finite element method for epitaxial growth with thermodynamic boundary conditions we develop an adaptive finite element method for island dynamics in epitaxial growth we study a step flow model which consists of an adatom adsorbed atom diffusion equation on terraces of different height thermodynamic boundary conditions on terrace boundaries including anisotropic line tension and the normal velocity law for the motion of such boundaries determined by a two sided flux together with the one dimensional anisotropic surface diffusion edge diffusion of edge adatoms along the step edges the problem is solved using independent meshes a two dimensional mesh for the adatom diffusion and one dimensional meshes for the boundary evolution a penalty method is used to incorporate the boundary conditions the evolution of the terrace boundaries includes both the weighted anisotropic mean curvature flow and the weighted anisotropic edge diffusion its governing equation is solved by a semi implicit front tracking method using parametric finite elements

partial non gaussian state space summary in this paper we suggest the use of simulation techniques to extend the applicability of the usual gaussian state space filtering and smoothing techniques to a class of nongaussian time series models this allows a fully bayesian or maximum likelihood analysis of some interesting models including outlier models discrete markov chain components multiplicative models and stochastic variance models finally we discuss at some length the use of a non gaussian model to seasonally adjust the published money supply figures

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

level set methods for the simulation of epitaxial phenomena we introduce a model for epitaxial phenomena based on the motion of island boundaries which is described by the level set method our model treats the growing film as a continuum in the lateral direction but retains atomistic discreteness in the growth direction an example of such an island dynamics model using the level set method is presented and compared with the corresponding rate equation description extensions of our methodology to more general settings are then discussed s1063 651x 98 50212 7

bayes inference in regression models with arma p q errors we develop practical and exact methods of analyzing arma p 4 regression error models in a bayesian framework by using the gibbs sampling and metropolis hastings algorithms and we prove that the kernel of the proposed markov chain sampler converges to the true density the procedures can be applied to pure arma time series models and to determine features of the likelihood function by choosing appropriate diffuse priors our results are unconditional on the initial observations we also show how the algorithm can be further simplified for the important special cases of stationary ar p and invertible ma q models recursive transformations developed in this paper to diagonalize the covariance matrix of the errors should prove useful in frequentist estimation examples with simulated and actual economic data are presented kpj ourls gibbs sampling metropolis hastings algorithm data augmentation time series arma processes markov chain bayesian statistics

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

bayesian inference in hidden markov models through the reversible jump markov chain monte carlo method hidden markov models form an extension of mixture models which provides a flexible class of models exhibiting dependence and a possibly large degree of variability we show how reversible jump markov chain monte carlo techniques can be used to estimate the parameters as well as the number of components of a hidden markov model in a bayesian framework we employ a mixture of zero mean normal distributions as our main example and apply this model to three sets of data from finance meteorology and geomagnetism

fuzzy sets and sheaves part ii sheaf theoretic foundations of fuzzy set theory with applications to algebra and topology this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

a boundary integral method for computing the dynamics of an epitaxial island in this paper we present a boundary integral method for computing the quasi steady evolution of an epitaxial island the problem consists of an adatom diffusion equation with desorption on terrace and a kinetic boundary condition at the step island boundary the normal velocity for step motion is determined by a two sided flux the integral formulation of the problem involves both double and single layer potentials due to the kinetic boundary condition numerical tests on a growing shrinking circular or a slightly perturbed circular island are in excellent agreement with the linear analysis demonstrating that the method is stable efficient and spectrally accurate in space nonlinear simulations for the growth of perturbed circular islands show that sharp tips and flat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium in particular bales zangwill instability is manifested in the form of wave like fronts meandering instability around the tip regions the numerical techniques presented here can be applied generally to a class of free moving boundary problems in physical and biological science

bayesian estimation of switching arma models abstract switching arma processes have recently appeared as an efficient modelling to nonlinear time series models because they can represent multiple or heterogeneous dynamics through simple components the levels of dependence between the observations are double at a first level the parameters of the model are selected by a markovian procedure at a second level the next observation is generated according to a standard time series model when the model involves a moving average structure the complexity of the resulting likelihood function is such that simulation techniques like those proposed by shephard 1994 biometrika 81 115 131 and billio and monfort 1998 journal of statistical planning and inference 68 65 103 are necessary to derive an inference on the parameters of the model we propose in this paper a bayesian approach with a non informative prior distribution developed in mengersen and robert 1996 bayesian statistics 5 oxford university press oxford pp 255 276 and robert and titterington 1998 statistics and computing 8 2 145 158 in the setup of mixtures of distributions and hidden markov models respectively the computation of the bayes estimates relies on mcmc techniques which iteratively simulate missing states innovations and parameters until convergence the performances of the method are illustrated on several simulated examples this work also extends the papers by chib and greenberg 1994 journal of econometrics 64 183 206 and chib 1996 journal of econometrics 75 1 79 97 which deal with arma and hidden markov models respectively

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

a finite element method for surface diffusion the parametric case surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for parametric surfaces with or without boundaries the method is semi implicit requires no explicit parametrization and yields a linear system of elliptic pde to solve at each time step we next develop a finite element method propose a schur complement approach to solve the resulting linear systems and show several significant simulations some with pinch off in finite time we introduce a mesh regularization algorithm which helps prevent mesh distortion and discuss the use of time and space adaptivity to increase accuracy while reducing complexity

stationarity of multivariate markov switching arma models in this article we consider multivariate arma models subject to markov switching in these models the parameters are allowed to depend on the state of an unobserved markov chain a natural idea when estimating these models is to impose local stationarity conditions i e stationarity within each regime in this article we show that the local stationarity of the observed process is neither sufficient nor necessary to obtain the global stationarity we derive stationarity conditions and we compute the autocovariance function of this nonlinear process interestingly it turns out that the autocovariance structure coincides with that of a standard arma some examples are proposed to illustrate the stationarity conditions using monte carlo simulations we investigate the consequences of accounting for the stationarity conditions in statistical inference

a numerical study of electro migration voiding by evolving level set functions on a fixed cartesian grid a numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed the mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes we use the modified immersed interface method with a fixed cartesian grid to solve for the electrical field and the fast local level set method to update the position of moving voids numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

bayesian estimation of an autoregressive model using markov chain monte carlo abstract we present a complete bayesian treatment of autoregressive model estimation incorporating choice of autoregressive order enforcement of stationarity treatment of outliers and allowance for missing values and multiplicative seasonality the paper makes three distinct contributions first we enforce the stationarity conditions using a very efficient metropolis within gibbs algorithm to generate the partial autocorrelations second we show how to carry out the gibbs sampler when the autoregressive order is unknown third we show how to combine the various aspects of fitting an autoregressive model giving a more comprehensive and efficient treatment than previous work we illustrate our methodology with a real example

level set approach to reversible epitaxial growth we generalize the level set approach to model epitaxial growth to include thermal detachment of atoms from island edges this means that islands do not always grow and island dissociation can occur we make no assumptions about a critical nucleus excellent quantitative agreement is obtained with kinetic monte carlo simulations for island densities and island size distributions in the submonolayer regime

membership functions for a fuzzy group from similarity relations abstract we show that the membership function x of a fuzzy group g satisfies the equation x e x where x y is a similarity relation on g which is invariant under left translation we also show that under some natural assumptions one can represent the elements x g as 1 1 onto transformations px on a suitable universe such that x equals the proportion of points in which are fixed points of px these results provide a deeper insight on the impact of group operation on the membership values x

finite element method for epitaxial growth with attachment detachment kinetics an adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth such problems consist of an adatom adsorbed atom diffusion equation on terraces of different height boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment and the normal velocity law for the motion of such boundaries determined by a two sided flux together with the one dimensional surface diffusion the problem is solved using two independent meshes a two dimensional mesh for the adatom diffusion and a one dimensional mesh for the boundary evolution the diffusion equation is discretized by the first order implicit scheme in time and the linear finite element method in space a technique of extension is used to avoid the complexity in the spatial discretization near boundaries all the elements are marked and the marking is updated in each time step to trace the terrace height the evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion its governing equation is solved by a semi implicit front tracking method using parametric finite elements simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem numerical tests on pure geometrical motion mass balance and the stability of a growing circular island demonstrate that the method is stable efficient and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period

covariance structure and convergence rate of the gibbs sampler with various scans this paper presents results on covariance structure and convergence for the gibbs sampler with both systematic and random scans it is shown that under conditions that guarantee the compactness of the markov forward operator and irreducibility of the corresponding chain the gibbs sampling scheme converges geometrically in terms of pearson 2 distance in particular for the random scan the autocovariance can be expressed as variances of iterative conditional expectations as a consequence the autocorrelations are all positive and decrease monotonically

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea