bayesian guided pattern search for robust local optimization optimization for complex systems in engineering often involves the use of expensive computer simulation by combining statistical emulation using treed gaussian processes with pattern search optimization we are able to perform robust local optimization more efficiently and effectively than when using either method alone our approach is based on the augmentation of local search patterns with location sets generated through improvement prediction over the input space we further develop a computational framework for asynchronous parallel implementation of the optimization algorithm we demonstrate our methods on two standard test problems and our motivating example of calibrating a circuit device simulator

ornstein zernike theory for finite range ising models above t c we derive a precise ornstein zernike asymptotic formula for the decay of the two point function 0 x in the general context of finite range ising type models on d the proof relies in an essential way on the a priori knowledge of the strict exponential decay of the two point function and by the sharp characterization of phase transition due to aizenman barsky and fernandez goes through in the whole of the high temperature region c as a byproduct we obtain that for every c the inverse correlation length is an analytic and strictly convex function of direction

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise

multivariate gaussian process emulators with nonseparable covariance structures the gaussian process regression model is a popular type of emulator used as a fast surrogate for computationally expensive simulators deterministic computer models for simulators with multivariate output common practice is to specify a separable covariance structure for the gaussian process though computationally convenient this can be too restrictive leading to poor performance of the emulator particularly when the different simulator outputs represent different physical quantities also treating the simulator outputs as independent can lead to inappropriate representations of joint uncertainty we develop nonseparable covariance structures for gaussian process emulators based on the linear model of coregionalization and convolution methods using two case studies we compare the performance of these covariance structures both with standard separable covariance structures and with emulators that assume independence between the outputs in each case study we find that only emulators with nons

post fisherian experimentation from physical to virtual fisher s pioneering work in design of experiments has inspired further work with broader applications especially in industrial experimentation this article discusses three topics in physical experiments principles of effect hierarchy sparsity and heredity for factorial designs a new method called conditional main effect cme for de aliasing aliased effects and robust parameter design i also review the recent emergence of virtual experiments on a computer some major challenges in computer experiments which must go beyond fisherian principles are outlined

harmonic pinnacles in the discrete gaussian model the 2d discrete gaussian model gives each height function eta mathbb z 2 to mathbb z a probability proportional to exp beta mathcal h eta where beta is the inverse temperature and mathcal h eta sum x sim y eta x eta y 2 sums over nearest neighbor bonds we consider the model at large fixed beta where it is flat unlike its continuous analog the discrete gaussian free field we first establish that the maximum height in an l times l box with 0 boundary conditions concentrates on two integers m m 1 with m sim sqrt 1 2 pi beta log l log log l the key is a large deviation estimate for the height at the origin in mathbb z 2 dominated by harmonic pinnacles integer approximations of a harmonic variational problem second in this model conditioned on eta geq 0 a floor the average height rises and in fact the height of almost all sites concentrates on levels h h 1 where h sim m sqrt 2 this in particular pins down the asymptotics and corrects the order in results of bricmont et al j stat phys 42 5 6 743 798 1986 where it was argued that the maximum and the height of the surface above a floor are both of order sqrt log l finally our methods extend to other classical surface models e g restricted sos featuring connections to p harmonic analysis and alternating sign matrices

emulation of utility functions over a set of permutations sequencing reliability growth tasks abstractwe consider bayesian design of experiments problems in which we maximise the prior expectation of a utility function over a set of permutations for example when sequencing a number of tasks to perform when the number of tasks is large and the expected utility is expensive to compute it may be unreasonable or infeasible to evaluate the expected utility of all permutations we propose an approach to emulate the expected utility using a surrogate function based on a parametric probabilistic model for permutations the surrogate function is fitted by maximising the correlation with the expected utility over a set of training points we propose a suitable transformation of the expected utility to improve the fit we provide results linking the correlation between the two functions and the number of expected utility evaluations to undertake the approach is applied to the sequencing of reliability growth tasks in the development of hardware systems in which there is a large number of potential tasks

interfaces for random cluster models a random cluster measure on mathbb z d that is not translationally invariant is constructed for d 3 the critical density pc and sufficiently large q the resulting measure is proven to be a gibbs state satisfying cluster model dlr equations

slab percolation for the ising model for the fk representation of the ising model we prove that the slab percolation threshold coincides with the critical temperature in any dimension d 3

estimation of a density from an imperfect simulation model uncertainty quantification of a technical system can be done using density estimation we usually start with a stochastic model which is fitted to the technical system and the density estimation is done using data from this stochastic model however in any application such a stochastic model will not be perfect and the estimation of the density should take into account the inadequacy of the stochastic model in this paper we show how observed data of the real system together with an imperfect simulation model can be used to derive confidence bands for the density of the technical system our main result is that the newly introduced confidence bands allow to derive lower and upper bounds on the probability of intervals in the technical system furthermore we present an upper bound on the area of the confidence band in case of a smooth density the results are illustrated by applying the estimates to simulated and real data

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

tightness of the recentered maximum of the two dimensional discrete gaussian free field we consider the maximum of the discrete two dimensional gaussian free field gff in a box and prove that its maximum centered at its mean is tight settling a longstanding conjecture the proof combines a recent observation by bolthausen deuschel and zeitouni with elements from bramson s results on branching brownian motion and comparison theorems for gaussian fields an essential part of the argument is the precise evaluation up to an error of order 1 of the expected value of the maximum of the gff in a box related gaussian fields such as the gff on a two dimensional torus are also discussed 2011 wiley periodicals inc

on local behaviour of the phase separation line in the 2d ising model the aim of this note is to discuss some statistical properties of the phase separation line in the 2d low temperature ising model we prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation the limiting gaussian measure corresponds to a scaled brownian bridge with direction dependent parameters up to the temperature factor the variances of local increments of this limiting process are inversely proportional to the stiffness

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

bayesian calibration of inexact computer models bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias the parameter in the predominant methodology is left undefined this results in an issue where the posterior of the parameter is suboptimally broad there has been no generally accepted alternatives to date this article proposes using bayesian calibration where the prior distribution on the bias is orthogonal to the gradient of the computer model problems associated with bayesian calibration are shown to be mitigated through analytic results in addition to examples supplementary materials for this article are available online

regression based inverse distance weighting with applications to computer experiments inverse distance weighting idw is a simple method for multivariate interpolation but has poor prediction accuracy in this article we show that the prediction accuracy of idw can be substantially improved by integrating it with a linear regression model this new predictor is quite flexible computationally efficient and works well in problems having high dimensions and or large datasets we also develop a heuristic method for constructing confidence intervals for prediction this article has supplementary material online

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations a bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation lco of a pitching airfoil in the transitional reynolds number regime at small angles of attack laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the lco the fluid structure interaction of the rigid but elastically mounted airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to lco the likelihood in the posterior parameter probability density function pdf is available semi analytically using the extended kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model the posterior parameter pdf is sampled using a parallel and adaptive markov chain monte carlo mcmc algorithm the posterior probability of each model is estimated using the chib jeliazkov method that directly uses the posterior mcmc samples for evidence marginal likelihood computation the bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind tunnel test data at various reynolds numbers

inference for deterministic simulation models the bayesian melding approach abstract deterministic simulation models are used in many areas of science engineering and policy making typically these are complex models that attempt to capture underlying mechanisms in considerable detail and they have many user specified inputs the inputs are often specified by some form of trial and error approach in which plausible values are postulated the corresponding outputs inspected and the inputs modified until plausible outputs are obtained here we address the issue of more formal inference for such models a probabilistic approach called bayesian synthesis was shown to suffer from the borel paradox according to which the results can depend on the parameterization of the model we propose a modified approach called bayesian melding which takes into full account information and uncertainty about both inputs and outputs to the model while avoiding the borel paradox this is done by recognizing the existence of two priors one implicit and one explicit on each input and output

advances in modeling model discrepancy comment on wu and browne 2015 wu and browne psychometrika 79 2015 have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent their contribution is related to ongoing developments in the field of uncertainty quantification uq on modeling and quantifying effects of model discrepancy we provide an overview of basic principles of uq and some relevant developments and we examine the wu browne work in that context we view the wu browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research

multi output separable gaussian process towards an efficient fully bayesian paradigm for uncertainty quantification computer codes simulating physical systems usually have responses that consist of a set of distinct outputs e g velocity and pressure that evolve also in space and time and depend on many unknown input parameters e g physical constants initial boundary conditions etc furthermore essential engineering procedures such as uncertainty quantification inverse problems or design are notoriously difficult to carry out mostly due to the limited simulations available the aim of this work is to introduce a fully bayesian approach for treating these problems which accounts for the uncertainty induced by the finite number of observations our model is built on a multi dimensional gaussian process that explicitly treats correlations between distinct output variables as well as space and or time the proper use of a separable covariance function enables us to describe the huge covariance matrix as a kronecker product of smaller matrices leading to efficient algorithms for carrying out inference and predictions the novelty of this work is the recognition that the gaussian process model defines a posterior probability measure on the function space of possible surrogates for the computer code and the derivation of an algorithmic procedure that allows us to sample it efficiently we demonstrate how the scheme can be used in uncertainty quantification tasks in order to obtain error bars for the statistics of interest that account for the finite number of observations

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

ornstein zernike theory for the bernoulli bond percolation on mathbb z d we derive a precise ornstein zernike asymptotic formula for the decay of the two point function p p 0 x of the bernoulli bond percolation on the integer lattice z d in any dimension d 2 in any direction x and for any subcritical value of p p c d

simulation based optimal bayesian experimental design for nonlinear systems the optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction particularly in situations where experiments are time consuming and expensive to conduct we propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation based models in particular we focus on finding sets of experiments that provide the most information about targeted sets of parameters our framework employs a bayesian statistical setting which provides a foundation for inference from noisy indirect and incomplete data and a natural mechanism for incorporating heterogeneous sources of information an objective function is constructed from information theoretic measures reflecting expected information gain from proposed combinations of experiments polynomial chaos approximations and a two stage monte carlo sampling method are used to evaluate the expected information gain stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high dimensional settings these algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics

ensemble evaluation of hydrological model hypotheses it is demonstrated for the first time how model parameter structural and data uncertainties can be accounted for explicitly and simultaneously within the generalized likelihood uncertainty estimation glue methodology as an example application 72 variants of a single soil moisture accounting store are tested as simplified hypotheses of runoff generation at six experimental grassland field scale lysimeters through model rejection and a novel diagnostic scheme the fields designed as replicates exhibit different hydrological behaviors which yield different model performances for fields with low initial discharge levels at the beginning of events the conceptual stores considered reach their limit of applicability conversely one of the fields yielding more discharge than the others but having larger data gaps allows for greater flexibility in the choice of model structures as a model learning exercise the study points to a leaking of the fields not evident from previous field experiments it is discussed how understanding observational uncertainties and incorporating these into model diagnostics can help appreciate the scale of model structural error

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

rigidity of the interface in percolation and random cluster models we study conditioned random cluster measures with edge parameter p and cluster weighting factor q satisfying q 1 the conditioning corresponds to mixed boundary conditions for a spin model interfaces may be defined in the sense of dobrushin and these are proved to be rigid in the thermodynamic limit in three dimensions and for sufficiently large values of p this implies the existence of non translation invariant conditioned random cluster measures in three dimensions the results are valid in the special case q 1 thus indicating a property of three dimensional percolation not previously noted

gaussian process single index models as emulators for computer experiments a single index model sim provides for parsimonious multidimensional nonlinear regression by combining parametric linear projection with univariate nonparametric nonlinear regression models we show that a particular gaussian process gp formulation is simple to work with and ideal as an emulator for some types of computer experiment as it can outperform the canonical separable gp regression model commonly used in this setting our contribution focuses on drastically simplifying reinterpreting and then generalizing a recently proposed fully bayesian gp sim combination favorable performance is illustrated on synthetic data and a real data computer experiment two r packages both released on cran have been augmented to facilitate inference under our proposed model s

factorial sampling plans for preliminary computational experiments a computational model is a representation of some physical or other system of interest first expressed mathematically and then implemented in the form of a computer program it may be viewed as a function of inputs that when evaluated produces outputs motivation for this article comes from computational models that are deterministic complicated enough to make classical mathematical analysis impractical and that have a moderate to large number of inputs the problem of designing computational experiments to determine which inputs have important effects on an output is considered the proposed experimental plans are composed of individually randomized one factor at a time designs and data analysis is based on the resulting random sample of observed elementary effects those changes in an output due solely to changes in a particular input advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs monotonicity of outputs with respect to inputs or ad

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

stochastic factorizations sandwiched simplices and the topology of the space of explanations we study the space of stochastic factorizations of a stochastic matrix v motivated by the statistical problem of hidden random variables we show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra and use this geometrical model to gain some insight into its structure and topology we prove theorems describing its homotopy type and in the case where the rank of v is 2 we give a complete description including bounds on the number of connected components and examples in which these bounds are attained we attempt to make the notions of topology accessible and relevant to statisticians

analysis of computer experiments using penalized likelihood in gaussian kriging models kriging is a popular analysis approach for computer experiments for the purpose of creating a cheap to compute meta model as a surrogate to a computationally expensive engineering simulation model the maximum likelihood approach is used to estimate the parameters in the kriging model however the likelihood function near the optimum may be flat in some situations which leads to maximum likelihood estimates for the parameters in the covariance matrix that have very large variance to overcome this difficulty a penalized likelihood approach is proposed for the kriging model both theoretical analysis and empirical experience using real world data suggest that the proposed method is particularly important in the context of a computationally intensive simulation model where the number of simulation runs must be kept small because collection of a large sample set is prohibitive the proposed approach is applied to the reduction of piston slap an unwanted engine noise due to piston secondary motion issu

inverse regression based uncertainty quantification algorithms for high dimensional models many uncertainty quantification uq approaches suffer from the curse of dimensionality that is their computational costs become intractable for problems involving a large number of uncertainty parameters in these situations the classic monte carlo often remains the preferred method of choice because its convergence rate o n 1 2 where n is the required number of model simulations does not depend on the dimension of the problem however many high dimensional uq problems are intrinsically low dimensional because the variation of the quantity of interest qoi is often caused by only a few latent parameters varying within a low dimensional subspace known as the sufficient dimension reduction sdr subspace in the statistics literature motivated by this observation we propose two inverse regression based uq algorithms iruq for high dimensional problems both algorithms use inverse regression to convert the original high dimensional problem to a low dimensional one which is then efficiently solved by building a response surface for the reduced model for example via the polynomial chaos expansion the first algorithm which is for the situations where an exact sdr subspace exists is proved to converge at rate o n 1 hence much faster than mc the second algorithm which doesn t require an exact sdr employs the reduced model as a control variate to reduce the error of the mc estimate the accuracy gain could still be significant depending on how well the reduced model approximates the original high dimensional one iruq also provides several additional practical advantages it is non intrusive it does not require computing the high dimensional gradient of the qoi and it reports an error bar so the user knows how reliable the result is

cohomology of colorings of cycles we compute the cohomology groups of the spaces of colorings of cycles i e of the prodsimplicial complexes tt hom c m k n we perform the computation first with bbb z 2 and then with integer coefficients the main technical tool is to use spectral sequences in conjunction with a detailed combinatorial analysis of a family of cubical complexes which we call it torus front complexes as an application of our method we demonstrate how to collapse each connected component of tt hom c m c n onto a garland of cubes

decay of correlations in surface models a convergent low temperature expansion for a variety of models of twodimensional surfaces is presented it yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz e in addition the estimates give exponential decay of truncated correlations which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue two particular examples included in the general framework are the solid on solid and discrete gaussian models

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

graph colorings spaces of edges and spaces of circuits abstract by lovasz proof of the kneser conjecture the chromatic number of a graph g is bounded from below by the index of the z 2 space hom k 2 g plus two we show that the cohomological index of hom k 2 g is also greater than the cohomological index of the z 2 space hom c 2 r 1 g for r 1 this gives a new and simple proof of the strong form of the graph coloring theorem by babson and kozlov which had been conjectured by lovasz and at the same time shows that it never gives a stronger bound than can be obtained by hom k 2 g the proof extends ideas introduced by ivaljevic in a previous elegant proof of a special case we then generalize the arguments and obtain conditions under which corresponding results hold for other graphs in place of c 2 r 1 this enables us to find an infinite family of test graphs of chromatic number 4 among the kneser graphs our main new result is a description of the z 2 homotopy type of the direct limit of the system of all the spaces hom c 2 r 1 g in terms of the z 2 homotopy type of hom k 2 g a corollary is that the coindex of hom k 2 g does not exceed the coindex of hom c 2 r 1 g by more then one if r is chosen sufficiently large thus the graph coloring bound in the theorem by babson and kozlov is also never weaker than that from lovasz proof of the kneser conjecture

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

proof of the lovasz conjecture to any two graphs g and h one can associate a cell complex horn g h by taking all graph multihomomorphisms from g to h as cells in this paper we prove the lovasz conjecture which states that if horn c 2r 1 g is k connected then g k 4 where r k z r 1 k 1 and c 2r 1 denotes the cycle with 2r 1 vertices the proof requires analysis of the complexes horn c 2r 1 k n for even n the obstructions to graph colorings are provided by the presence of torsion in h hom c 2r 1 k n z for odd n the obstructions are expressed as vanishing of certain powers of stiefel whitney characteristic classes of horn c 2r 1 k n where the latter are viewed as z 2 spaces with the involution induced by the reflection of c 2r 1

empirical bayesian analysis for computer experiments involving finite difference codes computer experiments are increasingly used in scientific investigations as substitutes for physical experiments in cases where the latter are difficult or impossible to perform a computer experiment consists of several runs of a computer model or code for the purpose of better understanding the input output relationship one practical difficulty in the use of these models is that a single run may require a prohibitive amount of computational resources in some situations a recent approach uses statistical approximations as less expensive surrogates for such computer codes these provide both point predictors and uncertainty characterization of the outputs a widely used class of computer codes is the finite difference solvers of differential equations which produce multivariate output e g time series the finite difference relationship underpins the statistical model proposed here and we show that this strategy has clear computational and accuracy advantages over a direct multivariate extension

experimental design and observation for large systems large systems require new methods of experimental designs suitable for the highly adaptive models which are employed to cope with complex non linear responses and high dimensionality of input spaces the area of computer experiments has started to provide such designs especially latin hypercube and lattice designs system decomposition prevalent in several branches of engineering can be employed to decrease complexity a combination of system decomposition using a sparse matrix method experimental design and modelling is applied to one example of an electrical circuit simulator producing a usable emulator of the circuit for use in optimization and sensitivity analysis

gaussian surrogates for computer models with time varying inputs and outputs computer models of dynamic systems produce outputs that are functions of time models that solve systems of differential equations often have this character in many cases time series output can be usefully reduced via principal components to simplify analysis time indexed inputs such as the functions that describe time varying boundary conditions are also common with such models however inputs that are functions of time often do not have one or a few characteristic shapes that are more common with output functions and so principal component representation has less potential for reducing the dimension of input functions in this article gaussian process surrogates are described for models with inputs and outputs that are both functions of time the focus is on construction of an appropriate covariance structure for such surrogates some experimental design issues and an application to a model of marrow cell dynamics

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

maximum likelihood estimation for stochastic differential equations using sequential gaussian process based optimization abstractstochastic differential equations sdes are used as statistical models in many disciplines however intractable likelihood functions for sdes make inference challenging and we need to resort to simulation based techniques to estimate and maximize the likelihood function while importance sampling methods have allowed for the accurate evaluation of likelihoods at fixed parameter values there is still a question of how to find the maximum likelihood estimate in this article we propose an efficient gaussian process based method for exploring the parameter space using estimates of the likelihood from an importance sampler our technique accounts for the inherent monte carlo variability of the estimated likelihood and does not require knowledge of gradients the procedure adds potential parameter values by maximizing the so called expected improvement leveraging the fact that the likelihood function is assumed to be smooth our simulations demonstrate that our method has significant computation

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

effect of environmental stress on regulation of gene expression in the yeast abstract several mathematical models have been proposed to predict the activation state of a transcription factor tf from the expression levels of its target genes this inference problem is complicated however due to the fact that different genes may be regulated by different activation schemes linear exponential sigmoidal etc in addition to transcription regulation the rate of gene expression at any instantaneous point in time is also determined by the independent rates of baseline production and degradation consequently the set of solutions to any model equations describe an infinite number of trajectories in probability space thus rendering the problem np hard in the current study we used a gaussian process gp approach to address this inverse problem experimental gene expression data were modeled by a putative linear activation scheme and discrepancy between theory and experiment was modeled by a gp model hyperparameters were calculated using maximum likelihood estimates to generate continuous tf state space profiles identifiability of model parameters was optimized by obtaining tf state space functions for multiple genes simultaneously we found that model parameters were sensitive to environmental stress conditions producing different state space profiles for different stresses

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

quantifying adventitious error in a covariance structure as a random effect we present an approach to quantifying errors in covariance structures in which adventitious error identified as the process underlying the discrepancy between the population and the structured model is explicitly modeled as a random effect with a distribution and the dispersion parameter of this distribution to be estimated gives a measure of misspecification analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation an algorithm is developed for numerical implementation of the procedure the consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard pitman drift assumption simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error two examples are also given as illustrations

calibration of computational models with categorical parameters and correlated outputs via bayesian smoothing spline anova it has become commonplace to use complex computer models to predict outcomes in regions where data do not exist typically these models need to be calibrated and validated using some experimental data which often consists of multiple correlated outcomes in addition some of the model parameters may be categorical in nature such as a pointer variable to alternate models or submodels for some of the physics of the system here we present a general approach for calibration in such situations where an emulator of the computationally demanding models and a discrepancy term from the model to reality are represented within a bayesian smoothing spline bss anova framework the bss anova framework has several advantages over the traditional gaussian process including ease of handling categorical inputs and correlated outputs and improved computational efficiency finally this framework is then applied to the problem that motivated its design a calibration of a computational fluid dynamics cfd model of

mechanistic hierarchical gaussian processes the statistics literature on functional data analysis focuses primarily on flexible black box approaches which are designed to allow individual curves to have essentially any shape while characterizing variability such methods typically cannot incorporate mechanistic information which is commonly expressed in terms of differential equations motivated by studies of muscle activation we propose a nonparametric bayesian approach that takes into account mechanistic understanding of muscle physiology a novel class of hierarchical gaussian processes is defined that favors curves consistent with differential equations defined on motor damper spring systems a gibbs sampler is proposed to sample from the posterior distribution and applied to a study of rats exposed to noninjurious muscle activation protocols although motivated by muscle force data a parallel approach can be used to include mechanistic information in broad functional data analysis applications

forcing function diagnostics for nonlinear dynamics this article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations odes i propose modeling lack of fit as a time varying correction to the right hand side of a proposed differential equation this correction can be described as being a set of additive forcing functions estimated from data representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements i derive lack of fit tests based on estimated forcing functions model building in partially observed systems of odes is particularly difficult and i consider the problem of identification of forcing functions in these systems the methods are illustrated with examples from computational neuroscience

a frequentist approach to computer model calibration summary the paper considers the computer model calibration problem and provides a general frequentist solution under the framework proposed the data model is semiparametric with a non parametric discrepancy function which accounts for any discrepancy between physical reality and the computer model in an attempt to solve a fundamentally important but often ignored identifiability issue between the computer model parameters and the discrepancy function the paper proposes a new and identifiable parameterization of the calibration problem it also develops a two step procedure for estimating all the relevant quantities under the new parameterization this estimation procedure is shown to enjoy excellent rates of convergence and can be straightforwardly implemented with existing software for uncertainty quantification bootstrapping is adopted to construct confidence regions for the quantities of interest the practical performance of the methodology is illustrated through simulation examples and an application to a computational fluid dynamics model

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach

letter to the editor probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen the objective in this paper is to analyze some stochastic models for estimating the ionization reaction rate constant of atomic nitrogen n e n 2e parameters of the models are identified by means of bayesian inference using spatially resolved absolute radiance data obtained from the electric arc shock tube east wind tunnel the proposed methodology accounts for uncertainties in the model parameters as well as physical model inadequacies providing estimates of the rate constant that reflect both types of uncertainties we present four different probabilistic models by varying the error structure either additive or multiplicative and by choosing different descriptions of the statistical correlation among data points in order to assess the validity of our methodology we first present some calibration results obtained with manufactured data and then proceed by using experimental data collected at east experimental facility in order to simulate the radiative signature emitted in the shock heated air plasma we use a one dimensional flow solver with park s two temperature model that simulates non equilibrium effects we also discuss the implications of the choice of the stochastic model on the estimation of the reaction rate and its uncertainties our analysis shows that the stochastic models based on correlated multiplicative errors are the most plausible models among the four models proposed in this study the rate of the atomic nitrogen ionization is found to be 6 2 3 3 x10 1 1cm 3mol 1s 1 at 10 000k

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l