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

dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

limited range multilinear extrapolation with applications to the bilinear hilbert transform we prove a limited range off diagonal extrapolation theorem that generalizes a number of results in the theory of rubio de francia extrapolation and use this to prove a limited range multilinear extrapolation theorem we give two applications of this result to the bilinear hilbert transform first we give sufficient conditions on a pair of weights w 1 w 2 for the bilinear hilbert transform to satisfy weighted norm inequalities of the form begin aligned bh l p 1 left w 1 p 1 right times l p 2 left w 2 p 2 right longrightarrow l p w p end aligned where w w 1w 2 and frac 1 p frac 1 p 1 frac 1 p 2 frac 3 2 this improves the recent results of culiuc et al by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to frac 2 3 the critical index from the unweighted theory of the bilinear hilbert transform second as an easy consequence of our method we obtain that the bilinear hilbert transform satisfies some vector valued inequalities with muckenhoupt weights this reproves and generalizes some of the vector valued estimates obtained by benea and muscalu in the unweighted case we also generalize recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators

two parameter version of bourgain s inequality rational frequencies our aim is to establish the first two parameter version of bourgain s maximal logarithmic inequality on l 2 r 2 for the rational frequencies we achieve this by introducing a variant of a two parameter rademacher menschov inequality the method allows us to control an oscillation seminorm as well

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

a multigrid solver for two dimensional stochastic diffusion equations steady and unsteady diffusion equations with stochastic diffusivity coefficient and forcing term are modeled in two dimensions by means of stochastic spectral representations problem data and solution variables are expanded using the polynomial chaos system the approach leads to a set of coupled problems for the stochastic modes spatial finite difference discretization of these coupled problems results in a large system of equations whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time to accelerate the convergence of the iterative technique a multigrid method based on spatial coarsening is implemented numerical experiments show good scaling properties of the method both with respect to the number of spatial grid points and the stochastic resolution level

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

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

nonparametric probabilistic approach of uncertainties for elliptic boundary value problem the paper is devoted to elliptic boundary value problems with uncertainties such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media model uncertainties are induced by the mathematical physical process which allows the boundary value problem to be constructed from the design system if experiments are not available the bayesian approach cannot be used to take into account model uncertainties recently a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non linear dynamical systems nevertheless the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem for instance for a linear elastostatic problem relative to an elastic bounded domain we then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems the theory and its validation are presented

lp estimates for non smooth bilinear littlewood paley square functions on mathbb r in this work we study some non smooth bilinear analogues of linear littlewood paley square functions on the real line we prove boundedness properties in lebesgue spaces for them let us consider the functions phi n satisfying widehat phi n xi bf 1 n n 1 xi and define the bilinear operator s n f g x int f x y g x y phi n y dy these bilinear operators are closely related to the bilinear hilbert transforms then for exponents p q r in 2 infty satisfying frac 1 p frac 1 q frac 1 r we prove that left left sum n in mathbb z left s n f g right 2 right 1 2 right l r mathbb r lesssim f l p mathbb r g l q mathbb r

truncations of multilinear hankel operators we extend to multilinear hankel operators the fact that some truncations of bounded hankel operators are still bounded we prove and use a continuity property of bilinear hilbert transforms on products of lipschitz spaces and hardy spaces

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

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

non intrusive low rank separated approximation of high dimensional stochastic models this work proposes a sampling based non intrusive approach within the context of low rank separated representations to tackle the issue of curse of dimensionality associated with the solution of models e g pdes odes with high dimensional random inputs under some conditions discussed in details the number of random realizations of the solution required for a successful approximation grows linearly with respect to the number of random inputs the construction of the separated representation is achieved via a regularized alternating least squares regression together with an error indicator to estimate model parameters the computational complexity of such a construction is quadratic in the number of random inputs the performance of the method is investigated through its application to three numerical examples including two ode problems with high dimensional random inputs

vector valued bilinear maximal operator and method of rotations abstract in this paper we obtain a bilinear analogue of fefferman stein s vector valued inequality for classical hardy littlewood maximal function also we prove the boundedness of bilinear hardy littlewood maximal operator from l p 1 r n l p 2 r n l 1 r n where 1 p 1 1 p 2 1 by applying the method of rotations

maximal multilinear operators we establish multilinear l p bounds for a class of maximal multilinear averages of functions of one variable reproving and generalizing the bilinear maximal function bounds of lacey 2000 as an application we obtain almost everywhere convergence results for these averages and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system

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

numerical methods for the discretization of random fields by means of the karhunen loeve expansion the computational efficiency of random field representations with the karhunen loeve kl expansion relies on the solution of a fredholm integral eigenvalue problem this contribution compares different methods that solve this problem focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions these include the nystrom method as well as collocation and galerkin projection methods among the galerkin methods we investigate the finite element method fem and propose the application of the finite cell method fcm this method is based on an extension to the fem but avoids mesh generation on domains of complex geometric shape the fcm was originally presented in parvizian et al 2007 17 for the solution of elliptic boundary value problems as an alternative to the l2l2 projection of the covariance function used in the galerkin method h1 2h1 2 projection and discrete projection are investigated it is shown that the expansion optimal linear estimation eole method proposed in li and der kiureghian 1993 18 constitutes a special case of the nystrom method it is found that the eole method is most efficient for the numerical solution of the kl expansion the fem and the fcm are more efficient than the eole method in evaluating a realization of the random field and therefore are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime e g in finite element reliability analysis

uniform estimates for bilinear hilbert transforms and bilinear maximal functions associated to polynomials we study the bilinear hilbert transforms and bilinear maximal functions associated to non flat polynomial curves and obtain uniform l r estimates for r d 1 over d where d is the degree of the polynomial in addition the lower bound of r is sharp up to the end point

uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials abstract this paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials at the microscale level the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used the geometry of the microstructure is random and characterized by a high number of random parameters the method is based on a deterministic non concurrent multiscale approach devoted to micro macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale cauchy green strain states to overcome the curse of dimensionality due to the high number of involved random variables the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion several strategies based on novel algorithms dedicated to high stochastic dimension are used and adapted for the class of multi modal random variables which may characterize the potential numerical examples at both small and large scales allow analyzing the efficiency of the approach through comparisons with classical methods

linear quadratic regulation of systems with stochastic parameter uncertainties in this paper we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters the framework is built on the generalized polynomial chaos theory in this framework the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space and the controller is designed in the expanded state space the proposed design framework results in a family of controllers parameterized by the associated random variables the theoretical results are applied to a controller design problem based on stochastic linear longitudinal f16 model the performance of the stochastic design shows excellent consistency in a statistical sense with the results obtained from monte carlo based designs

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

polynomial chaos expansion for nonlinear geophysical inverse problems abstractthere are lots of geophysical problems that include computationally expensive functions forward models polynomial chaos pc expansion aims to approximate such an expensive equation or system with a polynomial expansion on the basis of orthogonal polynomials evaluation of this expansion is extremely fast because it is a polynomial function this property of the pc expansion is of great importance for stochastic problems in which an expensive function needs to be evaluated thousands of times we have developed pc expansion as a novel technique to solve nonlinear geophysical problems to better evaluate the methodology we use pc expansion for automating the velocity analysis for this purpose we define the optimally picked velocity model as an optimizer of a variational integral in a semblance field however because computation of a variational integral with respect to a given velocity model is rather expensive it is impossible to use stochastic methods to search for the optimal velocity mo

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

supersensitivity due to uncertain boundary conditions summary we study the viscous burgers equation subject to perturbations on the boundary conditions two kinds of perturbations are considered deterministic and random for deterministic perturbations we show that small perturbations can result in o 1 changes in the location of the transition layer for random perturbations we solve the stochastic burgers equation using different approaches first we employ the jacobi polynomial chaos which is a subset of the generalized polynomial chaos for stochastic modeling converged numerical results are reported up to seven significant digits and we observe similar stochastic supersensitivity for the mean location of the transition layer subsequently we employ up to fourth order perturbation expansions we show that even with small random inputs the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output two types of distributions are considered uniform distribution and a truncated gaussian distribution with no tails various solution statistics including the spatial evolution of probability density function at steady state are studied copyright 2004 john wiley sons ltd

sampling free linear bayesian update of polynomial chaos representations we present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes described by possibly non gaussian distributions the description of the introduced random fields is given in a white noise framework which enables us to solve the stochastic forward problem through galerkin projection onto polynomial chaos with the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the baye s linear form of updating by introducing the hermite algebra this becomes a direct purely algebraic way of computing the posterior which is comparatively inexpensive to evaluate in addition we show that the well known kalman filter is the low order part of this update the proposed method is here tested on a stationary diffusion equation with prescribed source terms characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

generalized polynomial chaos and random oscillators we present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions which are more efficient than the classical wiener hermite expansions the approach is general but here we present results for linear oscillators only with random forcing or random coefficients in this context we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution we have also performed computational comparisons with monte carlo simulations which show that the new approach can be orders of magnitude faster especially for compact distributions

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

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

on solving elliptic stochastic partial differential equations abstract a model elliptic boundary value problem of second order with stochastic coefficients described by the karhunen loeve expansion is addressed this problem is transformed into an equivalent deterministic one the perturbation method and the method of successive approximations is analyzed rigorous error estimates in the framework of sobolev spaces are given

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

verification of stochastic models in uncertain environments using the constitutive relation error method abstract this paper deals with local stress errors which occur in linear elastic fe analysis in an uncertain environment material loads the estimation technique described here uses the concept of error in constitutive relation but it can be adapted to other estimators first we review the principle of the method then extend it to the stochastic case this approach yields bounds on the stresses associated with a given confidence level we illustrate this presentation through a test example

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

approximating infinity dimensional stochastic darcy s equations without uniform ellipticity we consider a stochastic darcy s pressure equation whose coefficient is generated by a white noise process on a hilbert space employing the ordinary rather than the wick product a weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf sup condition and well posedness of the problem we generalize the numerical approximations proposed in benth and theting stochastic anal appl 20 2002 pp 1191 1223 for wick stochastic partial differential equations to the ordinary product stochastic pressure equation we establish discrete inf sup conditions and provide a priori error estimates for a wide class of norms the proposed numerical approximation is based on wiener chaos finite element methods and yields a positive definite symmetric linear system we also improve and generalize the approximation results of benth and gjerde stochastics stochastics rep 63 1998 pp 313 326 and cao stochastics 78 2006 pp 179 187 when a generalized process is truncated by a finite wiener chaos expansion finally we present numerical experiments to validate the results

topology optimization under uncertainty via non intrusive polynomial chaos expansion abstract this paper presents a systematic approach for topology optimization under uncertainty that integrates non intrusive polynomial chaos expansion with design sensitivity analysis for reliability based and robust topology optimization uncertainty is introduced in loading and in geometry to address the manufacturing variability the manufacturing variability is modeled via a thresholding technique in which the threshold field is represented by a reduced dimensional random field response metrics such as compliance and volume are characterized as polynomial chaos expansions of the underlying uncertain parameters thus allowing accurate and efficient estimation of statistical moments failure probabilities and their sensitivities the number of simulations is reduced for linear structures under loading uncertainty by means of superposition efficiency of the non intrusive polynomial chaos approach is highlighted by comparison with the monte carlo method in terms of the number of simulations to demonstrate the effect of uncertainty optimized designs that consider uncertainty are compared to those that do not comparisons of polynomial chaos expansion to existing analytical methods on a benchmark numerical example are also provided for reliability based and worst case designs

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

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

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

stochastic design optimization application to reacting flows one concern in the optimization of realistic complex systems is to ensure that the optimal model response is reliable and robust with respect to the inherent uncertainties associated with the design variables constraints and the objective function traditional optimization techniques together with uncertainty analysis are computationally expensive and time consuming when it comes to identify what drives the response variability a stochastic optimization framework combining stochastic surrogate model representation and optimization algorithm is proposed a generalized polynomial chaos stochastic representation is used as the surrogate model this representation is obtained from a high order stochastic collocation method the surrogate model can then be put to use within an appropriate optimization algorithm and provides fast approximations at new design points this approach allows both sensitivity and optimization analysis the stochastic optimization method is applied to a multi layer reacting flow device the geometric configuration is assumed to be uncertain the structure design is optimized to maximize the energy transfer between the reacting flow and the device moving parts gradient descent and simulated annealing optimization techniques are successfully tested on the kinetic energy surrogate model of the device inner parts

a dual layer dimension wise fuzzy finite element method for structural analysis with epistemic uncertainties abstract epistemic uncertainty propagation is gaining momentum but remains a great challenge in the numerical analysis of a mechanical system a dual layer dimension wise analysis procedure for the fuzzy finite element method is proposed the core algorithm layer is to identify the minimal and maximal min max points of each slice of the response surface at the zero cut the data processing layer aims to filtrate the min max points of each response slice at each nonzero cut subsequently the min max input vectors at any alpha cut are dimension wisely assembled and the response interval at alpha cut is calculated by deterministic finite element analysis at the min max input vectors finally the fuzzy description of the system response is recomposed two examples are investigated to verify both the accuracy and efficiency of the proposed method

a multi scale spectral stochastic method for homogenization of multi phase periodic composites with random material properties in this work a spectral stochastic computational scheme is proposed that links the global properties of multi phase periodic composites to the geometry and random material properties of their microstructural components to propagate the uncertainties associated with the material properties to the microstructural response the scheme benefits from a combination of homogenization theory built into a finite element framework and the spectral representation of uncertainty based on hermite chaos where a probabilistic characterization of the solutions to a set of local problems defined on the period cell is first sought a full stochastic description of the global effective properties is then obtained by averaging the solutions to the forgoing set of local problems over the unit cell a representative subset of results is compared with the results obtained using monte carlo simulation to demonstrate the accuracy of the proposed procedure copyright 2010 john wiley sons ltd

on calderon s conjecture for the bilinear hilbert transform abstract we show that the bilinear hilbert transform defined by maps lp lq into lr for 1 p q 1 p 1 q 1 r and 2 3 r

generalizations of the carleson hunt theorem i the classical singularity case in this article we prove l p estimates for a general maximal operator which extend both the classical coifman meyer and carleson hunt theorems in harmonic analysis

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

double recurrence and almost sure convergence on repond de facon affirmative a une question posee par furstenberg sur la convergence presque sure de moyennes ergodiques de la forme 1 n n t n f t mn g avec 1 n ou n est un entier fixe t une transformation ergodique et preservant la mesure sur un espace de probabilite et f et g appartenant a l la somme ci dessus est un cas particulier d une recurrence multiple a laquelle peuvent etre facilement generalises les resultats obtenus ici pour la recurrence double

the bilinear maximal functions map into l p for 2 3 p 1 the bilinear maximal operator defined below maps lp x lq into lr provided 1 0 2t t in particular mfg is integrableif f and g are square integrable answering a conjecture posed by alberto calder6n

an extended stochastic finite element method for solving stochastic partial differential equations on random domains abstract recently a new strategy was proposed to solve stochastic partial differential equations on random domains it is based on the extension to the stochastic framework of the extended finite element method x fem this method leads by a direct calculus to an explicit solution in terms of the variables describing the randomness on the geometry it relies on two major points the implicit representation of complex geometries using random level set functions and the use of a galerkin approximation at both stochastic and deterministic levels in this article we detail the basis of this technique from theoretical and technical points of view several numerical examples illustrate the efficiency of this method and compare it to other approaches

on boundedness of discrete multilinear singular integral operators let m be a measurable locally bounded function defined in r2 let 1 p1 q1 p2 q2 such that pi 1 implies qi let also 0 p3 q3 and 1 p 1 p1 1 p2 1 p3 we prove the following transference result the operator cm f g x r rf g m e2 ix d d initially defined for integrable functions with compact fourier support extends to a bounded bilinear operator from lp1 q1 r lp2 q2 r into lp3 q3 r if and only if the family of operators dm t p a b n t1p 1212 1212p q m t t e2 in d d initially defined for finite sequences a ak1 k1 z b bk2 k2 z where p k1 zak1e 2 ik1 and q k2 zbk2e 2 ik2 extend to bounded bilinear operators from lp1 q1 z lp2 q2 z into lp3 q3 z with norm bounded by uniform constant for all t 0 we apply this result to prove boundedness of the discrete bilinear hilbert transforms and other related discrete multilinear singular integrals including the endpoints

point collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics this paper describes a point collocation nonintrusive polynomial chaos technique used for uncertainty propagation in computational fluid dynamics simulations the application of point collocation nonintrusive polynomial chaos to stochastic computational fluid dynamics is demonstrated with two examples 1 a stochastic expansion wave problem with an uncertain deflection angle geometric uncertainty and 2 a stochastic transonic wing case with uncertain freestream mach number and angle of attack for each problem input uncertainties are propagated with both the nonintrusive polynomial chaos method and monte carlo techniques to obtain the statistics of various output quantities confidence intervals for monte carlo statistics are calculated using the bootstrap method for the expansion wave problem a fourth degree polynomial chaos expansion which requires five deterministic computational fluid dynamics evaluations has been sufficient to predict the statistics within the confidence interval of 10 000 crude monte carlo simulations in the transonic wing case for various output quantities of interest it has been shown that a fifth degree point collocation nonintrusive polynomial chaos expansion obtained with hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 latin hypercube monte carlo simulations with a significantly lower computational cost overall the examples demonstrate that the point collocation nonintrusive polynomial chaos has a promising potential as an effective and computationally efficient uncertainty propagation technique for stochastic computational fluid dynamics simulations

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

fast fourier transforms for the rook monoid we define the notion of the fourier transform for the rook monoid also called the symmetric inverse semigroup and provide two efficient divide and conquer algorithms fast fourier transforms or ffts for computing it this paper marks the first extension of group ffts to nongroup semigroups

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

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

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

a fourier karhunen loeve discretization scheme for stationary random material properties in sfem in order to overcome the computational difficulties in karhunen loeve k l expansions of stationary random material properties in stochastic finite element method sfem analysis a fourier karhunen loeve f k l discretization scheme is developed in this paper by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical sfem simulations the proposed f k l discretization scheme exhibits a number of advantages over the widely used k l expansion scheme based on fe meshes including better computational efficiency in terms of memory and cpu time convenient a priori error control mechanism better approximation accuracy of random material properties explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes copyright 2007 john wiley sons ltd

on a biparameter maximal multilinear operator abstract it is well known that estimates for maximal operators and questions of pointwise convergence are strongly connected in recent years convergence properties of so called non conventional ergodic averages have been studied by a number of authors including assani austin host kra tao etc in particular much is known regarding convergence in l 2 of these averages but little is known about pointwise convergence in this spirit we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator over r thanks to a transference principle lacey in 15 and demeter tao and thiele in 6 have studied maximal multilinear operators previously however the maximal operator we develop has a novel biparameter structure which has not been previously encountered and cannot be estimated using their techniques we will carve this biparameter maximal multilinear operator using a certain taylor series and produce non trivial holder type estimates for one of the two main terms by treating it as a singular integral the symbol of which has singular set given by two intersecting planes similarly to that of the biest operator studied by muscalu tao and thiele in 24 and 25

discrete maximal functions in higher dimensions and applications to ergodic theory we establish a higher dimensional counterpart of bourgain s pointwise ergodic theorem along an arbitrary integer valued polynomial mapping we achieve this by proving variational estimates v r on l p spaces for all 1

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

geometrically non linear behavior of structural systems with random material property an asymptotic spectral stochastic approach abstract an asymptotic spectral stochastic approach is presented for computing the statistics of the equilibrium path in the post bifurcation regime for structural systems with random material properties the approach combines numerical implementation of koiter s asymptotic theory with a stochastic galerkin scheme and collocation in stochastic space to quantify uncertainties in the parametric representation of the load displacement relationship specifically in the form of uncertain post buckling slope post buckling curvature and a family of stochastic displacement fields using the proposed method post buckling response statistics for two plane frames are obtained and shown to be in close agreement with those obtained from monte carlo simulation provided a fine enough spectral representation is used to model the variability in the random dimension

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

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

endpoint bounds for the bilinear hilbert transform we study the behavior of the bilinear hilbert transform bht at the boundary of the known boundedness regionh a sample of our results is the estimate bht f1 f2 f3 c f1 3 4 f2 3 4 f3 1 2 log log ee f3 min f1 f2 valid for all tuples of sets f j r of finite measure and functions f j such that f j 1f j j 1 2 3 with the additional restriction that f3 be supported on a major subset f 3 of f3 that depends on f j j 1 2 3 the use of subindicator functions in this fashion is standard in the given context see 25 the double logarithmic term improves over the single logarithmic term obtained in 1 whether the double logarithmic term can be removed entirely as is the case for the quartile operator discussed in 7 remains open we employ our endpoint results to describe the blow up rate of weak type and strong type estimates for bht as the tuple approaches the boundary ofh we also discuss bounds on lorentz orlicz spaces near l 2 3 improving on results of 5 the main technical novelty in our article is an enhanced version of the multi frequency calderon zygmund decomposition of 26

extended stochastic finite element method for the numerical simulation of heterogeneous materials with random material interfaces an extended stochastic finite element method has been recently proposed for the numerical solution of partial differential equations defined on random domains this method is based on a marriage between the extended finite element method and spectral stochastic methods in this article we propose an extension of this method for the numerical simulation of random multi phased materials the random geometry of material interfaces is described implicitly by using random level set functions a fixed deterministic finite element mesh which is not conforming to the random interfaces is then introduced in order to approximate the geometry and the solution classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables which leads to a deterioration of the accuracy and convergence properties of the approximate solution in order to recover optimal convergence properties of the approximation we propose an extension of the partition of unity method to the spectral stochastic framework this technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure copyright 2010 john wiley sons ltd

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

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

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

lack of knowledge in structural model validation this paper deals with the structural modeling of a family of similar actual structures taking into account uncertainties and modeling errors only errors of the structural stiffness type are considered we develop a new theory in which what we call the lack of knowledge lok is defined through an internal variable whose upper and lower bounds are stochastic associated with each substructure two main questions are discussed the impact of the basic loks on the predicted structural response and the reduction of the basic loks through the use of additional information

lp estimates for the bilinear hilbert transform for the bilinear hilbert transform given by we announce the inequality h fg p3 kp1 p2 f p1 g p2 provided 2 p1 p2 1 p3 1 p1 1 p2 and 1 p3 2

wick stochastic finite element solution of reaction diffusion problems real life reaction diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters this paper presents a finite element solution of reaction diffusion equations of wick type using the wick product properties and the wiener ito chaos expansion the stochastic variational problem is reformulated to a set of deterministic variational problems to obtain the chaos coefficients in the corresponding deterministic reaction diffusion we implement the usual galerkin finite element method using standard techniques once this representation is computed the statistics of the numerical solution can be easily evaluated computational results are shown for one and two dimensional test examples