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

Wenrui Hao, Jan Hesthaven, Guang Lin, Bin Zheng

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Article number19
JournalJournal of Scientific Computing
Volume82
Issue number1
DOIs
StatePublished - Jan 1 2020

Fingerprint

Homotopy Continuation Method
Homotopy Method
Approximation Space
Multiple Solutions
Nonlinear Differential Equations
Differential equations
Differential equation
Spectral Approximation
Computing
Finite Element Discretization
Greedy Algorithm
Computational Cost
Finite Difference
Numerical Examples
Costs

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Cite this

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A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations. / Hao, Wenrui; Hesthaven, Jan; Lin, Guang; Zheng, Bin.

In: Journal of Scientific Computing, Vol. 82, No. 1, 19, 01.01.2020.

Research output: Contribution to journalArticle

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