On the ergodicity properties of some adaptive MCMC algorithms

In this paper we study the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms (MCMC) that have been recently proposed in the literature. We prove that under a set of verifiable conditions, ergodic averages calculated from the output of a so-called adaptive MCMC sampler converge to the required value and can even, under more stringent assumptions, satisfy a central limit theorem. We prove that the conditions required are satisfied for the independent Metropolis--Hastings algorithm and the random walk Metropolis algorithm with symmetric increments. Finally, we propose an application of these results to the case where the proposal distribution of the Metropolis--Hastings update is a mixture of distributions from a curved exponential family.Comment: Published at http://dx.doi.org/10.1214/105051606000000286 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$

Transverse exponential stability and applications

We investigate how the following properties are related to each other: i)-A manifold is "transversally" exponentially stable; ii)-The "transverse" linearization along any solution in the manifold is exponentially stable; iii)-There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property

Effect of a heterogeneous distribution of particles on the formation of banded grain structure in wrought Alloy 718

Alloy 718 is known to be sensitive to interdendritic segregation formed during ingot solidification. The occurrence of banded grain structures under heat treating conditions close to 1000 ° C related to interdendritic segregation is often reported. In order to have a better understanding of this microstructural evolution, an extensive experimental program has been carried out. Consequently, a model taking into account the selective dissolution of δ-phase (Ni3Nb) is proposed. A grain growth simulation by Monte-Carlo method is then used to illustrate the grain structure evolution in a banded particle distribution. By comparing experimental data and computer simulation, the relationship between the Monte-Carlo step and the real time is assessed and the range of parameters when heterogeneous microstructures appear is specified

Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism

When designing observers for nonlinear systems, the dynamics of the given system and of the designed observer are usually not expressed in the same coordinates or even have states evolving in different spaces. In general, the function, denoted $\tau$ (or its inverse, denoted $\tau^*$) giving one state in terms of the other is not explicitly known and this creates implementation issues. We propose to round this problem by expressing the observer dynamics in the the same coordinates as the given system. But this may impose to add extra coordinates, problem that we call augmentation. This may also impose to modify the domain or the range of the augmented" $\tau$ or $\tau^*$, problem that we call extension. We show that the augmentation problem can be solved partly by a continuous completion of a free family of vectors and that the extension problem can be solved by a function extension making the image of the extended function the whole space. We also show how augmentation and extension can be done without modifying the observer dynamics and therefore with maintaining convergence.Several examples illustrate our results.Comment: Submitted for publication in SIAM Journal of Control and Optimizatio

Experimental data about mechanical behaviour during compression tests for various matted fibres

A specific experimental device has been set up to test compressive mechanical behaviour of an assembly of fibres. Simple compression, as well as cyclic loading experiments and relaxation tests were performed. The experimental set up also allows to record the evolution of the mat fibre electrical resistance while testing. Experimental results are presented for a variety of fibrous materials. Despite the very different nature of each of these individual fibres, it appears that the mats exhibit a very similar mechanical behaviour. This common behaviour has been observed during monotonic single compression tests, as well as during cyclic or relaxation experiments. These experimental results are discussed in terms of different parameters such as the intrinsic mechanical properties of individual fibres and moreover the tangle intrinsic parameters (effect of fibre length, effect of geometrical position of fibres in the sample, fibre surface modifications. . .). The influence of the contact points between fibres is discussed in regard of the electric resistivity measurement

Quantitative convergence rates for sub-geometric Markov chains

We provide explicit expressions for the constants involved in the characterisation of ergodicity of sub-geometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The result is fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our result accommodates also some classes of inhomogeneous chains.Comment: 14 page