09:00 to 09:50 Registration 09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) 10:00 to 11:00 Ilya Mandel (University of Birmingham)Studying black holes with gravitational waves: Why GW astronomy needs you! Following the first direct observation of gravitational waves from a pair of merging black holes in September 2015, we are now entering the era of gravitational-wave astronomy, where gravitational waves are increasingly being used as a tool to explore topics ranging from astronomy (stellar and binary evolution) to fundamental physics (tests of the general theory of relativity). Future progress depends on addressing several key problems in statistical inference on gravitational-wave observations, including (i) rapidly growing computational cost for future instruments; (ii) noise characterisation; (iii) model systematics; and (iv) model selection. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:00 Paul Constantine (University of Colorado)Subspace-based dimension reduction for forward and inverse uncertainty quantification Many methods in uncertainty quantification suffer from the curse of dimensionality. I will discuss several approaches for identifying exploitable low-dimensional structure---e.g., active subspaces or likelihood-informed subspaces---that enable otherwise infeasible forward and inverse uncertainty quantification.Related Linkshttp://activesubspaces.org - Active subspaces INI 1 12:00 to 13:30 Buffet Lunch at INI 13:30 to 14:30 Christoph Schwab (ETH Zürich)Deterministic Multilevel Methods for Forward and Inverse UQ in PDEs We present the numerical analysis of Quasi Monte-Carlo methods for high-dimensional integration applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Emphasis will be placed on the role of parametric holomorphy of data-to-solution maps. We present corresponding results on deterministic quadratures in Bayesian Inversion of parametric PDEs, and the related bound on posterior sparsity and (dimension-independent) QMC convergence rates. Particular attention will be placed on Higher-Order QMC, and on the interplay between the structure of the representation system of the distributed uncertain input data (KL, splines, wavelets,...) and the structure of QMC weights. We also review stable and efficient generation of interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element PDE discretizations with applications to forward and inverse computational uncertainty quantification. QMC convergence rates will be compared with those afforded by Smolyak quadrature. Joint work with Robert Gantner and Lukas Herrmann and Jakob Zech (SAM, ETH) and Josef Dick, Thong LeGia and Frances Kuo (Sydney). References: [1] R. N. Gantner and L. Herrmann and Ch. Schwab Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights, SIAM J. Numer. Analysis, 56/1 (2018), pp. 111-135. [2] J. Dick and R. N. Gantner and Q. T. Le Gia and Ch. Schwab Multilevel higher-order quasi-Monte Carlo Bayesian estimation, Math. Mod. Meth. Appl. Sci., 27/5 (2017), pp. 953-995. [3] R. N. Gantner and M. D. Peters Higher Order Quasi-Monte Carlo for Bayesian Shape Inversion, accepted (2018) SINUM, SAM Report 2016-42. [4] J. Dick and Q. T. Le Gia and Ch. Schwab Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations, SIAM Journ. Uncertainty Quantification, 4/1 (2016), pp. 48-79 [5] J. Dick and F.Y. Kuo and Q.T. LeGia and Ch. Schwab Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J. Numer. Anal., 54/4 (2016), pp. 2541-2568 INI 1 14:30 to 15:00 Richard Nickl (University of Cambridge)Statistical guarantees for Bayesian uncertainty quantification in inverse problems We discuss recent results in mathematical statistics that provide objective statistical guarantees for Bayesian algorithms in (possibly non-linear) noisy inverse problems. We focus in particular on the justification of Bayesian credible sets as proper frequentist confidence sets in the small noise limit via so-called Bernstein - von Mises theorems', which provide Gaussian approximations to the posterior distribution, and introduce notions of such theorems in the infinite-dimensional settings relevant for inverse problems. We discuss in detail such a Bernstein-von Mises result for Bayesian inference on the unknown potential in the Schroedinger equation from an observation of the solution of that PDE corrupted by additive Gaussian white noise. See https://arxiv.org/abs/1707.01764 and also https://arxiv.org/abs/1708.06332 INI 1 15:00 to 15:30 Robert Scheichl (University of Bath)Low-rank tensor approximation for sampling high dimensional distributions High-dimensional distributions are notoriously difficult to sample from, particularly in the context of PDE-constrained inverse problems. In this talk, we will present general purpose samplers based on low-rank tensor surrogates in the tensor-train (TT) format, a methodology that has been exploited already for many years for scalable, high-dimensional function approximations in quantum chemistry. In the Bayesian context, the TT surrogate is built in a two stage process. First we build a surrogate of the entire PDE solution in the TT format, using a novel combination of alternating least squares and the TT cross algorithm. It exploits and preserves the block diagonal structure of the discretised operator in stochastic collocation schemes, requiring only independent PDE solutions at a few parameter values, thus allowing the use of existing high performance PDE solvers. In a second stage, we approximate the high-dimensional posterior density function also in TT format. Due to the particular structure of the TT surrogate, we can build an efficient conditional distribution method (or Rosenblatt transform) that only requires a sampling algorithm for one-dimensional conditionals. This conditional distribution method can also be used for other high-dimensional distributions, not necessarily coming from a PDE-constrained inverse problem. The overall computational cost and storage requirements of the sampler grow linearly with the dimension. For sufficiently smooth distributions, the ranks required for accurate TT approximations are moderate, leading to significant computational gains. We compare our new sampling method with established methods, such as the delayed rejection adaptive Metropolis (DRAM) algorithm, as well as with multilevel quasi-Monte Carlo ratio estimators. This is joint work with Sergey Dolgov (Bath), Colin Fox (Otago) and Karim Anaya-Izquierdo (Bath). INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Panel Discussion: Establishing UQ Challenge Problems & Supporting Infrastructure INI 1 17:00 to 18:00 Welcome Wine Reception at INI
 09:00 to 10:00 Youssef Marzouk (Massachusetts Institute of Technology)Optimal Bayesian experimental design: focused objectives and observation selection strategies I will discuss two complementary efforts in Bayesian optimal experimental design for inverse problems. The first focuses on evaluating an experimental design objective: we describe a new computational approach for focused'' optimal Bayesian experimental design with nonlinear models, with the goal of maximizing expected information gain in targeted subsets of model parameters. Our approach considers uncertainty in the full set of model parameters, but employs a design objective that can exploit learning trade-offs among different parameter subsets. We introduce a layered multiple importance sampling scheme that provides consistent estimates of expected information gain in this focused setting, with significant reductions in estimator bias and variance for a given computational effort. The second effort focuses on optimization of information theoretic design objectives---in particular, from the combinatorial perspective of observation selection. Given many potential experiments, one may wish to choose a most informative subset thereof. Even if the data have in principle been collected, practical constraints on storage, communication, and computational costs may limit the number of observations that one wishes to employ. We introduce methods for selecting near-optimal subsets of the data under cardinality constraints. Our methods exploit the structure of linear inverse problems in the Bayesian setting, and can be efficiently implemented using low-rank approximations and greedy strategies based on modular bounds. This is joint work with Chi Feng and Jayanth Jagalur-Mohan. INI 1 10:00 to 10:30 Claudia Schillings (Universität Mannheim)On the Convergence of Laplace's Approximation and Its Implications for Bayesian Computation Sampling methods for Bayesian inference show numerical instabilities in the case of concentrated posterior distributions. However, the concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. In this talk, we will discuss convergence results of Laplace’s approximation and analyze the use of the approximation within sampling methods. This is joint work with Bjoern Sprungk (U Goettingen) and Philipp Wacker (FAU Erlangen). INI 1 10:30 to 11:00 Morning Coffee 11:00 to 11:30 Hugo Maruri-Aguilar (Queen Mary University of London)Smooth metamodels Smooth supersaturated polynomials (Bates et al., 2014) have been proposed for emulating computer experiments. These models are simple to interpret and have spline-like properties inherited from the Sobolev-type of smoothing which is at the core of the method. An implementation of these models is available in the R package ssm. This talk aims to describe the method as well as discuss designs that could be performed with the help of smooth models. To illustrate the methodology, we use data from the fan blade assembly. This is joint work with H Wynn, R Bates and P Curtis. INI 1 11:30 to 12:00 Michael Goldstein (Durham University)Inverting the Pareto Boundary: Bayes linear decision support with a soft constraint We consider problems of decision support based around computer simulators, where we must take into account a soft constraint on our decision choices. This leads to the problem of identifying and inverting the Pareto boundary for the decision. We show how Bayes linear methods may be used for this purpose and how the sensitivity of the decision choices may be quantified and explored. The approach is illustrated with a problem on wind farm construction. This is joint work with Hailiang Du. INI 1 12:00 to 13:30 Buffet Lunch at INI 13:30 to 14:30 Andrew Stuart (University of Warwick)Large Graph Limits of Learning Algorithms Many problems in machine learning require the classification of high dimensional data. One methodology to approach such problems is to construct a graph whose vertices are identified with data points, with edges weighted according to some measure of affinity between the data points. Algorithms such as spectral clustering, probit classification and the Bayesian level set method can all be applied in this setting. The goal of the talk is to describe these algorithms for classification, and analyze them in the limit of large data sets. Doing so leads to interesting problems in the calculus of variations, Bayesian inverse problems and in Monte Carlo Markov Chain, all of which will be highlighted in the talk. These limiting problems give insight into the structure of the classification problem, and algorithms for it.     Collaboration with:   Andrea Bertozzi (UCLA) Michael Luo (UCLA) Kostas Zygalakis (Edinburgh) https://arxiv.org/abs/1703.08816   and   Matt Dunlop (Caltech) Dejan Slepcev (CMU) Matt Thorpe (Cambridge) (forthcoming paper) INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Tim Sullivan (Freie Universität Berlin); (Konrad-Zuse-Zentrum für Informationstechnik Berlin)Bayesian probabilistic numerical methods In this work, numerical computation - such as numerical solution of a PDE - is treated as a statistical inverse problem in its own right. The popular Bayesian approach to inversion is considered, wherein a posterior distribution is induced over the object of interest by conditioning a prior distribution on the same finite information that would be used in a classical numerical method. The main technical consideration is that the data in this context are non-random and thus the standard Bayes' theorem does not hold. General conditions will be presented under which such Bayesian probabilistic numerical methods are well-posed, and a sequential Monte-Carlo method will be shown to provide consistent estimation of the posterior. The paradigm is extended to computational `pipelines'', through which a distributional quantification of numerical error can be propagated. A sufficient condition is presented for when such propagation can be endowed with a globally coherent Bayesian interpretation, based on a novel class of probabilistic graphical models designed to represent a computational work-flow. The concepts are illustrated through explicit numerical experiments involving both linear and non-linear PDE models. This is joint work with Jon Cockayne, Chris Oates, and Mark Girolami. Further details are available in the preprint arXiv:1702.03673. INI 1 16:00 to 17:00 Poster Session