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The probability of trajectories of weakly diffusive processes to remain inthe tubular neighbourhood of a smooth path is given by theFreidlin-Wentzell-Graham theory of large deviations. The most probable pathbetween two states (the instanton) and the leading term in the logarithm of theprocess transition density (the quasipotential) are obtained from the minimumof the Freidlin-Wentzell action functional. Here we present a Ritz method thatsearches for the minimum in a space of paths constructed from a global basis ofChebyshev polynomials. The action is reduced, thereby, to a multivariatefunction of the basis coefficients, whose minimum can be found by nonlinearoptimization. For minimisation regardless of path duration, this procedure ismost effective when applied to a reparametrisation-invariant "on-shell" action,which is obtained by exploiting a Noether symmetry and is a generalisation ofthe scalar work [Olender and Elber, 1997] for gradient dynamics and thegeometric action [Heyman and Vanden-Eijnden, 2008] for non-gradient dynamics.Our approach provides an alternative to chain-of-states methods for minimumenergy paths and saddlepoints of complex energy landscapes and toHamilton-Jacobi methods for the stationary quasipotential of circulatoryfields. We demonstrate spectral convergence for three benchmark problemsinvolving the Muller-Brown potential, the Maier-Stein force field and the Eggerweather model.

Ritz method for transition paths and quasipotentials of rare diffusive events