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.

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Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

Transition pathways of stochastic dynamical systems are typically approximated by instantons. Here we show, using a dynamical system containing two competing pathways, that at low-tointermediate temperatures, instantons can fail to capture the most likely transition pathways. We construct an approximation which includes fluctuations around the instanton and, by comparing with the results of an accurate and efficient path-space Monte Carlo sampling method, find this approximation to hold for a wide range of temperatures. Our work delimits the applicability of large deviation theory and provides methods to probe these limits numerically.