multi element least square hdmr methods and their applications for stochastic multiscale model reduction stochastic multiscale modeling has become a necessary approach to quantify uncertainty and characterize multiscale phenomena for many practical problems such as flows in stochastic porous media the numerical treatment of the stochastic multiscale models can be very challengeable as the existence of complex uncertainty and multiple physical scales in the models to efficiently take care of the difficulty we construct a computational reduced model to this end we propose a multi element least square high dimensional model representation hdmr method through which the random domain is adaptively decomposed into a few subdomains and a local least square hdmr is constructed in each subdomain these local hdmrs are represented by a finite number of orthogonal basis functions defined in low dimensional random spaces the coefficients in the local hdmrs are determined using least square methods we paste all the local hdmr approximations together to form a global hdmr approximation to further reduce computational cost we present a multi element reduced least square hdmr which improves both efficiency and approximation accuracy in certain conditions to effectively treat heterogeneity properties and multiscale features in the models we integrate multiscale finite element methods with multi element least square hdmr for stochastic multiscale model reduction this approach significantly reduces the original model s complexity in both the resolution of the physical space and the high dimensional stochastic space we analyze the proposed approach and provide a set of numerical experiments to demonstrate the performance of the presented model reduction techniques multi element least square hdmr is proposed to treat stochastic models random domain is adaptively decomposed into some subdomains to obtain adaptive multi element hdmr least square reduced hdmr is proposed to enhance computation efficiency and approximation accuracy in certain conditions integrating msfem and multi element least square hdmr can significantly reduce computation complexity

an extension of sums of squares relaxations to polynomial optimization problems over symmetric cones this paper is based on a recent work by kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs let varepsilon and varepsilon be a finite dimensional real vector space and a symmetric cone embedded in varepsilon examples of varepsilon and varepsilon include a pair of the n dimensional euclidean space and its nonnegative orthant a pair of the n dimensional euclidean space and n dimensional second order cones and a pair of the space of m m real symmetric or complex hermitian matrices and the cone of their positive semidefinite matrices sums of squares relaxations are further extended to a polynomial optimization problem over varepsilon i e a minimization of a real valued polynomial a x in the n dimensional real variable vector x over a compact feasible region bf x b bf x in varepsilon where b x denotes an varepsilon valued polynomial in x it is shown under a certain moderate assumption on the varepsilon valued polynomial b x that optimal values of a sequence of sums of squares relaxations of the problem which are converted into a sequence of semidefinite programs when they are numerically solved converge to the optimal value of the problem

necessary conditions controllability and the value function for differential difference inclusions on deduit des conditions necessaires pour la commande optimale d un systeme contraint par des inclusions differentielles aux differences

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

nonparametric density estimation for randomly perturbed elliptic problems i computational methods a posteriori analysis and adaptive error control we consider the nonparametric density estimation problem for a quantity of interest computed from solutions of an elliptic partial differential equation with randomly perturbed coefficients and data our particular interest are problems for which limited knowledge of the random perturbations are known we derive an efficient method for computing samples and generating an approximate probability distribution based on lion s domain decomposition method and the neumann series we then derive an a posteriori error estimate for the computed probability distribution reflecting all sources of deterministic and statistical errors finally we develop an adaptive error control algorithm based on the a posteriori estimate

estimation and testing for lattice conditional independence models on euclidean jordan algebras in this paper we generalize the major results of andersson and perlman on lci models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized lci models that we define to this end we replace the cone h i r sitting inside the jordan algebra of symmetric real i x i matrices by the symmetric cone of an euclidean jordan algebra v we introduce the andersson perlman cone k which generalizes f k h i r we prove several characterizations and properties of k which allows us to recover though with different proofs the main results of andersson and perlman regarding p k the new lattice conditional independence models are defined assuming that the euclidean jordan algebra v has a symmetric representation using standard results from the theory of jordan algebras we can reduce the general model to the case where v is the jordan algebra of hermitian matrices over the real complex or quaternionic numbers and is the corresponding cone of positive definite matrices our main statistical result is a closed form formula for the estimate of the covariance matrix in the generalized lci model we also give the likelihood ratio test for testing a given model versus another one nested within the first

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

positive and doubly stochastic maps and majorization in euclidean jordan algebras abstract a positive map between euclidean jordan algebras is a symmetric cone order preserving linear map we show that the norm of such a map is attained at the unit element thus obtaining an analog of the operator matrix theoretic russo dye theorem a doubly stochastic map between euclidean jordan algebras is a positive unital and trace preserving map we relate such maps to jordan algebra automorphisms and majorization

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

commutation principle for variational problems on euclidean jordan algebras this paper establishes a commutation result for variational problems involving spectral sets and spectral functions the discussion takes places in the context of a general euclidean jordan algebra

scaling theory for quasibrittle structural failure this inaugural article has a twofold purpose i to present a simpler and more general justification of the fundamental scaling laws of quasibrittle fracture bridging the asymptotic behaviors of plasticity linear elastic fracture mechanics and weibull statistical theory of brittle failure and ii to give a broad but succinct overview of various applications and ramifications covering many fields many kinds of quasibrittle materials and many scales from 10 8 to 106 m the justification rests on developing a method to combine dimensional analysis of cohesive fracture with second order accurate asymptotic matching this method exploits the recently established general asymptotic properties of the cohesive crack model and nonlocal weibull statistical model the key idea is to select the dimensionless variables in such a way that in each asymptotic case all of them vanish except one the minimal nature of the hypotheses made explains the surprisingly broad applicability of the scaling laws

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

an upscaling method using coefficient splitting and its applications to elliptic pdes in this paper we develop an upscaling method using coefficient splitting techniques green s function is constructed using the differential operator associated with the first part of the splitting an effective upscaling coefficient is recursively calculated by green s function the computation of the upscaling process involves some independent steps combining the proposed upscaling method with the stochastic collocation method we present a stochastic space reduction collocation method where the stochastic collocation method is performed on a lower dimension stochastic space than the full dimension stochastic space we thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic pdes computation complexity is also addressed for the stochastic upscaling method a number of numerical tests are presented to confirm the convergence analysis

local soltutions for a nonlinear degenerate hyperbolic equation on etudie l existence et l unicite des solutions locales pour le probleme mixte associe a l equation hyperbolique degenere du type 2 u 2 m u x t 2 dx u 0 ou est un ouvert borne de r n a frontiere lisse et x est un vecteur de r n

spectral sets and functions on euclidean jordan algebras abstract spectral sets functions in euclidean jordan algebras are generalizations of permutation invariant sets respectively functions in r n in this article we study properties of such sets and functions and show how they are related to algebra automorphisms and majorization we show that spectral sets functions are indeed invariant under automorphisms but the converse may not hold unless the algebra is r n or simple we study schur convex spectral functions and provide some applications we also discuss the transfer principle and a related metaformula

parallel solution methods for stochastic finite element analysis using monte carlo simulation in the present paper innovative solution strategies for parallel computer implementation have been developed in connection with the monte carlo simulation mcs and the weighted integral method to produce efficient numerical handling of stochastic finite element analysis for 2d plane stress strain problems furthermore mcs in conjunction with the local average method is also used to extend the stochastic finite element analysis to 3d solid structures although mcs approaches have the major advantage that accurate solutions can be obtained for any type of problem whose deterministic solution is known either numerically or analytically their applicability is hindered by the high computational effort that is required the implementation of innovative parallel solution techniques in this study resulted in cost effective treatment of these highly computationally demanding problems one and two level domain decomposition methods have been implemented numerical results revealed that the proposed approaches permit an efficient treatment of stochastic finite element analysis for real scale 2d plane stress strain and 3d solid structures

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

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

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

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

the existence and decay of solutions of a damped kirchhoff carrier equation in banach spaces abstract this paper is concerned with the study of the existence and decay of solutions of the following initial value problem b u t m u t w a u t 1 k t u t d s 2 a u t 0 t 0 u 0 u 0 u 0 u 1 where v is a hilbert space with dual v a and b symmetric linear operators from v into v with b v v 0 v 0 and a v v v v 2 0 s a restriction of the operator a w a banach space m the real function m m 0 m 1 with m 0 0 and m 1 0 real numbers k a positive function and real numbers with 0 and 1 the successive approximation method the characterization of the derivative of m u t w and the arzela ascoli theorem allow us to obtain a local solution of the global solution follows by the prolongation method of solutions the exponential decay of the solution is derived by the perturbed energy method

existence results for some kirchhoff carrier problems abstract analyzing the viscoelastic problem for small vibrations of elastic strings kirchhoff and carrier proposed two different models of nonlinear partial differential equations by combining these two models we deal here with some nonlocal hyperbolic problems that cover a large class of kirchhoff and carrier type problems the existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established the proofs are based on the combination of the schauder fixed point theorem with some asymptotic method

permutation invariant proper polyhedral cones and their lyapunov rank abstract the lyapunov rank of a proper cone k in a finite dimensional real hilbert space is defined as the dimension of the space of all lyapunov like transformations on k or equivalently the dimension of the lie algebra of the automorphism group of k this rank measures the number of linearly independent bilinear relations needed to express a complementarity system on k that arises for example from a linear program or a complementarity problem on the cone motivated by the problem of describing spectral proper cones where the complementarity system can be expressed as a square system that is where the lyapunov rank is greater than equal to the dimension of the ambient space we consider proper polyhedral cones in r n that are permutation invariant for such cones we show that the lyapunov rank is either 1 in which case the cone is irreducible or n in which case the cone is isomorphic to r n in the latter case we show that the corresponding spectral cone is isomorphic to a symmetric cone

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

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

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

stochastic computation based on orthogonal expansion of random fields abstract in solving stochastic differential equations recently a random variable based polynomial chaos rv pc method has been developed as a major numerical solver for many realistic random media problems the rv pc method however confronts a critical challenge of curse of dimensionality since a random field is represented by random variables the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse of dimensionality distinguished from all the random variable based methods in this study a novel random field based orthogonal expansion rf oe method is proposed in aim to circumvent the curse of dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes e g seismic ocean wave wind load shock wave and geophysical media multiscale modeling of random media problems is selected as the benchmark problem to test the rf oe method especially the rf oe method provides a perfect matching with the higher order mehler s formula by replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field process the rf oe method reduces the number of dimensions of a stochastic differential equation exponentially in the first example the rf oe method is verified with monte carlo simulation on a lognormal random media flow transport problem in the second example the rf oe method is applied to a time domain problem involving orthogonal expansion of random excitations in the conclusion the items for further development of the rf oe method are identified

a stochastic computational method for evaluation of global and local behavior of random elastic media a stochastic computational method is developed to evaluate global effective properties and local probabilistic behavior of random elastic media the elasticity solution for random media is addressed with a method that combines a fast iterative numerical method with stochastic decomposition techniques stochastic homogenization problems are mathematically formulated with an ensemble average approximation scheme based on a concept of stochastic representative volume element srve probabilistic descriptors of local strain are generated based on convolution equations derived from the equilibrium equations of elasticity with the application of the galerkin formulation in probability space and the fourier transform of the convolution equations an efficient numerical scheme is implemented using an iterative algorithm that takes advantage of fast fourier transform two algorithms are provided for gaussian and non gaussian random media respectively with corresponding sufficient conditions for convergence derived finally numerical experiments are conducted for a gaussian and a non gaussian random medium respectively with verifications obtained from monte carlo simulation

existence of local solutions of the kirchhoff carrier equation in banach spaces abstract this paper is concerned with the study of the existence of a local solution of the problem b u t m u t w a u t 0 in v t 0 u 0 u 0 u 0 u 1 u 0 0 where v is a hilbert space with dual v a and b symmetric linear operators from v into v with a v v 0 and b v v 0 v 0 w a banach space with v continuously embedding in w a real number with 1 and m a function with m 0 m u 0 w 0 and smooth in a neighborhood of u 0 w the characterization of the derivative of the nonlinear term of the equation of and the arzela ascoli theorem allow us to obtain a solution u of defined in 0 t 0 where t 0 depends on u 0 u 1 and m

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

energy decay of solutions of a degenerate kirchhoff equation with a weak nonlinear dissipation abstract in this paper we study decay properties of the solutions to the degenerate kirchhoff equation with a weak nonlinear dissipative term

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

on the connectedness of spectral sets and irreducibility of spectral cones in euclidean jordan algebras abstract let v be a euclidean jordan algebra of rank n a set e in v is said to be a spectral set if there exists a permutation invariant set q in r n such that e 1 q where v r n is the eigenvalue map that takes x v to x the vector of eigenvalues of x written in the decreasing order if the above q is also a convex cone we say that e is a spectral cone this paper deals with connectedness and arcwise connectedness properties of spectral sets by relying on the result that in a simple euclidean jordan algebra every eigenvalue orbit x y y x is arcwise connected we show that if a permutation invariant set q is connected arcwise connected then 1 q is connected respectively arcwise connected a related result is that in a simple euclidean jordan algebra every pointed spectral cone is irreducible

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

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

spectral cones in euclidean jordan algebras abstract a spectral cone in a euclidean jordan algebra v of rank n is of the form k 1 q where q is a permutation invariant convex cone in r n and v r n is the eigenvalue map which takes x to x the vector of eigenvalues of x with entries written in the decreasing order in this paper we describe some properties of spectral cones we show for example that spectral cones are invariant under automorphisms of v that the dual of a spectral cone is a spectral cone when v is simple or carries the canonical inner product and characterize the pointedness solidness of a spectral cone we also show that for any spectral cone k in v dim k 0 1 m 1 m where dim k denotes the dimension of k and m is the dimension of v

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data