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 neighborhood 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 minimization regardless of path duration, this procedure is most effective when applied to a reparametrization-invariant "on-shell" action, which is obtained by exploiting a Noether symmetry and is a generalization of the scalar work [Olender and Elber, J. Mol. Struct: THEOCHEM 398, 63 (1997)] for gradient dynamics and the geometric action [Heyman and Vanden-Eijnden (2008)] for nongradient dynamics. Our approach provides an alternative to chain-of-states methods for minimum energy paths and saddle points 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 Müller-Brown potential, the Maier-Stein force field, and the Egger weather model.

An efficient jet marcher for computing the quasipotential for 2D SDEs

“…In practice, techniques conducting the minimizations of two types of action functionals are used. The Minimum Action Method (MAM) [11] and Adaptive Minimum Action Method (AMAM) [34] find MAPs by minimizing the Freidlin-Wentzell action, while the Geometric Minimum Action Method (GMAM) [15] and the recent Ritz method [17] do so by minimizing the geometric action. By design, these techniques are applicable to both finite and infinite dimensional problems.…”

“…The key decisions to be made concern how best to approximate the right hand side of equation (17). This requires determining (a) how to interpolate U (x λ ), since x λ in general lies between mesh points, (b) what path-space to take the inner minimization over, and (c) what quadrature rule to use to approximate the action integral.…”

“…This route is taken, for instance, by the Ordered Upwind Method (OUM) of [30]. With these approximations, (17) becomes…”

“…Update formula (18) in general leads to an O(h) convergence rate. It is perfectly valid for the quasipotential problem, however in EJM we will opt for higher order approximations of (17) in pursuit of a O(h 2 ) convergence rate.…”

“…The goal of neighborhood design is to assure that every mesh point y experiences at least 1 of the (17) updates with an x and z that straddle its MAP as in Figure 3. Since we do not a priori know where the MAPs will be coming from, many updates will necessarily have to be performed where x and z do not surround the MAP at all.…”