elements of a function analytic approach to probability we first provide a detailed motivation for using probability theory as a mathematical context in which to analyze engineering and scientific systems that possess uncertainties we then present introductory notes on the function analytic approach to probabilistic analysis emphasizing the connections to various classical deterministic mathematical analysis elements lastly we describe how to use the approach as a means to augment deterministic analysis methods in a particular hilbert space context and thus enable a rigorous framework for commingling deterministic and probabilistic analysis tools in an application setting

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

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

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

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

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

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

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition a new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented the eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis an efficient method for estimating the coefficients with respect to this basis is proposed the method uses a galerkin based approach by orthogonalizing the residual in the eigenvalue eigenvector equation to the subspace spanned by the basis functions used for approximation in this way the stochastic problem is framed as a system of deterministic non linear algebraic equations this system of equations is solved using a newton raphson algorithm although the proposed approach is not based on statistical sampling the efficiency of the proposed method can be significantly enhanced by initializing the non linear iterative process with a small statistical sample synthesized through a monte carlo sampling scheme the proposed method offers a number of advantages over existing methods based on statistical sampling first it provides an approximation to the complete probabilistic description of the eigensolution second it reduces the computational overhead associated with solving the statistical eigenvalue problem finally it circumvents the dependence of the statistical solution on the quality of the underlying random number generator copyright 2007 john wiley sons ltd

a matching algorithm for generation of statistically dependent random variables with arbitrary marginals simulation has gained acceptance in the operations research community as a viable method for analyzing complex problems while random generation of variables with various marginal distributions has been studied at length developing ability to preserve a given degree of statistical dependence among them has been lagging behind this paper includes a short summary of the previous work and a description of the proposed algorithm for efficient re arranging of generated random variables such that a desired product moment correlation matrix is induced the proposed approach is different from similar algorithms that induce a desired rank order correlation among random variables the algorithm is demonstrated using three numerical examples one of which also includes a comparison with risk commercial package its main features are simplicity ease of implementation and the ability to handle either theoretical or empirical distribution functions

karhunen loeve expansion revisited for vector valued random fields scaling errors and optimal basis due to scaling effects when dealing with vector valued random fields the classical karhunen loeve expansion which is optimal with respect to the total mean square error tends to favorize the components of the random field that have the highest signal energy when these random fields are to be used in mechanical systems this phenomenon can introduce undesired biases for the results this paper presents therefore an adaptation of the karhunen loeve expansion that allows us to control these biases and to minimize them this original decomposition is first analyzed from a theoretical point of view and is then illustrated on a numerical example

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

dynamics of a family of third order iterative methods that do not require using second derivatives the purpose of this article is to present results that amount to a description of the conjugacy classes of three third order root finding iterative methods that do not require the use of second derivatives for their formulation for complex polynomials of degrees two three and four for degrees two and three a full description of the conjugacy classes is accomplished in each case by a one parameter family of polynomials this is done in such a way that when one applies one of these three root finding iterative methods to the elements of these parametrized families a family of iterative methods is obtained in such a way that its dynamics represents up to conjugacy the dynamics of the corresponding iterative root finding method applied to any complex polynomial having the same degree for degree four analogous partial results are obtained

exhibiting chaos and fractals with a microcomputer abstract the microcomputer and its graphics capabilities are used to investigate chaos in newton s method for a complex valued quartic polynomial convergence maps that show fractal geometry are presented

a probabilistic graphical model based stochastic input model construction abstract model reduction techniques have been widely used in modeling of high dimensional stochastic input in uncertainty quantification tasks however the probabilistic modeling of random variables projected into reduced order spaces presents a number of computational challenges due to the curse of dimensionality the underlying dependence relationships between these random variables are difficult to capture in this work a probabilistic graphical model based approach is employed to learn the dependence by running a number of conditional independence tests using observation data thus a probabilistic model of the joint pdf is obtained and the pdf is factorized into a set of conditional distributions based on the dependence structure of the variables the estimation of the joint pdf from data is then transformed to estimating conditional distributions under reduced dimensions to improve the computational efficiency a polynomial chaos expansion is further applied to represent the random field in terms of a set of standard random variables this technique is combined with both linear and nonlinear model reduction methods numerical examples are presented to demonstrate the accuracy and efficiency of the probabilistic graphical model based stochastic input models

spectral power iterations for the random eigenvalue problem two computationally efficient algorithms are developed for solving the stochastic eigenvalue problem an algorithm based on the power iteration technique is proposed for the calculation of the dominant eigenpairs this algorithm is then extended to find other subdominant random eigenpairs the uncertainty in the operator is represented by a polynomial chaos expansion and a similar representation is considered for the random eigenvalues and eigenvectors the algorithms are distinguished due to their speed in converging to the true random eigenpairs and their ability to estimate a prescribed number of subdominant eigenpairs the algorithms are demonstrated on two examples with close agreement observed with the exact solution and a solution synthesized through monte carlo sampling

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

an approximate method for sampling correlated random variables from partially specified distributions this paper presents an algorithm for generating correlated vectors of random numbers the user need not fully specify the joint distribution function instead the user partially specifies only the marginal distributions and the correlation matrix the algorithm may be applied to any set of continuous strictly increasing distribution functions the marginal distributions need not all be of the same functional form the correlation matrix is first checked for mathematical consistency positive semi and adjusted if necessary then the correlated random vectors are generated using a combination of cholesky decomposition and gauss newton iteration applications are made to cost analysis where correlations are often present between cost elements in a work breakdown structure

a convergence study for spdes using combined polynomial chaos and dynamically orthogonal schemes we study the convergence properties of the recently developed dynamically orthogonal do field equations 1 in comparison with the polynomial chaos pc method to this end we consider a series of one dimensional prototype spdes whose solution can be expressed analytically and which are associated with both linear advection equation and nonlinear burgers equation problems with excitations that lead to unimodal and strongly bi modal distributions we also propose a hybrid approach to tackle the singular limit of the do equations for the case of deterministic initial conditions the results reveal that the do method converges exponentially fast with respect to the number of modes for the problems considered giving same levels of computational accuracy comparable with the pc method but in many cases with substantially smaller computational cost compared to stochastic collocation especially when the involved parametric space is high dimensional

dynamical aspects of some convex acceleration methods as purely iterative algorithm for newton s maps in this paper we define purely iterative algorithm for newton s maps which is a slight modification of the concept of purely iterative algorithm due to smale for this we use a characterization of rational maps which arise from newton s method applied to complex polynomials we prove the scaling theorem for purely iterative algorithm for newton s map then we focus our study in dynamical aspects of three root finding iterative methods viewed as a purely iterative algorithm for newton s map whittaker s iterative method the super halley iterative method and a modification of the latter we give a characterization of the attracting fixed points which correspond to the roots of a polynomial also numerical examples are included in order to show how to use the characterization of fixed points finally we give a description of the parameter spaces of the methods under study applied to a one parameter family of generic cubic polynomials

convergence study of the truncated karhunen loeve expansion for simulation of stochastic processes a random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients karhunen loeve k l series expansion is based on the eigen decomposition of the covariance function its applicability as a simulation tool for both stationary and non stationary gaussian random processes is examined numerically in this paper the study is based on five common covariance models the convergence and accuracy of the k l expansion are investigated by comparing the second order statistics of the simulated random process with that of the target process it is shown that the factors affecting convergence are a ratio of the length of the process over correlation parameter b form of the covariance function and c method of solving for the eigen solutions of the covariance function namely analytical or numerical comparison with the established and commonly used spectral representation method is made k l expansion has an edge over the spectral method for highly correlated processes for long stationary processes the spectral method is generally more efficient as the k l expansion method requires substantial computational effort to solve the integral equation the main advantage of the k l expansion method is that it can be easily generalized to simulate non stationary processes with little additional effort copyright 2001 john wiley sons ltd

spherically invariant vector random fields in space and time this paper is concerned with spherically invariant or elliptically contoured vector random fields in space and or time which are formulated as scale mixtures of vector gaussian random fields while a spherically invariant vector random field may or may not have second order moments a spherically invariant second order vector random field is determined by its mean and covariance matrix functions just like the gaussian one this paper explores basic properties of spherically invariant second order vector random fields and proposes an efficient approach to develop covariance matrix functions for such vector random fields

kernel principal component analysis for stochastic input model generation stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used these input models are often constructed from a set of experimental samples of the underlying random field to this end the karhunen loeve k l expansion also known as principal component analysis pca is the most popular model reduction method due to its uniform mean square convergence however it only projects the samples onto an optimal linear subspace which results in an unreasonable representation of the original data if they are non linearly related to each other in other words it only preserves the first order mean and second order statistics covariance of a random field which is insufficient for reproducing complex structures this paper applies kernel principal component analysis kpca to construct a reduced order stochastic input model for the material property variation in heterogeneous media kpca can be considered as a nonlinear version of pca through use of kernel functions kpca further enables the preservation of higher order statistics of the random field instead of just two point statistics as in the standard karhunen loeve k l expansion thus this method can model non gaussian non stationary random fields in this work we also propose a new approach to solve the pre image problem involved in kpca in addition polynomial chaos pc expansion is used to represent the random coefficients in kpca which provides a parametric stochastic input model thus realizations which are statistically consistent with the experimental data can be generated in an efficient way we showcase the methodology by constructing a low dimensional stochastic input model to represent channelized permeability in porous media

a copulas based approach to modeling dependence in decision trees this paper presents a general framework based on copulas for modeling dependent multivariate uncertainties through the use of a decision tree the proposed dependent decision tree model allows multiple dependent uncertainties with arbitrary marginal distributions to be represented in a decision tree with a sequence of conditional probability distributions this general framework could be naturally applied in decision analysis and real options valuations as well as in more general applications of dependent probability trees while this approach to modeling dependencies can be based on several popular copula families as we illustrate we focus on the use of the normal copula and present an efficient computational method for multivariate decision and risk analysis that can be standardized for convenient application

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

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

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

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

on the dynamics of the euler iterative function abstract the dynamics of euler s third order iterative method which is used to find roots of non linear equations applied to complex polynomials of degrees three and four is studied the conjugacy classes of this method are found explicitly

polynomial chaos expansions for optimal control of nonlinear random oscillators the polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems the procedure is demonstrated on a base excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its karhunen loeve expansion an adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement velocity norm and is applied to the identification of phase orbits of nonlinear oscillators this approximation is then integrated in the design of an optimal polynomial controller allowing for the efficient estimation of statistics and probability density functions of quantities of interest numerical investigations are carried out employing the polynomial chaos expansions and the lyapunov asymptotic stability condition based control policy the results reveal that the performance as gaged by probabilistic quantities of interest of the controlled oscillators is greatly improved a comparative study is also presented against the classical stochastic optimal control whereby statistical linearization based lqg is employed to design the optimal controller it is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators

polynomial chaos representation of spatio temporal random fields from experimental measurements two numerical techniques are proposed to construct a polynomial chaos pc representation of an arbitrary second order random vector in the first approach a pc representation is constructed by matching a target joint probability density function pdf based on sequential conditioning a sequence of conditional probability relations in conjunction with the rosenblatt transformation in the second approach the pc representation is obtained by having recourse to the rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target spearman s rank correlation coefficient srcc matrix both techniques are applied to model an experimental spatio temporal data set exhibiting strong non stationary and non gaussian features the data consists of a set of oceanographic temperature records obtained from a shallow water acoustics transmission experiment 1 the measurement data observed over a finite denumerable subset of the indexing set of the random process is treated as a collection of observed samples of a second order random vector that can be treated as a finite dimensional approximation of the original random field a set of properly ordered conditional pdfs that uniquely characterizes the target joint pdf in the first approach and a set of target marginal pdfs and a target srcc matrix in the second approach are estimated from available experimental data digital realizations sampled from the constructed pc representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy the relative advantages and disadvantages of the two proposed techniques are also highlighted

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

finite dimensional models for random functions abstract truncated karhunen loeve kl representations are used to construct finite dimensional fd models for non gaussian functions with finite variances the second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation polynomial chaos and translated polynomial chaos models referred to as t pc and pct models following theoretical considerations on kl representations and t pc and pct models three numerical examples are presented to illustrate the implementation and performance of these models the pct models inherit the desirable features of both t and pc models it approximates accurately all quantities of interest considered in these examples

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

julia sets and mandelbrot like sets associated with higher order schro der rational iteration functions a computer assisted study schroder iteration functions sm z a generalization of newton s method for which m 2 are constructed so that the sequence z i sm zn converges locally to a root z of g z 0 as o iz z in for g z a polynomial this involves the iteration of rational functions over the complex riemann sphere which is described by the classical theory of julia and fatou and subsequent developments the julia sets for the sm z as applied to the simple cases g z z 1 are examined for increasing m with the help of microcomputer plots the possible types of behavior of zn iteration sequences are catalogued by examining the orbits of free critical points of the sm z as applied to a one parameter family of cubic polynomials

spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning dynamical analysis tools are well established for deterministic models however for many biochemical phenomena in cells the molecule count is low leading to stochastic behavior that causes deterministic macroscale reaction models to fail the main mathematical framework representing these phenomena is based on jump markov processes that model the underlying stochastic reaction network conventional dynamical analysis tools do not readily generalize to the stochastic setting due to nondifferentiability and absence of explicit state evolution equations we developed a reduced order methodology for dynamical analysis that relies on the karhunen loeve decomposition and polynomial chaos expansions the methodology relies on adaptive data partitioning to obtain an accurate representation of the stochastic process especially in the case of multimodal behavior as a result a mixture model is obtained that represents the reduced order dynamics of the system the schlogl model is used as a prototype bistable process that exhibits time scale separation and leads to multimodality in the reduced order model

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

on torsors under elliptic curves and serre s pro algebraic structures let mathcal o k be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions k let x k be a torsor under an elliptic curve a k over k x the proper minimal regular model of x k over s hbox spec mathcal o k and j the identity component of the neron model of mathrm pic x k k 0 we study the canonical morphism q mathrm pic 0 x s rightarrow j which extends the natural isomorphism on generic fibres we show that q is pro algebraic in nature with a construction that recalls serre s work on local class field theory furthermore we interpret our results in relation to shafarevich s duality theory for torsors under abelian varieties

approximate methods for stochastic eigenvalue problems we consider the discretization and solution of eigenvalue problems of elliptic operators with random coefficients for solving the resulting systems of equations we present a new and efficient spectral inverse iteration based on the stochastic galerkin approach with respect to a polynomial chaos basis the curse of dimensionality inherent in normalization over parameter spaces is avoided by a solution of a non linear system of equations defining the galerkin coefficients for reference we also present an algorithm for adaptive stochastic collocation functionality of the algorithms is demonstrated by applying them on four examples of a given model problem convergence of the galerkin based method is analyzed and the results are tested against the collocated reference solutions and theoretical predictions

a computational inverse method for identification of non gaussian random fields using the bayesian approach in very high dimension this paper is devoted to the identification of bayesian posteriors for the random coefficients of the high dimension polynomial chaos expansions of non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to 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 must be solved to perform the identification of the random field a complete methodology is proposed to solve this very challenging problem in high dimension which consists in using the first four steps introduced in a previous paper followed by the identification of the posterior model the steps of the methodology are the following 1 introduction of a family of prior algebraic stochastic model pasm 2 identification of an optimal pasm in the constructed family using the partial experimental data 3 construction of a statistical reduced order optimal pasm 4 construction in high dimension of the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal pasm 5 substitution of these deterministic vector valued coefficients by random vector valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified 6 construction of the prior probability model of these random vector valued coefficients and finally 7 identification of the posterior probability model of these random vector valued coefficients using partial and limited experimental data through the stochastic boundary value problem two methods are proposed to carry out the identification of the posterior model the first one is based on the use of the classical bayesian method the second one is a new approach derived from the bayesian method which is more efficient in high dimension an application is presented for which several millions of random coefficients are identified

nested polynomial trends for the improvement of gaussian process based predictors the role of simulation has kept increasing for the sensitivity analysis and the uncertainty quantification of complex systems such numerical procedures are generally based on the processing of a huge amount of code evaluations when the computational cost associated with one particular evaluation of the code is high such direct approaches based on the computer code only can be not affordable surrogate models have therefore to be introduced to interpolate the information given by a fixed set of code evaluations to the whole input space when confronted to deterministic mappings the gaussian process based regression gpr or kriging presents a good compromise between complexity efficiency and error control such a method considers the quantity of interest of the system as a particular realization of a gaussian stochastic process which mean and covariance functions have to be identified from the available code evaluations in this context this work proposes an innovative parameterization of this mean function which is based on the composition of two polynomials this approach is particularly relevant for the approximation of strongly non linear quantities of interest from very little information after presenting the theoretical basis of this method this work compares its efficiency to alternative approaches on a series of examples

the effects of coefficient correlation structure in two dimensional knapsack problems on solution procedure performance this paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two dimensional knapsack problems 2kp the population correlation structure among 2kp coefficients the level of constraint slackness and the type of correlation product moment or rank are varied in this study representative branch and bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance population correlation structure and in particular the interconstraint component of the correlation structure is found to be a significant factor influencing the performance of both the algorithm and the heuristic in addition the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance

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

chessboard distributions and random vectors with specified marginals and covariance matrix there is a growing need for the ability to specify and generate correlated random variables as primitive inputs to stochastic models moti vated by this need several authors have explored the generation of random vectors with specified marginals together with a specified covariance matrix through the use of a transformation of a multivariate normal random vector the norta method a covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these characteristics we develop a computational approach for establishing whether a given covariance matrix is feasible for a given set of marginals the approach is used to rigorously establish that there are sets of marginals with feasible covariance matrix that the norta method cannot match in such cases we show how to modify the initialization phase of norta so that it will exactly match the marginals and approximately match the desired covariance matrix an important feature of our analysis is that we show that for almost any covariance matrix in a certain precise sense our computational procedure either explicitly provides a construction of a random vector with the required properties or establishes that no such random vector exists

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

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