Welcome to the webpage of the Warwick SIAM Student Chapter's conference on "High Dimensional Inference and Monte Carlo Techniques".
Professor George Deligiannidis, Jesus College, Oxford, Title: The Bouncy particle sampler and Randomized Hamiltonian Monte Carlo
Professor Gareth Roberts, University of Warwick, Title: An introduction to non-reversible MCMC and Piecewise Deterministic Markov processes
Dr Samuel Livingstone, UCL, Title: Ergodicity and kinetic energy choice in Hamiltonian Monte Carlo
Dr Murray Pollock, University of Warwick, Title: "Confusion"
Jonas Latz, TUM, Title: Fast sampling of parameterised Gaussian random fields
10:30-11 Registration: MRC, opposite the mathematics common, first floor
11-12 Gareth Roberts B3.02
12-13 George Deligiannidis B3.02
13-14 Lunch Break Maths Common Room
14-15 Jonas Latz MS.05
15-16 Samuel Livingstone MS.05
16:-16:30 Coffee Break Maths Common Room
16:30-17:30 Murray Pollock MS.05
17:30- ? Wine and Cheese Maths Common Room
University of Warwick, Zeeman Building, MS0.3 and MS0.5. Wednesday, 23rd of January 2019, Beginning 11, Registration 10:30
Professor George Deligiannidis
I will discuss some recent results on PDMCMC and in particular on the Bouncy Particle Sampler (BPS). In particular I will show that as the dimension of the target grows, at least for targets that factorise or satisfy some other weak dependence assumption, any finite collection of location and momentum coordinates of BPS converge weakly to the corresponding Randomized Hamiltonian Monte Carlo (RHMC). This is essentially a piecewise deterministic version of the well known HMC algorithm and therefore this establishes a close link between BPS and Hamiltonian dynamics. Next, I will go through some methods for obtaining dimension free convergence rates for RHMC and discuss its implications on the computational cost of BPS. This is joint work with D. Paulin, A. Bouchard-Côté and A. Doucet.
Dr Samuel Livingstone
I will discuss Hamiltonian Monte Carlo, and in particular focus on two areas: conditions under which the method will produce a geometrically ergodic Markov chain, and how algorithm behaviour is affected by the choice of kinetic energy. I will review some recent work by myself and co-authors, as well as related work by others in the area, with the aim of giving a good overall picture of the current state of knowledge. Some related papers are https://arxiv.org/abs/1601.08057 and https://arxiv.org/abs/1706.02649.
Gaussian random fields are popular models for spatially varying uncertainties, arising for instance in geotechnical engineering, hydrology or image processing. A Gaussian random field is fully characterised by its mean function and covariance operator. In more complex models these can also be partially unknown. In this case we need to handle a family of Gaussian random fields indexed with hyperparameters. Sampling for a fixed configuration of hyperparameters is already very expensive due to the nonlocal nature of many classical covariance operators. Sampling from multiple configurations increases the total computational cost severely. In this report we employ parameterised Karhunen-Lo\'eve expansions for sampling. To reduce the cost we construct a reduced basis surrogate built from snapshots of Karhunen-Lo\'eve eigenvectors. In particular, we consider Mat\'ern-type covariance operators with unknown correlation length and standard deviation. We suggest a linearisation of the covariance function and describe the associated online-offline decomposition. In numerical experiments we investigate the approximation error of the reduced eigenpairs. As an application we consider forward uncertainty propagation and Bayesian inversion with an elliptic partial differential equation where the logarithm of the diffusion coefficient is a parameterised Gaussian random field. In the Bayesian inverse problem we employ Markov chain Monte Carlo on the reduced space to generate samples from the posterior measure. All numerical experiments are conducted in 2D physical space, with non-separable covariance operators, and finite element grids with ~ 1E4 degrees of freedom.