Strong convergence rates of probabilistic integrators for ordinary differential equations

Abstract

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.

Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

“…Stability of the posterior with respect to the observed data y and the log-likelihood Φ was established for Gaussian priors by Stuart (2010) and for more general priors by many later contributions (Dashti et al, 2012;Hosseini, 2017;Hosseini and Nigam, 2017;Sullivan, 2017). (We note in passing that the stability of BIPs with respect to perturbation of the prior is possible but much harder to establish, particularly when the data y are highly informative and the normalisation constant Z(y) is close to zero; see e.g.…”

“…When such a probabilistic solver is used for the ODE, the likelihood becomes random in the sense considered in this paper. Random approximate solution of deterministic ODEs is an old idea (Diaconis, 1988;Skilling, 1992) that has received renewed attention in recent years (Conrad et al, 2016;Hennig et al, 2015;Lie et al, 2017;Schober et al, 2014). As random forward models, these probabilistic ODE solvers are amenable to the analysis of Section 3.…”

“…A fundamental task in numerical analysis is to determine sufficient conditions for convergence of the sequence (Z ′ k ) k∈[N ] to (z k ) k∈ [N ] . The investigations of Conrad et al (2016) and Lie et al (2017) concern a similar task in the context of uncertainty quantification. Given τ > 0, consider a collection (ξ k ) k∈[N ] of stochastic processes ξ k : Ω × [0, τ ] → R d having almost-surely continuous paths.…”

“…In the realm of "big data" a residual vector of prohibitively high dimension may be randomly subsampled or orthogonally projected onto a randomly-chosen lowdimensional subspace (Le et al, 2017;Nemirovski et al, 2008). In the field of probabilistic numerical methods (Hennig et al, 2015), a deterministic dynamical system may be solved stochastically, with the stochasticity representing epistemic uncertainty about the behaviour of the system below the temporal or spatial grid scale (Conrad et al, 2016;Lie et al, 2017).…”

A Role for Symmetry in the Bayesian Solution of Differential Equations

“…The challenge of designing a stochastic model for the sequence ( i ) n i=1 that reflects the highly structured nature of the error remains unresolved. On the other hand, the mathematical properties of this method are now wellunderstood [Lie et al, , 2019. The proposal of Abdulle and Garegnani [2018] was to instead consider randomisation of the inputs {x i } T i=0 in the context of a classical numerical method, also outside of the Bayesian framework.…”