a proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart young approach the proper generalized decomposition pgd is a methodology initially proposed for the solution of partial differential equations pde defined in tensor product spaces it consists in constructing a separated representation of the solution of a given pde in this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting in particular we introduce a generalization of eckart and young theorem which allows to prove the convergence of the so called progressive pgd for a large class of linear problems defined in tensor product hilbert spaces

compact and noncompact structures for a variant of kdv equation in higher dimensions in this paper we study two mathematical variants of kdv equation we show that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with cusps or infinite slopes a framework is developed for studying these problems in higher dimensions two distinct sets of general formulas for each type of variants are developed to present a fairly complete understanding of the compact and noncompact dispersive structures

prediction of apparent properties with uncertain material parameters using high order fictitious domain methods and pgd model reduction this contribution presents a numerical strategy to evaluate the effective properties of image based microstructures in the case of random material properties the method relies on three points i a high order fictitious domain method ii an accurate spectral stochastic model and iii an efficient model reduction method based on the proper generalized decomposition in order to decrease the computational cost introduced by the stochastic model a feedback procedure is proposed for an automatic estimation of the random effective properties with a given confidence numerical verifications highlight the convergence properties of the method for both deterministic and stochastic models the method is finally applied to a real 3d bone microstructure where the empirical probability density function of the effective behaviour could be obtained

optimal portfolio and consumption decisions for a small investor on a finite horizon a general consumption investment problem is considered for an agent whose actions cannot affect the market prices and who strives to maximize total expected discounted utility of both consumption

solving stochastic systems with low rank tensor compression abstract for parametrised equations which arise for example in equations dependent on random parameters the solution naturally lives in a tensor product space the application which we have in mind is a stochastic linear elliptic partial differential equation spde usual spatial discretisation leads to a potentially large linear system for each point in the parameter space approximating the parametric dependence by a galerkin ansatz the already large number of unknowns for a fixed parameter value is multiplied by the dimension of the galerkin subspace for the parametric dependence and thus can be very large therefore we try to solve the total system approximately on a smaller submanifold which is found adaptively through compression during the solution process by alternating iteration and compression we show that for general linearly converging stationary iterative schemes and general adaptation processes which can be seen as a modification or perturbation of the iteration the interlaced procedure still converges our proposed modification can be used for most stationary solvers for systems on tensor products we demonstrate this on an example of a discretised spde with a simple preconditioned iteration we choose a truncated singular value decomposition svd for the compression and give details of the implementation finishing with some examples

minimal dynamical systems from pdes using sobolev eigenfunctions abstract a modified karhunen loeve transform is derived using a weighted sobolev norm this approach optimizes the approximation of the higher derivative terms in numerical simulations of partial differential equation and reduces the sizes of the associated system of ordinary differential equations the kuramoto sivashinsky equation is used as an example to demonstrate how a galerkin procedure based on eigenfunctions determined by the sobolev optimality criterion improves the performance of the standard karhunen loeve procedure we present an example of a simulation which required a 19 mode expansion using the k l approach and show how this is reduced to a 7 mode simulation using the sobolev eigenfunctions

fuzzy stochastic fem based homogenization framework for materials with polymorphic uncertainties in the microstructure uncertainties in the macroscopic response of heterogeneous materials result from two sources the natural variability in the microstructure s geometry and the lack of sufficient knowledge regarding the microstructure the first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function the second type of uncertainty is denoted epistemic uncertainty this kind of uncertainty cannot be described using probabilistic methods models considering both sources of uncertainties are called polymorphic in the case of polymorphic uncertainties some combination of stochastic methods and fuzzy arithmetic should be used thus in the current work we examine a fuzzy stochastic finite element method based homogenization framework for materials with random inclusion sizes we analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element the stochastic finite element method is used to obtain the material response in the case of random inclusion radii due to unavoidable noise in experimental data an insufficient number of samples and limited accuracy of the fitting procedure the radii distribution density cannot be obtained exactly thus it is described in terms of fuzzy location and scale parameters the influence of fuzzy input on the homogenized stress measures is analyzed

transient response analysis of randomly parametrized finite element systems based on approximate balanced reduction a model order reduction scheme of the transient response of large scale randomly parametrized linear finite element system in state space form has been proposed the reduced order model realization is aimed at preserving the invariant properties of the dynamic system model based on the dominant coupling characteristics of the specified system inputs and outputs an a priori model reduction strategy based on the balanced truncation method has been proposed in conjunction with the stochastic spectral galerkin finite element method approximation of the dominant modes of the controllability gramian has been performed with iterative arnoldi scheme applied to lyapunov equations the reduced order representation of the randomly parametrized dynamical system has been obtained with arnoldi lyapunov vector basis using an implicit time stepping algorithm the performance and the computational efficacy of the proposed scheme has been illustrated with examples of randomly parametrized advection diffusion reaction problem under the action of transient external forcing functions the convergence of the proposed reduced order scheme has been shown with a posteriori error estimates

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

modified sub interval perturbation finite element method for 2d acoustic field prediction with large uncertain but bounded parameters abstract based on the sub interval perturbation analysis a modified sub interval perturbation finite element method is proposed to determine the bounds of sound pressure in the 2d acoustic field with large uncertain but bounded parameters in the proposed method the inversion of the invertible sub interval dynamic stiffness matrix is approximated by a modified approximate interval value sherman morrison woodbury formula to overcome the drawbacks arising from the dependency phenomenon of parameters and the unpredictable effect of neglecting the higher order terms in neumann series the modified sub interval perturbation procedures are implemented in a numerical finite element framework numerical results on a 2d acoustic tube and a 2d acoustic cavity of a car with large uncertain but bounded parameters evidence the remarkable accuracy and effectiveness of the proposed method

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

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

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

model reduction based on proper generalized decomposition for the stochastic steady incompressible navier stokes equations in this paper we consider a proper generalized decomposition method to solve the steady incompressible navier stokes equations with random reynolds number and forcing term the aim of such a technique is to compute a low cost reduced basis approximation of the full stochastic galerkin solution of the problem at hand a particular algorithm inspired by the arnoldi method for solving eigenproblems is proposed for an efficient greedy construction of a deterministic reduced basis approximation this algorithm decouples the computation of the deterministic and stochastic components of the solution thus allowing reuse of preexisting deterministic navier stokes solvers it has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m dimensional reduced basis rather than m coupled problems of the full stochastic galerkin approximation space with m ll m up to one order of magnitude for the problem at hand in this work

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

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

stochastic isogeometric analysis in linear elasticity abstract a new stochastic method integrating spline dimensional decomposition sdd and isogeometric analysis iga is proposed for solving stochastic boundary value problems from linear elasticity the method referred to as sdd iga involves galerkin isogeometric analysis as a deterministic solver for governing partial differential equations and a novel fourier like orthogonal spline expansion generated from the analysis of variance decomposition of a high dimensional function for the stochastic part of the sdd iga method an innovative dimension reduction integration technique is presented for efficiently calculating the expansion coefficients analytical formulae have been derived to calculate the second moment properties of an sdd iga approximation for a general output random variable of interest numerical examples demonstrate the capability of a low order sdd iga in efficiently delivering probabilistic solutions with an approximation quality as good as if not better than that obtained from a high order polynomial dimensional decomposition the proposed sdd iga method is most suitable in the presence of locally nonlinear or nonsmooth behavior commonly found in applications

on a multiscale computational strategy with time and space homogenization for structural mechanics a new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale this strategy which involves homogenization in space as well as in time could replace in several domains of application the standard homogenization techniques which are generally limited to the space domain it is an iterative strategy which calls for the resolution of problems on both a micro fine scale and a macro homogenized scale in this paper we review the bases of this approach and present improved approximation techniques to solve the micro and macro problems

parallel domain decomposition strategies for stochastic elliptic equations part b accelerated monte carlo sampling with local pc expansions solving stochastic partial differential equations spdes can be a computationally intensive task particularly when the underlying parametrization of the stochastic input field involves a large nu

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

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

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

proper generalized decomposition of time multiscale models summary models encountered in computational mechanics could involve many time scales when these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies in some cases the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval in this paper we considered an alternative approach that lies in separating the time axis one dimensional in nature in a multidimensional time space then for circumventing the resulting curse of dimensionality the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration copyright 2011 john wiley sons ltd

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

continuous time mean variance portfolio selection with random horizon in an incomplete market in this paper we consider a continuous time mean variance portfolio selection problem with random market parameters and random time horizon in an incomplete market this problem will be formulated as a linearly constrained stochastic linear quadratic lq optimal control problem the solvability of this lq problem will be reduced to the global solvability of two backward stochastic differential equations bsdes one is conventionally called a stochastic riccati equation sre and the other is referred to as an auxiliary bsde we shall apply the martingales of bounded mean oscillation briefly called bmo martingales to provide a direct and simplified proof of the solvability of the two bsdes we also derive closed form expressions for both the optimal portfolios and the efficient frontier in terms of the solutions of the two bsdes

advanced simulation of models defined in plate geometries 3d solutions with 2d computational complexity abstract many models in polymer processing and composites manufacturing are defined in degenerated three dimensional domains involving plate or shell geometries the reduction of models from 3d to 2d is not obvious when complex physics are involved the hypotheses to be introduced for reaching this dimensionality reduction are sometimes unclear and most of possible proposals will have a narrow interval of validity the only getaway is to explore new discretization strategies able to circumvent or at least alleviate the drawbacks related to mesh based discretizations of fully 3d models defined in plate or shell domains an in plane out of plane separated representation of the involved fields within the context of the proper generalized decomposition allows solving the fully 3d model by keeping a 2d characteristic computational complexity moreover the pgd features allow the introduction of many extra coordinates as for example the orientation of the different laminate plies without affecting the solvability of the resulting multidimensional model

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

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

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

a fully symmetric nonlinear biorthogonal decomposition theory for random fields abstract we present a general approach for nonlinear biorthogonal decomposition of random fields the mathematical theory is developed based on a fully symmetric operator framework that unifies different types of expansions and allows for a simple formulation of necessary and sufficient conditions for their completeness the key idea of the method relies on an equivalence between nonlinear mappings of hilbert spaces and local inner products i e inner products that may be functionals of the random field being decomposed this extends previous work on the subject and allows for an effective formulation of field dependent and field independent representations the proposed new methodology can be applied in many areas of mathematical physics for stochastic low dimensional modelling of partial differential equations and dimensionality reduction of complex nonlinear phenomena an application to a transient stochastic heat conduction problem in a one dimensional infinite medium is presented and discussed

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

partitioned treatment of uncertainty in coupled domain problems a separated representation approach abstract this work is concerned with the propagation of uncertainty across coupled domain problems with high dimensional random inputs a stochastic model reduction approach based on low rank separated representations is proposed for the partitioned treatment of the uncertainty space the construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub domain and not the combined dimensionality hence drastically reducing the overall computational cost the coupling between the sub domain solutions is done via the classical finite element tearing and interconnecting feti method thus providing a well suited framework for parallel computing two high dimensional stochastic problems a 2d elliptic pde with random diffusion coefficient and a stochastic linear elasticity problem have been considered to study the performance and accuracy of the proposed stochastic coupling approach

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

on the verification of model reduction methods based on the proper generalized decomposition in this work we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition pgd approximation this estimator which is based on the constitutive relation error enables to capture all error sources i e those coming from space and time numerical discretizations from the truncation of the pgd decomposition etc and leads to guaranteed bounds on the exact error the specificity of the associated method is a double approach i e a kinematic approach and a unusual static approach for solving the parameterized problem by means of pgd this last approach makes straightforward the computation of a statically admissible solution which is necessary for robust error estimation an attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes it thus represents a practical and relevant tool for driving algorithms carried out in pgd being possibly used as a stopping criterion or as an adaptation indicator numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

galerkin reduced order modeling scheme for time dependent randomly parametrized linear partial differential equations summary in this paper we consider the problem of constructing reduced order models of a class of time dependent randomly parametrized linear partial differential equations our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates parameter space and time the proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions the unknown basis functions in the decomposition are estimated using an alternating iterative galerkin projection scheme numerical studies on the time dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos based galerkin projection schemes comparison studies are also made against nouy s generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages copyright 2012 john wiley sons ltd

a grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains abstract a tensor based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains it provides an accurate solution which is explicit with respect to parameters defining the shape of the domain thus allowing efficient a posteriori probabilistic or parametric analyses in the proposed method a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain yielding a weak formulation in a tensor product space product of space functions and parametric functions the paper is limited to the case of neumann conditions on uncertain parts of the boundary the proper generalized decomposition method is then introduced for the construction of a tensor product approximation separated representation of the solution it can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution this tensor based method is made computationally tractable by introducing separated representations of variational forms resulting from separated representations of the parameterized indicator function of the uncertain domain for this purpose a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well posedness of problems associated with approximate indicator functions moreover a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

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

general linear quadratic optimal stochastic control problems with random coefficients linear stochastic hamilton systems and backward stochastic riccati equations consider the minimization of the following quadratic cost functional j u e langle mx t x t rangle e int 0 t langle q sx s x s rangle langle n su s u s rangle ds where x is the solution of the following linear stochastic control system eq dx t a tx t b tu t dt sum i 1 d c t ix t d t iu t dw t i cr x 0 h cr u is a square integrable adapted process the problem is conventionally called the stochastic lq the abbreviation of linear quadratic problem we are concerned with the following general case the coefficients a b ci di q n and m are allowed to be adapted processes or random matrices we prove the existence and uniqueness result for the associated riccati equation which in our general case is a backward stochastic differential equation with the generator the drift term being highly nonlinear in the two unknown variables this solves bismut and peng s long standing open problem for the case of a brownian filtration which was initially proposed by the french mathematician j m bismut in seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 we also provide a rigorous derivation of the riccati equation from the stochastic hamilton system this completes the interrelationship between the riccati equation and the stochastic hamilton system as two different but equivalent tools for the stochastic lq problem there are two key points in our arguments the first one is to connect the existence of the solution of the riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system actually we establish their equivalence as a consequence we can construct solutions to a sequence of suitably modified riccati equations in terms of the associated stochastic hamilton systems and the optimal controls the second key point is to establish a new type of a priori estimate for solutions of riccati equations with which we show that the sequence of constructed solutions has a limit which is a solution to the original riccati equation

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

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

space time proper generalized decompositions for the resolution of transient elastodynamic models in this paper we investigate ability of proper generalized decomposition pgd to solve transient elastodynamic models in space time domain classical methods use time integration schemes and an incremental resolution process we propose here to use standard time integration methods in a non incremental strategy as a result pgd gives a separated representation of the space time solution as a sum of tensorial products of space and time vectors that we interpret as space time modes recent time integration schemes are based on multi field formulations in this case separated representation can be constructed using state vectors in space and same vectors in time however we have experienced bad convergence order using this decomposition furthermore temporal approximation must be the same for all fields thus we propose an extension of classical separated representation for multi field problems this multi field pgd mf pgd uses space and time vectors that are different for each field calculation of decomposition is done using a monolithic approach in space and time potentially allowing the use of different approximations in space and time finally several simulations are performed with the transient elastodynamic problem with one dimension in space different approximations in time are investigated newmark scheme single field time discontinuous galerkin method and two fields time continuous and discontinuous galerkin methods

enforcing positivity in intrusive pc uq methods for reactive ode systems we explore the relation between the development of a non negligible probability of negative states and the instability of numerical integration of the intrusive galerkin ordinary differential equation system describing uncertain chemical ignition to prevent this instability without resorting to either multi element local polynomial chaos pc methods or increasing the order of the pc representation in time we propose a procedure aimed at modifying the amplitude of the pc modes to bring the probability of negative state values below a user defined threshold this modification can be effectively described as a filtering procedure of the spectral pc coefficients which is applied on the fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold we demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor this is carried out by comparing different observables and error measures as obtained by non intrusive monte carlo and gauss quadrature integration and the filtered intrusive procedure the filtering procedure has been shown to effectively stabilize divergent intrusive solutions and also to improve the accuracy of stable intrusive solutions which are close to the stability limits

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of 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 treats easily a wide 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 of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

optimization via separated representations and the canonical tensor decomposition abstract we introduce a new quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format the computational complexity of the algorithm is linear in the dimension of the tensor we show how to use this algorithm to find global maxima of non convex multivariate functions in separated form we demonstrate the performance of the new algorithms on several examples

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

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

a kind of optimal investment problem under inflation and uncertain time horizon abstract in this paper we study a kind of optimal investment problem under inflation and uncertain time horizon it can be generally formulated into a stochastic optimal control problem in particular for the constant relative risk aversion utility we employ the method of completion of squares to give an explicit form of optimal portfolio and maximum utility by the solution of a stochastic riccati equation whose wellposedness is obtained and also of significance in its own right the most distinguishing result of our work is that the randomness of exit time actually affects not only the optimal portfolio but also the maximum utility in the case of stochastic market parameters moreover we present several numerical examples to show the application of theoretical results and further discuss the influence of inflation and random time horizon from the economic viewpoint

quadratic hedging and mean variance portfolio selection with random parameters in an incomplete market this paper concerns the problems ofquadratic hedging andpricing andmean variance portfolio selection in anincomplete market setting with continuous trading multiple assets and brownian information in particular we assume throughout that the parameters describing the market model may be random processes we approach these problems from the perspective oflinear quadratic lq optimal control and backward stochastic differential equations bsdes that is we focus on the so calledstochastic riccati equation sre associated with the problem excepting certain special cases solvability of the sre remains an open question our primary theoretical contribution is a proof of existence and uniqueness of solutions of the sre associated with the quadratic hedging and mean variance problems in addition we derive closed form expressions for the optimal portfolios and efficient frontier in terms of the solution of the sre a generalization of themutual fund theorem is also obtained

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty the problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method a randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered the discretized parametric fields are interpolated within each element with multidimensional lagrange polynomials and integrated into the weak formulation the proposed discretized random field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables the description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain a method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral galerkin framework is studied in detail the treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems the convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme

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

computational aspects for constructing realizations of polynomial chaos in high dimension this paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension the method proposed consists of 1 constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and 2 performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the gram schmidt orthogonalization algorithm which is not stable in high dimension a brief review of polynomial chaos expansion with arbitrary measure is given the statistically independent realizations of multivariate monomials are introduced the centered statistically independent realizations of orthonormal multivariate polynomials are developed finally a quantification of the errors induced by the usual methods is given

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

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

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

multi index stochastic collocation for random pdes in this work we introduce the multi index stochastic collocation method misc for computing statistics of the solution of a pde with random data misc is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data we propose an optimization procedure to select the most effective mixed differences to include in the misc estimator such optimization is a crucial step and allows us to build a method that provided with sufficient solution regularity is potentially more effective than other multi level collocation methods already available in literature we then provide a complexity analysis that assumes decay rates of product type for such mixed differences showing that in the optimal case the convergence rate of misc is only dictated by the convergence of the deterministic solver applied to a one dimensional problem we show the effectiveness of misc with some computational tests comparing it with other related methods available in the literature such as the multi index and multilevel monte carlo multilevel stochastic collocation quasi optimal stochastic collocation and sparse composite collocation methods

a posteriori error estimation and adaptive strategy for pgd model reduction applied to parametrized linear parabolic problems abstract we define an a posteriori verification procedure that enables to control and certify pgd based model reduction techniques applied to parametrized linear elliptic or parabolic problems using the concept of constitutive relation error it provides guaranteed and fully computable global goal oriented error estimates taking both discretization and pgd truncation errors into account splitting the error sources it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of pgd approximations the focus of the paper is on two technical points i construction of equilibrated fields required to compute guaranteed error bounds ii error splitting and adaptive process when performing pgd based model reduction performances of the proposed verification and adaptation tools are shown on several multi parameter mechanical problems

time inconsistent stochastic linear quadratic control in this paper we formulate a general time inconsistent stochastic linear quadratic lq control problem the time inconsistency arises from the presence of a quadratic term of the expected state as well as a state dependent term in the objective functional we define an equilibrium instead of optimal solution within the class of open loop controls and derive a sufficient condition for equilibrium controls via a flow of forward backward stochastic differential equations when the state is one dimensional and the coefficients in the problem are all deterministic we find an explicit equilibrium control as an application we then consider a mean variance portfolio selection model in a complete financial market where the risk free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes applying the general sufficient condition we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques