Tim Sullivan

Junior Professor in Applied Mathematics:
Risk and Uncertainty Quantification

Strong convergence rates of probabilistic integrators for ordinary differential equations

Preprint: Strong convergence rates of probabilistic integrators for ODEs

Han Cheng Lie, Andrew Stuart and I have just uploaded a preprint of our latest paper, “Strong convergence rates of probabilistic integrators for ordinary differential equations” to the arXiv. This paper is a successor to the convergence results presented by Conrad et al. (2016). We consider a generic probabilistic solver for an ODE, using a fixed time step of length \(\tau > 0\), where the mean of the solver has global error of order \(\tau^{q}\) and the variance of the truncation error model has order \(\tau^{1 + 2 p}\). Whereas Conrad et al. showed, for a Lipschitz driving vector field, that the mean-square error between the numerical solution \(U_{k}\) and the true solution \(u_{k}\) is bounded uniformly in time as

\( \displaystyle \max_{0 \leq k \leq T / \tau} \mathbb{E} [ \| U_{k} - u_{k} \|^{2} ] \leq C \tau^{2 \min \{ p, q \}} \)

(i.e. has the same order of convergence as the underlying deterministic method), we are able to relax the regularity assumptions on the vector field / flow an obtain a stronger mode of convergence (mean-square in the uniform norm) with the same convergence rate:

\( \displaystyle \mathbb{E} \left[ \max_{0 \leq k \leq T / \tau} \| U_{k} - u_{k} \|^{2} \right] \leq C \tau^{2 \min \{ p, q \}} \)

Abstract. Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced 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., 2016), to establish mean-square 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.

Published on Monday 13 March 2017 at 13:00 UTC #publication #preprint #prob-num

Bayesian probabilistic numerical methods

Preprint: Bayesian probabilistic numerical methods

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Bayesian probabilistic numerical methods” to the arXiv. Following on from our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems”, our aim is to provide some rigorous theoretical underpinnings for the emerging field of probabilistic numerics, and in particular to define what it means for such a method to be “Bayesian”, by connecting with the established theories of Bayesian inversion and disintegration of measures.

Abstract. The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented.

Published on Tuesday 14 February 2017 at 12:00 UTC #publication #preprint #prob-num

Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Preprint: Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic numerical methods for PDE-constrained Bayesian inverse problems” to the arXiv. This paper is intended to complement our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” and to give a more concise presentation of the main ideas, aimed at a general audience.

Published on Wednesday 18 January 2017 at 12:00 UTC #publication #preprint #prob-num

Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors

Preprint: Bayesian inversion with heavy-tailed stable priors

A revised version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has been released on arXiv today. Among other improvements, the revised version incorporates additional remarks on the connection to the existing literature on stable distributions in Banach spaces, and generalises the results of the previous version of the paper to quasi-Banach spaces, which are like complete normed vector spaces in every respect except that the triangle inequality only holds in the weakened form

\( \| x + y \| \leq C ( \| x \| + \| y \| ) \)

for some constant \( C \geq 1 \).

Published on Monday 21 November 2016 at 11:30 UTC #publication #preprint

Cameron-Martin theorems for Cauchy-distributed random sequences

Preprint: Cameron-Martin theorems for Cauchy-distributed random sequences

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv. Inspired by questions of prior robustness left unanswered in this earlier paper on α-stable Banach space priors, this paper addresses the basic probabilistic question: when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation? In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite. We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\). More precisely, if the Cauchy distribution on the nth term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| < \infty \)

and, with the usual convention that \(0 \log 0 = 0\),

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \log \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| \right| < \infty . \)

We also discuss similar results for dilation of the scale parameters, i.e. \(\gamma_{n} \mapsto \sigma_{n} \gamma_{n}\) for some real sequence \(\sigma = (\sigma_n)_{n = 1}^{\infty}\).

Published on Monday 15 August 2016 at 10:00 UTC #publication #preprint

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