Dr. Philipp Hennig, Max Planck Institute for Intelligent Systems, Tübingen.

Title: Numerical Computation as Inference

To set the scene for the workshop, I will briefly (and superficially) outline the general connection between statistical inference and numerical computation: Because numerical methods estimate an unknown quantity from tractable computations ("observations"), they can be cast as inference rules. Several elementary numerical methods, from linear algebra to quadrature to nonlinear optimization, can be identified precisely with probabilistic inference under specific Gaussian probabilistic models. One aim of this workshop is to discuss and clarify this relationship, and the theoretical and practical opportunities arising from it, in the area of differential equations.

Prof. Martin Hairer, University of Warwick.

Title: Numerical approximation of singular SPDEs

Dr. Catherine Powell, University of Manchester.

Title: Energy Norm A Posteriori Error Estimation for Elliptic PDEs with Uncertain Data

Abstract: Stochastic Galerkin approximation is a popular approach for the numerical solution of elliptic PDEs with correlated random data. A typical strategy is to combine conventional (h-type) finite element approximation on the spatial domain with spectral (p-type) approximation on a finite-dimensional manifold in the (stochastic) parameter domain. In this talk, we focus on the efficient solution of the forward problem only and assume that the diffusion coefficient is represented as a linear expansion involving infinitely many independent random variables (with assumed distributions). This expansion is not truncated a priori.

A novel a posteriori estimator for the energy error that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the Galerkin approximation space. This estimator is provably reliable and efficient. Moreover, two of its components can be used to establish two-sided estimates of the energy error reduction that would be achieved by enriching either the finite element part or the polynomial part of the approximation space. Since the choice of the polynomial part determines which random variables are actually incorporated into the numerical solution, the estimator may be used to drive an adaptive solver which 'learns' which of the variables are (in the energy norm sense) the most important.

This is joint work with Alex Bespalov (Birmingham) and David Silvester (Manchester)

13:00

Dr. Søren Hauberg, Technical University of Denmark.

Title: Measuring with no tape (or a diffuse problem, a hack, and some pretty pictures)

Abstract: Statistics rely on measured distances! Often the distance function is itself estimated from data (e.g. Mahalanobis' distance), and should, as a consequence, be treated as a stochastic variable. We consider computing geodesics (shortest paths) under a Riemannian metric estimated from noisy data, i.e. a "random Riemannian metric". When the Riemannian metric is deterministic, geodesics are governed by a system of ordinary differential equations (ODEs). When the metric is stochastic we can only make "noisy evaluations" of this ODE, and the solution geodesics will be stochastic objects. We approximate the distribution of the solution geodesics with a Gaussian Process, and use probabilistic numerics for estimating this distribution. As a guiding example, we consider estimating tracts in diffusion weighted MRI brain scans.

This is work in progress and in collaboration with Michael Schober (MPI-IS), Philipp Hennig (MPI-IS), Aasa Feragen (DIKU), Niklas Kasenburg (DIKU), Matt Liptrot (DIKU/DTU).

14:00

Coffee break with cakes.

14:30

Dr. Patrick Conrad, University of Warwick.

Title: Probability Measures on Numerical Solutions of ODEs and PDEs for Uncertainty Quantification and Inference

Abstract: Deterministic ODE and PDE solvers are widely used, but characterizing discretization error in numerical solutions within a coherent statistical framework is challenging. We successfully address this problem by constructing a probability measure over functions consistent with the solution that provably contracts to a Dirac measure on the unique solution at rates determined by an underlying deterministic solver. These measures can be used to rigorously include effects of discretization error into statistical analyses, such as inference, where discretization can be an important source of statistical bias. The effect of the measures on statistical inference and uncertainty propagation are illustrated with example systems, including a large scale shallow water component of a global climate model.

15:30