Ritz method for transition paths and quasipotentials of rare diffusive events

Abstract

The probability of trajectories of weakly diffusive processes to remain in the tubular neighbourhood of a smooth path is given by the Freidlin-Wentzell-Graham theory of large deviations. The most probable path between two states (the instanton) and the leading term in the logarithm of the process transition density (the quasipotential) are obtained from the minimum of the Freidlin-Wentzell action functional. Here we present a Ritz method that searches for the minimum in a space of paths constructed from a global basis of Chebyshev polynomials. The action is reduced, thereby, to a multivariate function of the basis coefficients, whose minimum can be found by nonlinear optimization. For minimisation regardless of path duration, this procedure is most effective when applied to a reparametrisation-invariant "on-shell" action, which is obtained by exploiting a Noether symmetry and is a generalisation of the scalar work [Olender and Elber, 1997] for gradient dynamics and the geometric action [Heyman and Vanden-Eijnden, 2008] for non-gradient dynamics. Our approach provides an alternative to chain-of-states methods for minimum energy paths and saddlepoints of complex energy landscapes and to Hamilton-Jacobi methods for the stationary quasipotential of circulatory fields. We demonstrate spectral convergence for three benchmark problems involving the Muller-Brown potential, the Maier-Stein force field and the Egger weather model.