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

TLDR: In this article, the authors 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.

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.