Next week Jon Cockayne (University of Warwick) will give a talk on “Probabilistic Numerics for Partial Differential Equations”.
Time and Place. Friday 14 October 2016, 12:00–13:00, ZIB Seminar Room 2006, Zuse Institute Berlin, Takustraße 7, 14195 Berlin
Abstract. Probabilistic numerics is an emerging field which constructs probability measures to capture uncertainty arising from the discretisation which is often necessary to solve complex problems numerically. We explore probabilistic numerical methods for Partial differential equations (PDEs). We phrase solution of PDEs as a statistical inference problem, and construct probability measures which quantify the epistemic uncertainty in the solution resulting from the discretisation .
We analyse these probability measures in the context of Bayesian inverse problems, parameter inference problems whose dynamics are often constrained by a system of PDEs. Sampling from parameter posteriors in such problems often involves replacing an exact likelihood with an approximate one, in which a numerical approximation is substituted for the true solution of the PDE. Such approximations have been shown to produce biased and overconfident posteriors when error in the forward solver is not tightly controlled. We show how the uncertainty from a probabilistic forward solver can be propagated into the parameter posteriors, thus permitting the use of coarser discretisations while still producing valid statistical inferences.
 Jon Cockayne, Chris Oates, Tim Sullivan, and Mark Girolami. “Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems.” arXiv preprint, 2016. arXiv:1605.07811
There will be a workshop on Probabilistic Numerics at this year's MCQMC conference at Stanford University. The workshop will be held on Thursday, 18 August 2016, 15:50–17:50, at the Li Ka Shing Center on the Stanford University campus. Speakers include:
- Mark Girolami (University of Warwick & Alan Turing Institute) — Probabilistic Numerical Computation: A New Concept?
- François-Xavier Briol (University of Warwick & University of Oxford) — Probabilistic Integration: A Role for Statisticians in Numerical Analysis?
- Chris Oates (University of Technology Sydney) — Probabilistic Integration for Intractable Distributions
- Jon Cockayne (University of Warwick) — Probabilistic meshless methods for partial differential equations and Bayesian inverse problems
Update, 19 August 2016. The slides from the talks can be found here, on Chris Oates' website.
Last Friday 8 July 2016, as part of this year's Turing Lectures on The Intersection of Mathematics, Statistics and Computation, Mark Girolami gave a lecture on “Probabilistic Numerical Computation: A New Concept?”, containing some of our joint work on probabilistic numerics. The video can be found on the Alan Turing Institute's Youtube channel.
Time and Place. Tuesday 14 June 2016, 12:15–13:15, ZIB Seminar Room 2006, Zuse Institute Berlin, Takustraße 7, 14195 Berlin
Abstract. In an ongoing push to construct probabilistic extensions of classic ODE solvers for application in statistics and machine learning, two recent papers have provided distinct methods that return probability measures instead of point estimates, based on sampling and filtering respectively. While both approaches leverage classical numerical analysis, by building on well-studied solutions of existing seminal solvers, the different constructions of probability measures strike a divergent balance between a formal quantification of epistemic uncertainty and a low computational overhead.
On the one hand, Conrad et al. proposed to randomise existing non-probabilistic one-step solvers by adding suitably scaled Gaussian noise after every step and thereby inducing a probability measure over the solution space of the ODE which contracts to a Dirac measure on the true unknown solution in the order of convergence of the underlying classic numerical method. But the computational cost of these methods is significantly above that of classic solvers.
On the other hand, Schober et al. recast the estimation of the solution as state estimation by a Gaussian (Kalman) filter and proved that employing a integrated Wiener process prior returns a posterior Gaussian process whose maximum likelihood (ML) estimate matches the solution of classic Runge–Kutta methods. In an attempt to amend this method's rough uncertainty calibration while sustaining its negligible cost overhead, we propose a novel way to quantify uncertainty in this filtering framework by probing the gradient using Bayesian quadrature.
Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” to the arXiv. This paper forms part of the push for probabilistic numerics in scientific computing.
Abstract. This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.