Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples

Long term dynamics of stochastic evolution equations

Stochastic differential equations with delay are the inspiration for this thesis. Examples of such equations arise in population models, control systems with delay and noise, lasers, economical models, neural networks, environmental pollution and in many other situations. In such models we are often interested in the evolution of a particular quantity, for example the size of a population, or the amount of pollution in a particular area, changing in time. A differential equation with delay, or delay equation, is a differential equation in which the change in time of such a quantity is expressed as a function of the value of that quantity at different points in time, in the past as well as in the present. This is in contrast with an ordinary differential equation, in which the change in time of the quantity at a specific time is expressed as a function of that quantity at that specific time only.UBL - phd migration 201

Long term dynamics of stochastic evolution equations

Stochastic differential equations with delay are the inspiration for this thesis. Examples of such equations arise in population models, control systems with delay and noise, lasers, economical models, neural networks, environmental pollution and in many other situations. In such models we are often interested in the evolution of a particular quantity, for example the size of a population, or the amount of pollution in a particular area, changing in time. A differential equation with delay, or delay equation, is a differential equation in which the change in time of such a quantity is expressed as a function of the value of that quantity at different points in time, in the past as well as in the present. This is in contrast with an ordinary differential equation, in which the change in time of the quantity at a specific time is expressed as a function of that quantity at that specific time only

Adaptive schemes for piecewise deterministic Monte Carlo algorithms

The Bouncy Particle sampler (BPS) and the Zig-Zag sampler (ZZS) are continuous time, non-reversible Monte Carlo methods based on piecewise deterministic Markov processes. Experiments show that the speed of convergence of these samplers can be affected by the shape of the target distribution, as for instance in the case of anisotropic targets. We propose an adaptive scheme that iteratively learns all or part of the covariance matrix of the target and takes advantage of the obtained information to modify the underlying process with the aim of increasing the speed of convergence. Moreover, we define an adaptive scheme that automatically tunes the refreshment rate of the BPS or ZZS. We prove ergodicity and a law of large numbers for all the proposed adaptive algorithms. Finally, we show the benefits of the adaptive samplers with several numerical simulations.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Statistic

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

In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible 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 n1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n3/4 for LMH and n3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.Statistic

Infinite dimensional Piecewise Deterministic Markov Processes

In this paper we aim to construct infinite dimensional versions of well established Piecewise Deterministic Monte Carlo methods, such as the Bouncy Particle Sampler, the Zig-Zag Sampler and the Boomerang Sampler. In order to do so we provide an abstract infinite dimensional framework for Piecewise Deterministic Markov Processes (PDMPs) with unbounded event intensities. We further develop exponential convergence to equilibrium of the infinite dimensional Boomerang Sampler, using hypocoercivity techniques. Furthermore we establish how the infinite dimensional Boomerang Sampler admits a finite dimensional approximation, rendering it suitable for computer simulation.Statistic

Limit theorems for the zig-zag process

Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis-Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.Accepted author manuscriptStatistic

Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds

Contains fulltext : 167916.pdf (preprint version ) (Open Access)32 p

The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.Statistic