adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

polynomial meta models with canonical low rank approximations the growing need for uncertainty analysis of complex computational models has led to an expanding use of meta models across engineering and sciences the efficiency of meta modeling techniques relies on their ability to provide statistically equivalent analytical representations based on relatively few evaluations of the original model polynomial chaos expansions pce have proven a powerful tool for developing meta models in a wide range of applications the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials the classical pce approach nevertheless faces the curse of dimensionality namely the exponential increase of the basis size with increasing input dimension to address this limitation the sparse pce technique has been proposed in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm an alternative for developing meta models with polynomial functions in high dimensional problems is offered by the newly emerged low rank approximations lra approach by exploiting the tensor product structure of the multivariate basis lra can provide polynomial representations in highly compressed formats through extensive numerical investigations we herein first shed light on issues relating to the construction of canonical lra with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions specifically we examine the selection of optimal rank stopping criteria in the updating of the polynomial coefficients and error estimation in the sequel we confront canonical lra to sparse pce in structural mechanics and heat conduction applications based on finite element solutions canonical lra exhibit smaller errors than sparse pce in cases when the number of available model evaluations is small with respect to the input dimension a situation that is often encountered in real life problems by introducing the conditional generalization error we further demonstrate that canonical lra tend to outperform sparse pce in the prediction of extreme model responses which is critical in reliability analysis

sequential subspace reliability method with univariate revolving integration a new computational framework called the sequential subspace reliability method ssrm is presented this method decomposes the multidimensional random space into multiple two dimensional subspaces in this manner ssrm is able to approximate bivariate interaction effects when the reliability estimate contribution is calculated subspace by subspace the final assessment is updated in a progressive manner the iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest in a decision making situation the flexibility of the proposed ssrm to provide iterative updates on reliability estimation becomes especially valuable in dealing with large scale and complex problems under the constraints of limited time and resources to calculate the individual subspace contributions a novel univariate revolving integration uri method is proposed the uri method takes advantage of the axisymmetric nature of a joint pr

a robust and efficient stepwise regression method for building sparse polynomial chaos expansions abstract polynomial chaos pc expansions are widely used in various engineering fields for quantifying uncertainties arising from uncertain parameters the computational cost of classical pc solution schemes is unaffordable as the number of deterministic simulations to be calculated grows dramatically with the number of stochastic dimension this considerably restricts the practical use of pc at the industrial level a common approach to address such problems is to make use of sparse pc expansions this paper presents a non intrusive regression based method for building sparse pc expansions the most important pc contributions are detected sequentially through an automatic search procedure the variable selection criterion is based on efficient tools relevant to probabilistic method two benchmark analytical functions are used to validate the proposed algorithm the computational efficiency of the method is then illustrated by a more realistic cfd application consisting of the non deterministic flow around a transonic airfoil subject to geometrical uncertainties to assess the performance of the developed methodology a detailed comparison is made with the well established lar based selection technique the results show that the developed sparse regression technique is able to identify the most significant pc contributions describing the problem moreover the most important stochastic features are captured at a reduced computational cost compared to the lar method the results also demonstrate the superior robustness of the method by repeating the analyses using random experimental designs

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

strain and stress computations in stochastic finite element methods this paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element method based on the polynomial chaos expansion it identifies the cases where this objective can be achieved by analytical means using the orthogonality property of the chaos polynomials and those where it requires a numerical integration technique to this effect the applicability and efficiency of several numerical integration schemes are considered these include the gauss hermite quadrature with the direct tensor product also known as the kronecker product smolyak s approximation of such a tensor product monte carlo sampling and the latin hypercube sampling method an algorithm for reducing the dimensionality of integration under a direct tensor product is also explored for optimizing the computational cost and complexity the convergence rate and algorithmic complexity of all of these methods are discussed and illustrated with the non deterministic linear stress analysis of a plate copyright 2007 john wiley sons ltd

a new specification of generalized linear models for categorical responses many regression models for categorical responses have been introduced motivated by different paradigms but it is difficult to compare them because of their different specifications in this paper we propose a unified specification of regression models for categorical responses based on a decomposition of the link function into an inverse continuous cumulative distribution function and a ratio of probabilities this allows us to define a new family of reference models for nominal responses comparable to the families of adjacent cumulative and sequential models for ordinal responses a new equivalence between cumulative and sequential models is shown invariances under permutations of the categories are studied for each family of models we introduce a reversibility property that distinguishes adjacent and cumulative models from sequential models the new family of reference models is tested on three benchmark classification datasets

adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation the polynomial dimensional decomposition pdd is employed in this work for the global sensitivity analysis and uncertainty quantification uq of stochastic systems subject to a moderate to large number of input random variables due to the intimate connection between the pdd and the analysis of variance anova approaches pdd is able to provide a simpler and more direct evaluation of the sobol sensitivity indices when compared to the polynomial chaos expansion pc unfortunately the number of pdd terms grows exponentially with respect to the size of the input random vector which makes the computational cost of standard methods unaffordable for real engineering applications in order to address the problem of the curse of dimensionality this work proposes essentially variance based adaptive strategies aiming to build a cheap meta model i e surrogate model by employing the sparse pdd approach with its coefficients computed by regression three levels of adaptivity are carried out in this paper 1 the truncated dimensionality for anova component functions 2 the active dimension technique especially for second and higher order parameter interactions and 3 the stepwise regression approach designed to retain only the most influential polynomials in the pdd expansion during this adaptive procedure featuring stepwise regressions the surrogate model representation keeps containing few terms so that the cost to resolve repeatedly the linear systems of the least squares regression problem is negligible the size of the finally obtained sparse pdd representation is much smaller than the one of the full expansion since only significant terms are eventually retained consequently a much smaller number of calls to the deterministic model is required to compute the final pdd coefficients

an adaptive least squares global sensitivity method and application to a plasma coupled combustion prediction with parametric correlation abstract we introduce an efficient non intrusive surrogate based methodology for global sensitivity analysis and uncertainty quantification modified covariance based sensitivity indices mcov si are defined for outputs that reflect correlated effects the overall approach is applied to simulations of a complex plasma coupled combustion system with disparate uncertain parameters in sub models for chemical kinetics and a laser induced breakdown ignition seed the surrogate is based on an analysis of variance anova expansion such as widely used in statistics with orthogonal polynomials representing the anova subspaces and a polynomial dimensional decomposition pdd representing its multi dimensional components the coefficients of the pdd expansion are obtained using a least squares regression which both avoids the direct computation of high dimensional integrals and affords an attractive flexibility in choosing sampling points this facilitates importance sampling using a bayesian calibrated posterior distribution which is fast and thus particularly advantageous in common practical cases such as our large scale demonstration for which the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense effort instead is focused on efficient finite resolution sampling standard covariance based sensitivity indices cov si are employed to account for correlation of the uncertain parameters magnitude of cov si is unfortunately unbounded which can produce extremely large indices that limit their utility alternatively mcov si are then proposed in order to bound this magnitude 0 1 the polynomial expansion is coupled with an adaptive anova strategy to provide an accurate surrogate as the union of several low dimensional spaces avoiding the typical computational cost of a high dimensional expansion it is also adaptively simplified according to the relative contribution of the different polynomials to the total variance the approach is demonstrated for a laser induced turbulent combustion simulation model which includes parameters with correlated effects

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

a polynomial dimensional decomposition for stochastic computing this article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences the method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the coefficients of the expansion the new decomposition method does not require sample points as in the previous version yet it generates a convergent sequence of lower variate estimates of the probabilistic characteristics of a generic stochastic response the results of five numerical examples indicate that the proposed decomposition method provides accurate convergent and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems

polynomial chaos expansion with latin hypercube sampling for estimating response variability a computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented to deal with the random nature of input parameters of structural models several efficient probabilistic methods are investigated specifically the polynomial chaos expansion with latin hypercube sampling is used to represent the response of an uncertain system latin hypercube sampling is employed for evaluating the generalized fourier coefficients of the polynomial chaos expansion because the key challenge in uncertainty analysis is to find the most significant components that drive response variability analysis of variance is employed to find the significant parameters of the approximation model several analytical examples and a large finite element model of a joined wing are used to verify the effectiveness of this procedure

weighted discrete least squares polynomial approximation using randomized quadratures we discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation the problem stems from uncertainty quantification uq where the independent variables of the functions are random variables with specified probability measure we propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product gaussian quadrature points we analyze the stability properties of this method and prove that the method is asymptotically stable provided that the number of points scales linearly up to a logarithmic factor with the cardinality of the polynomial space specific results in both bounded and unbounded domains are obtained along with a convergence result for chebyshev measure numerical examples are provided to verify the theoretical results

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

computational uncertainty analysis in multiresolution materials via stochastic constitutive theory abstract a stochastic constitutive theory is proposed in this work to propagate microstructure uncertainties in computational multiscale continuum models to bulk multiresolution material behavior ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized in detail and this research transmits these uncertainties to coarser material resolutions by introducing a stochastic constitutive theory deduced from volume element simulations to implement the stochastic upscaling process two advanced uncertainty quantification methods are examined statistical copula functions and random process polynomial chaos expansion both methods confront the mathematical difficulty in randomizing constitutive laws by capturing the marked correlation among constitutive parameters seen in complex materials thus the results proffer a more accurate probabilistic estimation of constitutive material behavior the contribution of this work is twofold uncertainty is propagated from heterogeneous material structure to material property via the stochastic constitutive theory and rigorous data driven mathematics are formalized to represent complicated dependence structures in multivariate statistical distributions to the authors knowledge this is the first work in multiresolution mechanics that presents an approach to computationally derive correlation functions from numerical experiments as opposed for instance to assuming one a priori the method put forth in this research though quite general is applied to a mathematical example and plastic high strength steel alloy for demonstration results include stochastic constitutive curve confidence intervals for the material stress strain response and qualitative comparisons of the two stochastic methods detailed herein

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

prediction of uncertain frequency response function bounds using polynomial chaos expansion a probabilistic method is developed to predict the uncertainty bounds on frequency response functions frfs developed from finite element models a non intrusive polynomial chaos expansion pce method is used to predict uncertainty regression models of the various parameters that make up a curvefit of the frf natural frequencies damping ratios complex amplitudes mass and stiffness residuals by making use of an efficient latin hypercube technique these uncertainty models are then combined to efficiently determine pdfs of the parameters and also the uncertainty bounds of the frfs the approach is demonstrated using two examples a simple beam containing uncertainty in young s modulus and a full scale aircraft composite wing model containing uncertainties in both young s modulus and the shear modulus the results were compared with monte carlo simulation mcs and it was found that the parameter pdfs and frf error bounds obtained using a 2nd order pce model agreed very well whilst requiring significantly less computation

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc