Stability of Relative Equilibria in the Planar N-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.Comment: 23 pages, 3 figure

Optimal scaling for partially updating MCMC algorithms

In this paper we shall consider optimal scaling problems for high-dimensional Metropolis--Hastings algorithms where updates can be chosen to be lower dimensional than the target density itself. We find that the optimal scaling rule for the Metropolis algorithm, which tunes the overall algorithm acceptance rate to be 0.234, holds for the so-called Metropolis-within-Gibbs algorithm as well. Furthermore, the optimal efficiency obtainable is independent of the dimensionality of the update rule. This has important implications for the MCMC practitioner since high-dimensional updates are generally computationally more demanding, so that lower-dimensional updates are therefore to be preferred. Similar results with rather different conclusions are given for so-called Langevin updates. In this case, it is found that high-dimensional updates are frequently most efficient, even taking into account computing costs.Comment: Published at http://dx.doi.org/10.1214/105051605000000791 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A piecewise deterministic scaling limit of Lifted Metropolis-Hastings in the Curie-Weiss model

In Turitsyn, Chertkov, Vucelja (2011) a non-reversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals $n^{1/2}$ for LMH, which should be compared to $n$ for MH. At the critical temperature the required jump rate equals $n^{3/4}$ for LMH and $n^{3/2}$ for MH, in agreement with experimental results of Turitsyn, Chertkov, Vucelja (2011). The scaling limit of LMH turns out to be a non-reversible piecewise deterministic exponentially ergodic `zig-zag' Markov process

SET for success : the supply of people with science, technology, engineering and mathematics skills : the report of Sir Gareth Roberts' review

In March 2001, Sir Gareth Roberts was asked by the Chancellor of the Exchequer and the Secretaries of State at the Department of Trade and Industry and at the Department for Education and Skills to undertake a review into the supply of science and engineering skills in the UK. The review was commissioned as part of the Government's productivity and innovation strategy. Sir Gareth Roberts' final report was published on 15 April. The report sets out a series of recommendations to the Government, employers and others with an interest in fostering science, engineering and innovation in the UK. The Government is currently considering Sir Gareth's report and recommendations. The full report is available below in Adobe Acrobat Portable Document Format (PDF). If you do not have Adobe Acrobat installed on your computer you can download the software free of charge from the Adobe website

Scalable Importance Tempering and Bayesian Variable Selection

We propose a Monte Carlo algorithm to sample from high dimensional probability distributions that combines Markov chain Monte Carlo and importance sampling. We provide a careful theoretical analysis, including guarantees on robustness to high dimensionality, explicit comparison with standard Markov chain Monte Carlo methods and illustrations of the potential improvements in efficiency. Simple and concrete intuition is provided for when the novel scheme is expected to outperform standard schemes. When applied to Bayesian variable-selection problems, the novel algorithm is orders of magnitude more efficient than available alternative sampling schemes and enables fast and reliable fully Bayesian inferences with tens of thousand regressors.Comment: Online supplement not include