Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat. Comput., 2017), to establish meansquare convergence in the uniform norm on discrete-or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.
Trajectory-or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality-their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov state model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge. Introduction.In statistical mechanics, a molecule is described by a suitably chosen ensemble of systems having the same equations of motion but different initial conditions. The ensemble is specified by a probability density function which gives the probability that a system lies in a certain subset of the phase space. If the dynamical behavior of a system exhibits two or more time scales, then we can consider the metastabilities of each system in the following way: in the fine scale (smaller time scale) dynamics, the system fluctuates but relaxes for certain given initial conditions to subsets which are almost stable; in the coarse scale (larger time scale) dynamics, the system appears to be a Markovian process jumping between the almost stable subsets, where the transition probability depends only on the current state. A subset is almost stable if the expected exit time of a system from the subset is large, relative to the fluctuations occurring at the short time scale. One sometimes uses the terms metastabilities or conformations to refer to the almost stable subsets of the phase space. In Figure 1.1 we see a trajectory of a system exhibiting the kind of multiscale dynamics we have described. The identification of the metastabilities of a system and the study of its coarse scale dynamics are among the primary goals of coarse-graining and multiscale methods [3,13,16].Methods based on Markov state models [24,26,15] seek to identify the metastabilities and study the coarse scale dynamics by constructing a Markov chain on a finite
We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.
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We prove the Fréchet differentiability with respect to the drift of Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen-and singular values and the corresponding eigen-and singular functions of the stochastic Perron-Frobenius and Koopman operators. *
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