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An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

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Abstract

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

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Data Availibility Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Authors and Affiliations

  1. School of Mathematics and Physics and Hunan Key Laboratory for Mathematical Modeling and Scientific Computing, University of South China, Hengyang, 421001, China

    Lin Li

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Li-Lian Wang

  3. State Key Laboratory of Computer Science/Laboratory of Parallel Computing Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Huiyuan Li

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  1. Lin Li
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  2. Li-Lian Wang
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  3. Huiyuan Li
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Contributions

All authors contributed to the study conception and design. The computations and the first draft were prepared by the first author. All authors read and approved the final manuscript.

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Correspondence to Li-Lian Wang.

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L. Li: This work of this author is partially supported by the Science Foundations of Hunan Province (Nos: 2020JJ5464, 20C1595).

L-L. Wang: The research of this author is partially supported by Singapore MOE AcRF Tier 1 Grant: RG15/21.

L. Li would like to thank the hospitality of Nanyang Technological University for the visit to finalize this work.

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Li, L., Wang, LL. & Li, H. An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions. J Sci Comput 95, 32 (2023). https://doi.org/10.1007/s10915-023-02154-0

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  • DOI: https://doi.org/10.1007/s10915-023-02154-0

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