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

addendum to on the stability of functional equations on square symmetric groupoid abstract let x be a square symmetric groupoid and y d a complete metric divisible square symmetric groupoid in this paper we obtain the hyers ulam stability problem of functional inequality d g x y g x g y e x y for approximate mapping g x y of functional equation f x y f x f y differing by e x 2 0 in particular we have investigated the case of f x f y h f x 1 t f y 1 t on some set y

an enhanced hybrid method for the simulation of highly skewed non gaussian stochastic fields abstract in this paper an enhanced hybrid method ehm is presented for the simulation of homogeneous non gaussian stochastic fields with prescribed target marginal distribution and spectral density function the presented methodology constitutes an efficient blending of the deodatis micaletti method with a neural network based function approximation precisely the function fitting ability of neural networks based on the resilient back propagation rprop learning algorithm is employed to approximate the unknown underlying gaussian spectrum the resulting algorithm can be successfully applied for simulating narrow banded fields with very large skewness at a fraction of the computing time required by the existing methods its computational efficiency is demonstrated in three numerical examples involving fields that follow the beta and lognormal distributions

long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos abstract polynomial chaos pc is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems in the case of a structural system under dynamic loading the pc method is combined with a time integration scheme the pc method with galerkin projection converts the stochastic pde to a set of simultaneous deterministic pde which can be solved using any standard time integration scheme however the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response the probability density functions of the responses are not of constant type as the time progresses however in the case of generalized pc gpc these are approximated using initially considered gpc thus loses its efficiency time dependent general polynomial chaos tdgpc can be considered to overcome these difficulties where polynomials are generated on the fly to match with the pdf of the responses the present study considered an investigation of structural mechanics responses with both gaussian and non gaussian random system properties under dynamic loading using tdgpc an adaptive updation scheme based on the rms value of the coefficients of pc expansion is investigated for a proper updation of pc expansion the curse of dimensionality of pc expansion is addressed by considering only the dominant components of the responses evaluated using kl expansion further starting with lower order pc the order of pc expansion is increased only after the first updation thus reducing computational complexities

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

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 conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

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

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

wiener hermite polynomial expansion for multivariate gaussian probability measures abstract this paper introduces a new transformation free generalized polynomial chaos expansion pce comprising multivariate hermite orthogonal polynomials in dependent gaussian random variables the second moment properties of hermite polynomials reveal a weakly orthogonal system when obtained for a general gaussian probability measure still the exponential integrability of norm allows the hermite polynomials to constitute a complete set and hence a basis in a hilbert space the completeness is vitally important for the convergence of the generalized pce to the correct limit the optimality of the generalized pce and the approximation quality due to truncation are discussed new analytical formulae are proposed to calculate the mean and variance of a generalized pce approximation of a general output variable in terms of the expansion coefficients and statistical properties of hermite polynomials however unlike in the classical pce calculating the coefficients of the generalized pce requires solving a coupled system of linear equations besides the variance formula of the generalized pce contains additional terms due to statistical dependence among gaussian variables the additional terms vanish when the gaussian variables are statistically independent reverting the generalized pce to the classical pce numerical examples illustrate the generalized pce approximation in estimating the statistical properties of various output variables

independent component analysis to enhance performances of karhunen loeve expansions for non gaussian stochastic processes application to uncertain systems abstract in the context of engineering systems an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters for such input parameters modeled as stochastic processes karhunen loeve expansion is a classical approach providing efficient representations using a set of uncorrelated but generally statistically dependent random variables the dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications this simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state thus limiting the effectiveness of karhunen loeve expansion in certain cases in this paper independent component analysis is exploited to linearly transform the random variables used in karhunen loeve expansion resulting into a set of random variables exhibiting higher order decorrelation the stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non gaussian stochastic process under the assumption of independence among the random variables used in the karhunen loeve expansion and independent component analysis representations the latter provides more accurate statistical characterization of the output process for the specific cases examined

adaptive sparse polynomial dimensional decomposition methods for high dimensional stochastic computing this article presents two novel adaptive sparse polynomial dimensional decomposition pdd methods for solving high dimensional uncertainty quantification problems in computational science and engineering the methods entail global sensitivity analysis for retaining important pdd component functions and a full or sparse grid dimension reduction integration or quasi monte carlo simulation for estimating the pdd expansion coefficients a unified algorithm endowed with two distinct ranking schemes for grading component functions was created for their numerical implementation the fully adaptive sparse pdd method is comprehensive and rigorous leading to the second moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish a partially adaptive sparse pdd method obtained through regulated adaptivity and sparsity is economical and is therefore expected to solve practical problems with numerous variables compared with past developments the adaptive sparse pdd methods do not require their truncation parameter s to be assigned a priori or arbitrarily the numerical results reveal that an adaptive sparse pdd method achieves a desired level of accuracy with considerably fewer coefficients compared with existing pdd approximations for a required accuracy in calculating the probabilistic response characteristics the new bivariate adaptive sparse pdd method is more efficient than the existing bivariately truncated pdd method by almost an order of magnitude finally stochastic dynamic analysis of a disk brake system was performed demonstrating the ability of the new methods to tackle practical engineering problems

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

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

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

stability analysis of the nonlinear galerkin method our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations and to study the stability of the schemes spatial discretization can be performed by either spectral or pseudospectral methods finite elements or finite differences time discretization is done by two level schemes partly or fully explicit the algorithms that we present stem from the study of the evolution equations from the dynamical systems point of view they are based on a differentiated treatment of the small and large wave lengths and they are particularly adapted to the integration of such equations on large intervals of time introduction the long time integration of dissipative evolution equations has emerged as a new type of problem in numerical analysis and its appearance is due to the considerable increase in computing power that we have seen during the last years much effort has been devoted in the past to the approximation of evolution equations on finite intervals of time or in the dynamically simple case where the solution converges to a steady stationary state as time goes to infinity however in many physically relevant situations the solutions to a dissipative evolution equation do not converge to a stationary solution they rather remain time dependent and they converge to a compact attractor that encompasses the nonlinear dynamics hence and whether this is addressed implicitly or explicitly the numerical integration of the evolution equation is then closely related to the approximation of the attractor that may be a complicated set even a fractal inertial algorithms are new integration algorithms that are based on the approximation of the attractor by smooth simple manifolds namely the approximate inertial manifolds these algorithms will be recalled below but at this point let us point out that one of their main features is a differentiated treatment of small and large wave lengths an inertial manifold is an exact law of dependence of small wave lengths in terms of the large ones the small wave received april 3 1990 1980 mathematics subject classification 1985 revision primary 65f10 65f35 65n20 35a40 1991 american mathematical society 0025 5718 91 1 00 25 perpage

a polynomial chaos expansion in dependent random variables abstract this paper introduces a new generalized polynomial chaos expansion pce comprising measure consistent multivariate orthonormal polynomials in dependent random variables unlike existing pces whether classical or generalized no tensor product structure is assumed or required important mathematical properties of the generalized pce are studied by constructing orthogonal decomposition of polynomial spaces explaining completeness of orthogonal polynomials for prescribed assumptions exploiting whitening transformation for generating orthonormal polynomial bases and demonstrating mean square convergence to the correct limit analytical formulae are proposed to calculate the mean and variance of a truncated generalized pce for a general output variable in terms of the expansion coefficients an example derived from a stochastic boundary value problem illustrates the generalized pce approximation in estimating the statistical properties of an output variable for 12 distinct non product type probability measures of input variables

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

convergence and stability of finite element nonlinear galerkin method for the navier stokes equations in this article we present a new fully discrete finite element nonlinear galerkin method which are well suited to the long time integration of the navier stokes equations spatial discretization is based on two grid finite element technique time discretization is based on euler explicit scheme with variable time step size moreover we analyse the boundedness convergence and stability condition of the finite element nonlinear galerkin method our discussion shows that the time step constraints of the method depend only on the coarse grid parameter h and the time step constraints of the finite element galerkin method depend on the fine grid parameter h h under the same convergence accuracy

bayesian sparse polynomial chaos expansion for global sensitivity analysis polynomial chaos expansions are frequently used by engineers and modellers for uncertainty and sensitivity analyses of computer models they allow representing the input output relations of computer models usually only a few terms are really relevant in such a representation it is a challenge to infer the best sparse polynomial chaos expansion of a given model input output data set in the present article sparse polynomial chaos expansions are investigated for global sensitivity analysis of computer model responses a new bayesian approach is proposed to perform this task based on the kashyap information criterion for model selection the efficiency of the proposed algorithm is assessed on several benchmarks before applying the algorithm to identify the most relevant inputs of a double diffusive convection model

a formulation for stochastic finite element analysis of plate structures with uncertain poisson s ratio up to now the young s modulus is mainly dealt within the analysis of response variability however since the poisson s ratio is the other material constant which influences the behavior of structures the independent evaluation of the effects of this parameter on the response variability is of importance in this paper a formulation to determine the response variability in plate structure due to the randomness of poisson s ratio is given to filter out the independent contributions of randomness in poisson s ratio to the response variability the constitutive matrix has to be decomposed into several sub matrices in order to include the poisson s ratio in the constitutive relation as a non linear parameter a polynomial expansion of poisson s ratio is introduced to demonstrate the validity of the proposed formulation an example is chosen and the results are compared with those obtained by means of monte carlo simulation through the formulation proposed in this paper it becomes possible for the non statistical weighted integral stochastic approach to deal with all the uncertain material parameters in its application

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded