-2

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  • DOI: https://doi.org/10.1007/s11425-022-2149-2

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