Measuring the irreversibility of numerical schemes for reversible stochastic differential equations

Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th

Universality of phonon transport in surface-roughness dominated nanowires

We analyze, both theoretically and numerically, the temperature dependent thermal conductivity \k{appa} of two-dimensional nanowires with surface roughness. Although each sample is characterized by three independent parameters - the diameter (width) of the wire, the correlation length and strength of the surface corrugation - our theory predicts that there exists a universal regime where \k{appa} is a function of a single combination of all three model parameters. Numerical simulations of propagation of acoustic phonons across thin wires confirm this universality and predict a d 1/2 dependence of \k{appa} on the diameter d

Supervised exercise training and increased physical activity to reduce cardiovascular disease risk in women with polycystic ovary syndrome: Study protocol for a randomized controlled feasibility trial

Background: Polycystic ovary syndrome (PCOS) affects up to 20% of women and is characterised by higher amounts of visceral fat, obesity, insulin resistance, dyslipidemia and reproductive and cardiometabolic complications. Increased oxidised low-density lipoprotein (LDL) concentrations have been associated with an increased risk of cardiovascular disease (CVD)-related events. Oxidised LDL is rarely used as a marker for CVD risk in PCOS-related studies despite its widely accepted role in atherogenesis and the increased risk factors associated with PCOS. Additionally, prolonged periods of sedentary behaviour can negatively affect metabolic health. No studies have specifically examined the effects of reducing sedentary behaviour on CVD risk in PCOS with a lifestyle physical activity intervention. The aim of the current study is to measure the feasibility of a randomised controlled trial (RCT) examining the effects of supervised exercise and reducing sedentary behaviour in women with PCOS on CVD risk. Methods/design: A feasibility, exploratory RCT will be conducted. Fifty-one pre-menopausal females will be randomly allocated between an exercise group (EG), a lifestyle physical activity group (LPAG) and a control group. Participants in the EG will undertake a 12-week supervised aerobic exercise programme. The LPAG will aim to increase daily physical activity and reduce sedentary behaviour for 12 weeks. The control group will not take part in any intervention. Primary outcomes are feasibility and acceptability of the intervention and procedures. Secondary outcomes are oxidised LDL, aerobic fitness, blood lipid profile, fasting glucose and insulin, testosterone and inflammatory markers. Discussion: PCOS is associated with various increased risk factors for CVD, including hypertension, dyslipidemia, abdominal obesity, insulin resistance, and inflammation. Whether oxidised LDL has a role in this increased risk is not yet known. The present study aims to measure the feasibility of implementing structured exercise training and/or increased lifestyle physical activity in women with PCOS, so that a subsequent adequately powered RCT can be designed. The results from the study will be used to refine the interventions and determine the acceptability of the study design. A limitation is that some self-monitoring in the lifestyle physical activity group may not be reliable or replicable, for example inputting information about time spent cleaning/gardening

A parallel Jacobson-Oksman optimization algorithm

A gradient-dependent optimization technique which exploits the vector-streaming or parallel-computing capabilities of some modern computers is presented. The algorithm, derived by assuming that the function to be minimized is homogeneous, is a modification of the Jacobson-Oksman serial minimization method. In addition to describing the algorithm, conditions insuring the convergence of the iterates of the algorithm and the results of numerical experiments on a group of sample test functions are presented. The results of these experiments indicate that this algorithm will solve optimization problems in less computing time than conventional serial methods on machines having vector-streaming or parallel-computing capabilities

Spatial multi-level interacting particle simulations and information theory-based error quantification

We propose a hierarchy of multi-level kinetic Monte Carlo methods for sampling high-dimensional, stochastic lattice particle dynamics with complex interactions. The method is based on the efficient coupling of different spatial resolution levels, taking advantage of the low sampling cost in a coarse space and by developing local reconstruction strategies from coarse-grained dynamics. Microscopic reconstruction corrects possibly significant errors introduced through coarse-graining, leading to the controlled-error approximation of the sampled stochastic process. In this manner, the proposed multi-level algorithm overcomes known shortcomings of coarse-graining of particle systems with complex interactions such as combined long and short-range particle interactions and/or complex lattice geometries. Specifically, we provide error analysis for the approximation of long-time stationary dynamics in terms of relative entropy and prove that information loss in the multi-level methods is growing linearly in time, which in turn implies that an appropriate observable in the stationary regime is the information loss of the path measures per unit time. We show that this observable can be either estimated a priori, or it can be tracked computationally a posteriori in the course of a simulation. The stationary regime is of critical importance to molecular simulations as it is relevant to long-time sampling, obtaining phase diagrams and in studying metastability properties of high-dimensional complex systems. Finally, the multi-level nature of the method provides flexibility in combining rejection-free and null-event implementations, generating a hierarchy of algorithms with an adjustable number of rejections that includes well-known rejection-free and null-event algorithms.Comment: 34 page