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A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations

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Abstract

The homotopy continuation method has been widely used to compute multiple solutions of nonlinear differential equations, but the computational cost grows exponentially based on the traditional finite difference and finite element discretizations. In this work, we presented a new method by constructing a spectral approximation space adaptively based on a greedy algorithm for nonlinear differential equations. Then multiple solutions were computed by the homotopy continuation method on this low-dimensional approximation space. Various numerical examples were given to illustrate the feasibility and the efficiency of this new approach.

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Acknowledgements

W. Hao’s research was supported by the American Heart Association (Grant 17SDG33660722) and National Science Foundation (Grant DMS-1818769). GL would like to gratefully acknowledge the support from National Science Foundation (DMS-1555072, DMS-1736364 and DMS-1821233). B. Zheng would like to acknowledge the support by Beijing Institute for Scientific and Engineering Computing and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials and a Laboratory Directed Research and Development (LDRD) Program from Pacific Northwest National Laboratory. The PNNL is operated by Battelle for the US Department of Energy under Contract DE-AC05-76RL01830.

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Correspondence to Bin Zheng.

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Hao, W., Hesthaven, J., Lin, G. et al. A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations. J Sci Comput 82, 19 (2020). https://doi.org/10.1007/s10915-020-01123-1

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Keywords

  • Multiple solutions
  • Nonlinear differential equations
  • Polynomial systems
  • Homotopy continuation