An Efficient Laguerre Minimum Action Method for Computing Quasi-Potentials
For researchers studying rare transitions in dynamical systems, this method offers a more accurate and efficient way to compute quasi-potentials, though it is an incremental improvement over existing minimum action methods.
The paper develops an efficient Laguerre spectral minimum action method (LMAM) for computing quasi-potentials in small-noise-driven dynamical systems, addressing time truncation and parameter sensitivity issues. Numerical experiments on Allen-Cahn and Navier-Stokes equations demonstrate improved accuracy and efficiency.
Minimum action methods provide a powerful framework for analyzing rare transitions in small-noise-driven dynamical systems, but their practical performance is often limited by time truncation and parameter sensitivity in infinite-horizon problems. In this paper, we develop an efficient Laguerre spectral minimum action method (LMAM) for computing quasi-potentials associated with fixed points of dynamical systems. Based on the large deviation framework, the method computes minimum action paths by formulating the problem on a semi-infinite time interval and discretize the temporal direction using Laguerre functions. An appropriate time rescaling strategy is proposed to enhance accuracy and convergence of the Laguerre spectral approximation. To efficiently handle nonlinear terms, we employ an improved procedure for evaluating Laguerre--Gauss--Radau quadrature, which enables stable and accurate double-precision computations with a large number of Laguerre modes. Precise numerical analysis for the linear problem and a local result for the nonlinear case are developed. Numerical experiments including both ordinary and partial differential equations (Allen-Cahn and Navier-Stokes) are presented to illustrate the accuracy and efficiency of the proposed method.