topology optimization of continuum structures under uncertainty a polynomial chaos approach abstract a computational method for topology optimization in the presence of uncertainty is proposed the method combines the spectral stochastic approach for the representation and propagation of uncertainties with an existing deterministic topology optimization technique the idea in spectral stochastic formulations is to add an extra dimension a random dimension to the problem where the stochastic variability of the input parameters and outputs of interest can be modeled the resulting compact representations for the response quantities allow for efficient and accurate calculation of sensitivities of response statistics with respect to the design variables which are then fed into a gradient based optimizer that searches for the optimum design from a variety of frameworks for uncertainty informed optimization robust topology optimization is chosen to demonstrate the applicability of the method examples from continuum topology optimization under uncertainty in material properties are presented it is also shown that results obtained from the proposed method are in excellent agreement with those obtained from a monte carlo based optimization algorithm

some orthogonal polynomials arising from coherent states we explore in this paper some orthogonal polynomials which are naturally associated with certain families of coherent states often referred to as nonlinear coherent states in the quantum optics literature some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

asymptotic normality with small relative errors of posterior probabilities of half spaces some examples of half spaces of interest in parameter spaces are in a clinical trial of a treatment versus placebo the half spaces where the treatment is a helpful or b harmful thus one may not only want to test the hypothesis that the treatment c makes no difference but to assign posterior probabilities to a b and c under conditions as unrestrictive as possible on the choice of prior probabilities e g dudley and haughton 2001 more generally we have in mind applications to model selections as in the bic criterion of schwarz 1978 and its extensions poskitt 1987 haughton 1988 specifically to one sided models and multiple data sets dudley and haughton 1997

a probabilistic construction of model validation we describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty the proposed approach is a functional analysis based probabilistic approach for which we represent random quantities using polynomial chaos expansions pces the approach permits the formulation of the uncertainty assessment in validation a significant component of the process as a problem of approximation theory it has two essential parts first a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model based measurements such a calibration technique employs pces to represent the inherent uncertainty of the model parameters based on the asymptotic behavior of the statistical parameter estimator the associated pce coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information second a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem the above validation path is implemented for the case of dynamical system validation challenge exercise

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

polynomial chaos representation of databases on manifolds characterizing the polynomial chaos expansion pce of a vector valued random variable with probability distribution concentrated on a manifold is a relevant problem in data driven settings the probability distribution of such random vectors is multimodal in general leading to potentially very slow convergence of the pce in this paper we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold the proposed methodology constructs a pce of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold the method is robust and remains efficient for high dimension and large datasets the resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data driven queries

on asymptotic properties of maximum likelihood estimators for parabolic stochastic pde s we investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic spdes of the form du t x a 0 theta a 1 u t x dt dw t x wherea 0 anda 1 are partial differential operators andw is a cylindrical brownian motion we introduce a spectral method for computing mles based on finite dimensional approximations to solutions of such systems and establish criteria for consistency asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity we derive the asymptotic properties of the mle from a condition on the order of the operators in particular the mle is consistent if and only if ord a 1 1 2 ord a 0 a 1 d

probabilistic equivalence and stochastic model reduction in multiscale analysis this paper presents a probabilistic upscaling of mechanics models a reduced order probabilistic model is constructed as a coarse scale representation of a specified fine scale model whose probabilistic structure can be accurately determined equivalence of the fine and coarse scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained a significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine and coarse scale representations the upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations an illustration in stochastic structural dynamics is provided to demonstrate the proposed framework

generalization of jeffreys divergence based priors for bayesian hypothesis testing summary we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models we call them diver gence based db priors db priors have simple forms and desirable properties like information finite sample consistency and are often similar to other existing proposals like intrinsic priors moreover in normal linear model scenarios they reproduce the jeffreys zellner siow priors exactly most importantly in challenging scenarios such as irregular models and mixture mod els db priors are well defined and very reasonable whereas alternative proposals are not we derive approximations to the db priors as well as markov chain monte carlo and asymptotic expressions for the associated bayes factors

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

a completely monotone function related to the gamma function abstract we show that the reciprocal of the function f z log z 1 z log z z c 0 is a stieltjes transform as a corollary we obtain that the derivative of f is completely monotone in the sense that 1 n 1f n x 0 for all n 1 and all x 0 this answers a question raised by dimitar dimitrov at the fifth international symposium on orthogonal polynomials special functions and applications held in patras in september 1999 to prove the result we examine the imaginary part of 1 f in the upper half plane in particular close to the negative real axis where stirling s formula is not valid

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

objective bayesian comparison of constrained analysis of variance models in the social sciences we are often interested in comparing models specified by parametric equality or inequality constraints for instance when examining three group means mu 1 mu 2 mu 3 through an analysis of variance anova a model may specify that mu 1 mu 2 mu 3 while another one may state that mu 1 mu 3 mu 2 and finally a third model may instead suggest that all means are unrestricted this is a challenging problem because it involves a combination of nonnested models as well as nested models having the same dimension we adopt an objective bayesian approach requiring no prior specification from the user and derive the posterior probability of each model under consideration our method is based on the intrinsic prior methodology suitably modified to accommodate equality and inequality constraints focussing on normal anova models a comparative assessment is carried out through simulation studies we also present an application to real data collected in a psychological experiment

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

a unified conditional frequentist and bayesian test for fixed and sequential simple hypothesis testing preexperimental frequentist error probabilities are arguably inadequate as summaries of evidence from data in many hypothesis testing settings the conditional frequentist may respond to this by identifying certain subsets of the outcome space and reporting a conditional error probability given the subset of the outcome space in which the observed data lie statistical methods consistent with the likelihood principle including bayesian methods avoid the problem by a more extreme form of conditioning in tits paper we prove that the conditional frequentist s method can be made exactly equivalent to the bayesian s in simple versus simple hypothesis testing specifically we find a conditioning strategy for which the conditional frequentist s reported conditional error probabilities are the same as the bayesian s posterior probabilities of error a conditional frequentist who uses such a strategy can exploit other features of the bayesian approachfor example the validity of sequential hypothesis tests including versions of the sequential probability ratio test or sprt even if the stopping rule is incompletely specified

the moment problem associated with the stieltjes wigert polynomials abstract we consider the indeterminate stieltjes moment problem associated with the stieltjes wigert polynomials after a presentation of the well known solutions we study a transformation t of the set of solutions all the classical solutions turn out to be fixed under this transformation but this is not the case for the so called canonical solutions based on generating functions for the stieltjes wigert polynomials expressions for the entire functions a b c and d from the nevanlinna parametrization are obtained we describe t n for n n when 0 is a particular n extremal solution and explain in detail what happens when n

the cube of a normal distribution is indeterminate it is established that if x is a stochastic variable with a normal distribution then x 2n 1 has an indeterminate distribution for n geq 1 furthermore the distribution of x alpha is determinate for 0 4

bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives testing the fit of data to a parametric model can be done by embedding the parametric model in a nonparametric alternative and computing the bayes factor of the parametric model to the nonparametric alternative doing so by specifying the nonparametric alternative via a polya tree process is particularly attractive from both theoretical and methodological perspectives among the benefits is a degree of computational simplicity that even allows for robustness analyses to be implemented default nonsubjective versions of this analysis are developed herein in the sense that recommended choices are provided for the many features of the polya tree process that need to be specified considerable discussion of these features is also provided to assist those who might be interested in subjective choices a variety of examples involving location scale models are studied finally it is shown that the resulting procedure can be viewed as a conditional frequentist test resulting in data dependent reported err

algebraic and topological structure of some spaces of set valued maps in this paper we consider possible topological and algebraic structures on some spaces of set valued maps in particular we introduce algebraic operations on the set m x y of all minimal upper semi continuous compact valued maps from a topological space x into a topological group y it is shown that under suitable assumptions on the spaces x and y we may equip the set m x y with a group structure this structure extends the usual pointwise operations on the set of point valued continuous functions we also introduce convergence structures on certain sets of set valued maps in particular we consider the continuous convergence structure on sets of upper semi continuous maps as well as a convergence structure on m x y derived through it which is compatible with the mentioned algebraic structure it is also shown that the generalized compact open topology is compatible with the algebraic structure introduced on m x y

simultaneous bayesian frequentist sequential testing of nested hypotheses summary conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed and have been shown to be equivalent to conventional bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses these results are herein extended to sequential testing and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule seeming to lend frequentist support to the stopping rule principle

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

automatic bayes factors for testing equality and inequality constrained hypotheses on variances in comparing characteristics of independent populations researchers frequently expect a certain structure of the population variances these expectations can be formulated as hypotheses with equality and or inequality constraints on the variances in this article we consider the bayes factor for testing such in equality constrained hypotheses on variances application of bayes factors requires specification of a prior under every hypothesis to be tested however specifying subjective priors for variances based on prior information is a difficult task we therefore consider so called automatic or default bayes factors these methods avoid the need for the user to specify priors by using information from the sample data we present three automatic bayes factors for testing variances the first is a bayes factor with equal priors on all variances where the priors are specified automatically using a small share of the information in the sample data the second is the fractional bayes factor where a fraction of the likelihood is used for automatic prior specification the third is an adjustment of the fractional bayes factor such that the parsimony of inequality constrained hypotheses is properly taken into account the bayes factors are evaluated by investigating different properties such as information consistency and large sample consistency based on this evaluation it is concluded that the adjusted fractional bayes factor is generally recommendable for testing equality and inequality constrained hypotheses on variances

the index of determinacy for measures and the norm of orthonormal polynomials for determinate measures having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of it z for z a complex number using this index of determinacy we solve the problem of determining for which z e c the sequence p m z n m e n belongs to t2 where pn n is the sequence of orthonormal polynomials associated with the measure u

operating characteristics and extensions of the false discovery rate procedure summary we investigate the operating characteristics of the benjamini hochberg false discovery rate procedure for multiple testing this is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected the paper provides a framework for understanding more about this procedure we first study the asymptotic properties of the deciding point d that determines the critical p value from this we obtain explicit asymptotic expressions for a particular risk function we introduce the dual notion of false non rejections and we consider a risk function that combines the false discovery rate and false non rejections we also consider the optimal procedure with respect to a measure of conditional risk

multiple bayes factors for testing hypotheses abstract partial and multiple bayes factors are introduced to obtain pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space robust bayesian analyses are performed by introducing suitable classes of priors and by calculating lower and upper bounds of bayes factors and posterior probabilities classes of intuitively meaningful priors are introduced including unimodal densities without the constraint of symmetry for the case of precise hypotheses procedures for the corresponding optimizations are specified and examples are given

p values for composite null models abstract the problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters the most frequently used measures of compatibility are p values based on statistics t for which large values are deemed to indicate incompatibility of the data and the model when the null model has unknown parameters p values are not uniquely defined the proposals for computing a p value in such a situation include the plug in and similar p values on the frequentist side and the predictive and posterior predictive p values on the bayesian side we propose two alternatives the conditional predictive p value and the partial posterior predictive p value and indicate their advantages from both bayesian and frequentist perspectives

invariant subspaces of the quasinilpotent dt operator abstract in 4 we introduced the class of dt operators which are modeled by certain upper triangular random matrices and showed that if the spectrum of a dt operator is not reduced to a single point then it has a nontrivial closed hyperinvariant subspace in this paper we prove that also every dt operator whose spectrum is concentrated on a single point has a nontrivial closed hyperinvariant subspace in fact each such operator has a one parameter family of them it follows that every dt operator generates the von neumann algebra l f 2 of the free group on two generators

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

reconciling bayesian and frequentist evidence in the one sided testing problem abstract for the one sided hypothesis testing problem it is shown that it is possible to reconcile bayesian evidence against h 0 expressed in terms of the posterior probability that h 0 is true with frequentist evidence against h 0 expressed in terms of the p value in fact for many classes of prior distributions it is shown that the infimum of the bayesian posterior probability of h 0 is equal to the p value in other cases the infimum is less than the p value the results are in contrast to recent work of berger and sellke 1987 in the two sided point null case where it was found that the p value is much smaller than the bayesian infimum some comments on the point null problem are also given

parametrization of random vectors in polynomial chaos expansions via optimal transportation polynomial chaos pc expansions are used for the propagation of uncertainty through dynamical systems as an alternative to monte carlo methods model parameters in a given dynamical system are assumed to have known expansions which correspond to simple standard distributions and one is usually interested in the polynomial expansion of the system solution we are concerned with the problem of estimating the pc expansion of a parameter vector when only realizations from its distribution are given to this end we apply ideas from optimal transportation theory and network optimization

behavior of the generalized rosenblatt process at extreme critical exponent values the generalized rosenblatt process is obtained by replacing the single critical exponent characterizing the rosenblatt process by two different exponents living in the interior of a triangular region what happens to that generalized rosenblatt process as these critical exponents approach the boundaries of the triangle we show by two different methods that on each of the two symmetric boundaries the limit is non gaussian on the third boundary the limit is brownian motion the rates of convergence to these boundaries are also given the situation is particularly delicate as one approaches the corners of the triangle because the limit process will depend on how these corners are approached all limits are in the sense of weak convergence in c 0 1 c 0 1 these limits cannot be strengthened to convergence in l2 l2

generating functions via integral transforms abstract in this paper we use some integral transforms to derive for a polynomial sequence p n x n 0 generating functions of the type g x t n 0 n p n x t n starting from a generating function of type g x t n 0 p n x t n where n n 0 is a real numbers sequence independent on x and t that allows us to unify the treatment of a generating function problem for many well known polynomial sequences in the literature

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

dt operators and decomposability of voiculescu s circular operator the dt operators are introduced one for every pair c consisting of a compactly supported borel probability measure on the complex plane and a constant c 0 these are operators on hilbert space that are defined as limits in moments of certain upper triangular random matrices the dt operators include voiculescu s circular operator and elliptic deformations of it as well as the circular free poisson operators we show that every dt operator is strongly decomposable we also show that a dt operator generates a ii1 factor whose isomorphism class depends only on the number and sizes of atoms of those dt operators that are also r diagonal are identified for a quasi nilpotent dt operator t we find the distribution of t t and a recursion formula for general moments of t

the moment problem for polynomial forms in normal random variables let y be a random variable defined by a polynomial p w of degree n in finitely many normally distributed variables this paper studies which such variables y are determinate i e have probability laws uniquely determined by their moments extending results of berg which applied to powers of a single normal variable we prove that a y is determinate if n 1 2 or if n 4 with the essential support of the law of y strictly smaller than the real line and b y is not determinate either if n is odd 2 3 or if n is even 2 6 such that p w attains a finite minimum value some other polynomials y p w with even degree n 2 4 are proved not to be determinate

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

on the use of non local prior densities in bayesian hypothesis tests we examine philosophical problems and sampling deficiencies that are associated with current bayesian hypothesis testing methodology paying particular attention to objective bayes methodology because the prior densities that are used to define alternative hypotheses in many bayesian tests assign non negligible probability to regions of the parameter space that are consistent with null hypotheses resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses but only sublinear accumulation of evidence in favour of true null hypotheses thus it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis even when moderately large sample sizes have been obtained we review asymptotic convergence rates of bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most bayesian tests using members of these classes we obtain analytic expressions for bayes factors in linear models and derive approximations to bayes factors in large sample settings

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

solving large classes of nonlinear systems of pdes it is shown that large classes of nonlinear systems of pdes with possibly associated initial and or boundary value problems can be solved by the method of order completion the solutions obtained can be assimilated with hausdorff continuous functions the usual navier stokes equations as well as their various modifications aiming at a realistic modeling are included as particular cases the same holds for the critically important constitutive relations in various branches of continuum mechanics the solution method does not involve functional analysis nor various sobolev or other spaces of distributions or generalized functions the general and type independent existence and regularity results regarding solutions presented here have recently been introduced in the literature provided also if need be that the notion of a solution shall be suitably extended cited from hilbert s 20th problem

length biasing and laws equivalent to the log normal abstract let x 0 denote a generic lifetime of a renewal process having unit mean lifetime let hat x denote the stationary total lifetime and let q 0 1 be a fixed constant we consider anew the scale invariance problem for which laws does q hat x have the same distribution as x our setting is more probabilistic than those presented hitherto and we explore connections with the log normal moment problem in particular it is shown that all explicitly known laws which have log normal moments solve our problem the notion of remaining lifetime is generalized and its scaling invariance is investigated using the notion of total lifetime two moment equivalent laws of askey are shown to have a simple representation in terms of laws equivalent to the log normal the representation involves a q gamma law which we explore in its own right an affine extension of our basic scale invariance relation arising in the theory of orthogonal polynomials is shown to be equivalent to the latter

training samples in objective bayesian model selection central to several objective approaches to bayesian model selection is the use of training samples subsets of the data so as to allow utilization of improper objective priors the most common prescription for choosing training samples is to choose them to be as small as possible subject to yielding proper posteriors these are called minimal training samples when data can vary widely in terms of either information content or impact on the improper priors use of minimal training samples can be inadequate important examples include certain cases of discrete data the presence of censored observations and certain situations involving linear models and explanatory variables such situations require more sophisticated methods of choosing training samples a variety of such methods are developed in this paper and successfully applied in challenging situations

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

the truncated stieltjes moment problem solved by using kernel density functions in this work the problem of the approximate numerical determination of a semi infinite supported continuous probability density function pdf from a finite number of its moments is addressed the target space is carefully defined and an approximation theorem is proved establishing that the set of all convex superpositions of appropriate kernel density functions kdfs is dense in this space a solution algorithm is provided based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics the solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution a parsimonious formulation of the proposed solution procedure based on a novel sequentially adaptive scheme is developed enabling a very efficient moment data inversion the whole methodology is fully illustrated by numerical examples

a simultaneous estimation and variable selection rule a new data based method of estimation and variable selection in linear statistical models is proposed this method is based on a generalized maximum entropy formalism and makes use of both sample and non sample information in determining a basis for coefficient shrinkage and extraneous variable identification in contrast to tradition shrinkage and variable selection are achieved on a coordinate by coordinate basis and the procedure works well for both ill and well posed statistical models analytical asymptotic results are presented and sampling experiments are used as a basis for determining finite sample behavior and comparing the sampling performance of the new estimation rule with traditional competitors solution algorithms for the non linear inversion problem that results are simple to implement

an iterative polynomial chaos approach for solution of structural mechanics problem with gaussian material property abstract a new approach for solution of stochastic structural mechanics problem with random coefficient in the framework of polynomial chaos pc expansion is proposed the basic strategy of the proposed method is to iteratively construct a pc expansion with reduced numbers of unknown coefficients the method is based on orthogonal expansion of stochastic responses and generation of an iterative pc based on the responses of the previous iteration the polynomials are evaluated using gram schmidt orthogonalization process the number of random variables in pc expansion is reduced by considering only the dominant components of the response characteristics which is evaluated using karhunen loeve kl expansion in case of random material field problem kl expansion is used to discretize the random field the usefulness of the proposed method in terms of accuracy and computational efficiency is examined the results of linear static structural mechanics problems are compared with mcs and pc method the proposed method is observed to be computationally more efficient and accurate than pc method

bayesian hypothesis testing in two arm trials with dichotomous outcomes summary this article is motivated by an interest in comparing inferences made when using a bayesian or frequentist statistical approach the article addresses the study of one sided superiority and noninferiority bayesian tests these tests are stated in terms of the posterior probability that the null hypothesis is true for the binomial distribution and in terms of one sided credible limits we restrict our considerations to conjugate beta priors with integer parameters under this assumption the posterior probabilities of tested hypotheses can be transformed into the frequentist probabilities of bernoulli trials with an adjusted number of events and population sizes the method resembles a standard frequentist problem formulation by using an appropriate choice of prior parameters the posterior probabilities of the null hypothesis can be made smaller or larger than the p values of frequentist tests

stochastic elastic plastic finite elements abstract a computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic plastic materials uncertain elastic plastic material properties are modeled as random fields which appear as the coefficient term in the governing partial differential equation of mechanics a spectral stochastic elastic plastic finite element method with fokker planck kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non linear elastic plastic partial differential equation with stochastic coefficient to this end the random field material properties are discretized in both spatial and stochastic dimension into finite numbers of independent basic random variables using karhunen loeve expansion those random variables are then propagated through the elastic plastic constitutive rate equation using fokker planck kolmogorov equation approach to obtain the evolutionary material properties as the material plastifies the unknown displacement solution random field is then assembled using polynomial chaos as a function of known basic random variables and unknown deterministic coefficients which are obtained by minimizing the error of finite representation by galerkin technique

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved

asymptotic sampling distribution for polynomial chaos representation from data a maximum entropy and fisher information approach a procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos pc coefficients of a second order nonstationary and non gaussian random process by using a collection of observations the random process represents a physical quantity of interest and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector mathcal y representing a finite dimensional projection of the random process the karhunen loeve decomposition and a scaling transformation are employed to produce a reduced order model mathcal z of mathcal y the pc expansion of mathcal z is next determined by having recourse to the maximum entropy principle the metropolis hastings markov chain monte carlo algorithm and the rosenblatt transformation the resulting pc expansion has random coefficients where the random characteristics of the pc coefficients can be attributed to the limited data available from the experiment the estimators of the pc coefficients of mathcal y obtained from that of mathcal z are found to be maximum likelihood estimators as well as consistent and asymptotically efficient computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the pc coefficients of mathcal y requires knowledge of the fisher information matrix fim the fim is evaluated here by using a numerical integration scheme as well as a sampling technique the resulting confidence interval on the pc coefficient estimators essentially reflects the effect of incomplete information due to data limitation on the characterization of the stochastic process this asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

default bayes factors for nonnested hypothesis testing abstract bayesian hypothesis testing for nonnested hypotheses is studied using various default bayes factors such as the fractional bayes factor the median intrinsic bayes factor and the encompassing and expected intrinsic bayes factors the different default methods are first compared with each other and with the p value in normal one sided testing to illustrate the basic issues general results for one sided testing in location and scale models are then presented the default bayes factors are also studied for specific models involving multiple hypotheses in particular a multiple hypothesis testing example involving a sequential clinical trial is discussed in most of the examples presented we also derive the intrinsic prior this is the prior distribution which if used directly would yield answers asymptotically equivalent to those for the given default bayes factor

reduced chaos decomposition with random coefficients of vector valued random variables and random fields we develop a stochastic functional representation that is adapted to problems involving various forms of epistemic uncertainties including modeling error and data paucity the new representation builds on the polynomial chaos decomposition and eventually yields a karhunen loeve expansion with random multiplicative coefficients in this expansion one set of uncertainty is captured in the usual manner as uncorrelated scalar random variables another component of the uncertainty statistically independent from the first is captured by constructing the usually deterministic functions in the kl expansion as random functions we think of the first set of uncertainties as associated with a coarse scale model and of the second set as associated with subscale fluctuations not captured in the coarse scale description

on thermomechanical restrictions of continua the central idea proposed here is that in entropy producing processes a specific choice from among a competing class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production when attention is restricted to quadratic forms for the rate of entropy production the assumption leads to results that are fully in keeping with linear phenomenological relations that satisfy the onsager relations in other words the usual linear evolution laws such as fourier s law of heat conduction fick s law darcy s law newton s law of viscosity etc all corroborate this assumption we clarify the difference between the maximum rate of entropy production criterion that characterizes choices among constitutive relations and the minimum entropy production theorem due to onsager 1931 that characterizes steady states for special choices of the rate of entropy production we then show that for other forms of entropy production that are not quadratic for which the onsager relations and related theorems cannot be applied we can use the procedure described here to obtain nonlinear laws we demonstrate by means of an example that even yield type phenomena can be accommodated within this framework while they cannot within the framework of onsager we also discuss issues concerning constraints especially in thermoelasticity within the context of our ideas

on powers of stieltjes moment sequences ii we consider the set of stieltjes moment sequences for which every positive power is again a stieltjes moment sequence and prove an integral representation of the logarithm of the moment sequence in analogy to the levy khintchine representation we use the result to construct product convolution semigroups with moments of all orders and to calculate their mellin transforms as an application we construct a positive generating function for the orthonormal hermite polynomials