Gibbs flow for approximate transport with applications to Bayesian computation

Let $\pi_{0}$ and $\pi_{1}$ be two distributions on the Borel space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. Any measurable function $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $Y=T(X)\sim\pi_{1}$ if $X\sim\pi_{0}$ is called a transport map from $\pi_{0}$ to $\pi_{1}$. For any $\pi_{0}$ and $\pi_{1}$, if one could obtain an analytical expression for a transport map from $\pi_{0}$ to $\pi_{1}$, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution $\pi_{0}$ to the target distribution $\pi_{1}$ using this transport map. Although it is usually impossible to obtain an explicit transport map for complex target distributions, we show here how to build a tractable approximation of a novel transport map. This is achieved by moving samples from $\pi_{0}$ using an ordinary differential equation with a velocity field that depends on the full conditional distributions of the target. Even when this ordinary differential equation is time-discretized and the full conditional distributions are numerically approximated, the resulting distribution of mapped samples can be efficiently evaluated and used as a proposal within sequential Monte Carlo samplers. We demonstrate significant gains over state-of-the-art sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.Comment: Significantly revised with new methodology and numerical example

Energetics and switching of quasi-uniform states in small ferromagnetic particles

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size

Remarks on drift estimation for diffusion processes

In applications such as molecular dynamics it is of interest to fit Smoluchowski and Langevin equations to data. Practitioners often achieve this by a variety of seemingly ad hoc procedures such as fitting to the empirical measure generated by the data, and fitting to properties of auto-correlation functions. Statisticians, on the other hand, often use estimation procedures which fit diffusion processes to data by applying the maximum likelihood principle to the path-space density of the desired model equations, and through knowledge of the properties of quadratic variation. In this note we show that these procedures used by practitioners and statisticians to fit drift functions are, in fact, closely related and can be thought of as two alternative ways to regularize the (singular) likelihood function for the drift. We also present the results of numerical experiments which probe the relative efficacy of the two approaches to model identification and compare them with other methods such as the minimum distance estimator

Fitting stochastic differential equations to molecular dynamics data

The thesis consists of three main parts. Firstly, a molecular dynamics and potential energy minimisation package that has been implemented is described in detail All potential and force interactions are described and tested successfully. Compound tests on minimal energies for clusters of water molecules, the radial distribution function for liquid argon and the equilibrium distribution for the dihedral angle in Butane under Langevin dynamics are performed and the presence of multiple time scales is noted for Butane as well as for a simplified protein model due to Grubmiiller and Tavan. Secondly, fitting stochastic differential equations (SDEs) to time series is studied. Initially, I consider the well-understood case of non-degenerate diffusions, where all components of the process are driven directly by Brownian motion. An SDE with constant diffusivity and trigonometric force expression is fitted to trajectories obtained from simulations of Butane by maximum likelihood methods and fitted diffusion and drift parameters depend strongly on the timescale considered. Hypoelliptic diffusion processes are considered next. Here, the unexpected failure of simple estimators necessitates the use of carefully chosen approximate likelihoods. For the case of only partial observations being available, a compound algorithm is designed and numerically seen to be asymptotically consistent. It is applied to the same Butane sample path and found to equilibrate, although the fitted SDE fails to reproduce the free energy landscape Thirdly, connections between maximum likelihood estimators (MLEs) and practitioners’ methods are investigated. Analytical links are found for reversible processes and for second order Langevin processes. In the case of ID processes, MLE and practitioners’ methods for the drift are found to yield estimators identical up to lower order terms even for finite times of observation

Drift Models on Complex Projective Space for Electron-Nuclear Double Resonance

ENDOR spectroscopy is an important tool to determine the complicated three-dimensional structure of biomolecules and in particular enables measurements of intramolecular distances. Usually, spectra are determined by averaging the data matrix, which does not take into account the significant thermal drifts that occur in the measurement process. In contrast, we present an asymptotic analysis for the homoscedastic drift model, a pioneering parametric model that achieves striking model fits in practice and allows both hypothesis testing and confidence intervals for spectra. The ENDOR spectrum and an orthogonal component are modeled as an element of complex projective space, and formulated in the framework of generalized Fr\'echet means. To this end, two general formulations of strong consistency for set-valued Fr\'echet means are extended and subsequently applied to the homoscedastic drift model to prove strong consistency. Building on this, central limit theorems for the ENDOR spectrum are shown. Furthermore, we extend applicability by taking into account a phase noise contribution leading to the heteroscedastic drift model. Both drift models offer improved signal-to-noise ratio over pre-existing models.Comment: 68 pages, 10 figure

Fitting stochastic differential equations to molecular dynamics data

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Non parametric Bayesian drift estimation for one-dimensional diffusion processes

We consider diffusions on the circle and establish a Bayesian estimator for the drift function based on observing the local time and using Gaussian priors. Given a standard Girsanov likelihood, we prove that the procedure is well-defined and that the posterior enjoys robustness against small deviations of the local time. A simple method for estimating the local time from high-frequency discrete time observations yielding control of the L2 error is proposed. Complemented by a finite element implementation this enables error-control for a fixed random sample all the way from high-frequency discrete observation to the numerical computation of the posterior mean and covariance. An empirical Bayes procedure is suggested which allows automatic selection of the smoothness of the prior in a given family. Some numerical experiments extend our observations to subsets of the real line other than circles and exhibit more probabilistic convergence properties such as rates of posterior contraction

Inference for partially observed Riemannian Ornstein-Uhlenbeck diffusions of covariance matrices

We construct a generalization of the Ornstein-Uhlenbeck processes on the cone of covariance matrices endowed with the Log-Euclidean and the Affine-Invariant metrics. Our development exploits the Riemannian geometric structure of symmetric positive definite matrices viewed as a differential manifold. We then provide Bayesian inference for discretely observed diffusion processes of covariance matrices based on an MCMC algorithm built with the help of a novel diffusion bridge sampler accounting for the geometric structure. Our proposed algorithm is illustrated with a real data financial application.Comment: 39 pages, 14 figures and 3 table