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 method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

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

the multisublinear maximal type operators in banach function lattices abstract the main aim of this paper is to study a general multisublinear operators generated by quasi concave functions between weighted banach function lattices these operators in particular generalize the hardy littlewood and fractional maximal functions playing an important role in harmonic analysis we prove that under some general geometrical assumptions on banach function lattices two weight weak type and also strong type estimates for these operators are true to derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators as special cases we provide boundedness results for fractional maximal operators in concrete function spaces

adaptive reduced basis generation for reduced order modeling for the solution of stochastic nondestructive evaluation problems a novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented more specifically the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty in order to achieve this computational efficiency the approach combines reduced basis reduced order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters provided the distributions are known the reduced order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique to then create the reduced order basis the capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty the first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing then the second case study considered characterization of an unknown temperature dependent thermal conductivity of a solid from simulated thermal testing overall the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

domain decomposition using spectral expansions of steklov poincare operators ii a matrix formulation we describe a matrix formulation of the iterative domain decomposition method in natarajan siam j sci comput 16 1995 pp 470 495 from one point of view this method can be regarded as a preconditioning technique for the interface schur complement operator obtained from a decomposition into nonoverlapping subdomains from another point of view it can be viewed as a method of the schwarz type for overlapping subdomains but with an overlap between the physical space in one subdomain and the leading components of the eigenspace induced by the steklov poincare operator in the complementary domain

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

scalable domain decomposition solvers for stochastic pdes in high performance computing abstract stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non linear problems with billions of unknowns for stochastic modeling it is therefore essential to design robust parallel and scalable algorithms that can efficiently utilize high performance computing to tackle such large scale systems domain decomposition based iterative solvers can handle such systems although these algorithms exhibit excellent scalabilities significant algorithmic and implementational challenges exist to extend them to solve extreme scale stochastic systems using emerging computing platforms intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in house implementation sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi level iterative solvers parallel sparse matrix vector operations are used to reduce the floating point operations and memory requirements numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non gaussian stochastic process scalability of the solvers with respect to the number of random variables is also investigated

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

addressing the curse of dimensionality in ssfem using the dependence of eigenvalues in kl expansion on domain size this work is aimed at reducing the dimensionality in the spectral stochastic finite element method ssfem thus the computational cost through a domain decomposition dd method this reduction hinges on some new mathematical results on domain size dependence of the karhunen loeve kl expansion it has been reported in the literature that for few covariance kernels a lower domain size leads to a faster convergence of the kl eigenvalues this observation leads to an early truncation of the kl expansion and this reduction in stochastic dimensionality brings down the total computational cost in this work first we mathematically show the generalization of this faster convergence that is for any arbitrary covariance kernel this is achieved via developing a bound on eigenvalues as a function of the domain size then we prove that for a chosen number of terms in the kl expansion with any kernel for a one dimensional process the approximation error in the trace norm reduces with the domain size based on this domain size dependence we propose an algorithm for solving a stochastic elliptic equation in a dd framework the computational cost gain is demonstrated by a numerical study and is observed that the serial implementation of the proposed algorithm is about an order of magnitude faster compared to the existing method the cost saving increases with the stochastic dimensionality in the global domain the structure of this algorithm provides scope for parallelization which would help in efficiently solving large scale problems the sharpness of the proposed eigenvalue bounds is also tested for gaussian and exponential kernels the generalization opens avenues for developing further dd based ssfem solvers c 2016 elsevier b v all rights reserved

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

a domain decomposition approach for uncertainty analysis this paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations pdes the system is split into local components using domain decomposition our domain decomposed uncertainty quantification dduq approach performs uncertainty analysis independently on each local component in an offline phase and then assembles global uncertainty analysis results using precomputed local information in an online phase at the heart of the dduq approach is importance sampling which weights the precomputed local pde solutions appropriately so as to satisfy the domain decomposition coupling conditions to avoid global pde solves in the online phase a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions

schwarz preconditioners for stochastic elliptic pdes abstract increasingly the spectral stochastic finite element method ssfem has become a popular computational tool for uncertainty quantification in numerous practical engineering problems for large scale problems however the computational cost associated with solving the arising linear system in the ssfem still poses a significant challenge the development of efficient and robust preconditioned iterative solvers for the ssfem linear system is thus of paramount importance for uncertainty quantification of large scale industrially relevant problems in the context of high performance computing the preconditioner must scale to a large number of processors therefore in this paper a two level additive schwarz preconditioner is described for the iterative solution of the ssfem linear system the proposed preconditioner can be viewed as a generalization of the mean based block diagonal preconditioner commonly used in the literature for the numerical illustrations two dimensional steady state diffusion and elasticity problems with spatially varying random coefficients are considered the performance of the algorithm is investigated with respect to the geometric parameters strength of randomness dimension and order of the stochastic expansion

percolation on shopping and cashback electronic commerce networks abstract many realistic networks live in the form of multiple networks including interacting networks and interdependent networks here we study percolation properties of a special kind of interacting networks namely shopping and cashback electronic commerce networks scecns we investigate two actual scecns to extract their structural properties and develop a mathematical framework based on generating functions for analyzing directed interacting networks then we derive the necessary and sufficient condition for the absence of the system wide giant in and out component and propose arithmetic to calculate the corresponding structural measures in the sub critical and supercritical regimes we apply our mathematical framework and arithmetic to those two actual scecns to observe its accuracy and give some explanations on the discrepancies we show those structural measures based on our mathematical framework and arithmetic are useful to appraise the status of scecns we also find that the supercritical regime of the whole network is maintained mainly by hyperlinks between different kinds of websites while those hyperlinks between the same kinds of websites can only enlarge the sizes of in components and out components

cost reduction of stochastic galerkin method by adaptive identification of significant polynomial chaos bases for elliptic equations abstract one widely used and computationally efficient method for uncertainty quantification using spectral stochastic finite element is the stochastic galerkin method here the solution is represented in polynomial chaos expansion and the residual of the discretized governing equation is projected on the polynomial chaos bases this results in a system of deterministic algebraic equations with the polynomials chaos coefficients as unknown however one impediment for its large scale applications is the curse of dimensionality that is the exponential growth of the number of polynomial chaos bases with the stochastic dimensionality and degree of expansion here for a stochastic elliptic problem an adaptive selection of polynomial chaos bases is proposed accordingly during the first few iterations in the preconditioned conjugate gradient method for solving the system of linear algebraic equations the chaos bases with maximal contribution in an appropriately defined metric to the solution are first identified subsequently only these bases are retained for further iterations until convergence is achieved using numerical studies a three times cost saving over the existing method is observed furthermore for enhancing the computational cost gain the stochastic galerkin method is reformulated as a generalized sylvester equation this step allowed efficient usage of the sparsity of moments of product of polynomial chaos bases through numerical studies on problems with large stochastic dimensionality an additional cost saving of up to one order of magnitude twenty times is observed this amounts to sixty times speedup over the existing method when adaptive selection and generalized sylvester equation formulation are used together the proposed methodology can be easily incorporated in an existing standard stochastic galerkin method solver for elliptic problems

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

convexity and concavity constants in lorentz and marcinkiewicz spaces abstract we provide here the formulas for the q convexity and q concavity constants for function and sequence lorentz spaces associated to either decreasing or increasing weights it yields also the formula for the q convexity constants in function and sequence marcinkiewicz spaces in this paper we extent and enhance the results from g j o jameson the q concavity constants of lorentz sequence spaces and related inequalities math z 227 1998 129 142 and a kaminska a m parrish the q concavity and q convexity constants in lorentz spaces in banach spaces and their applications in analysis conference in honor of nigel kalton may 2006 walter de gruyter berlin 2007 pp 357 373

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

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

corrigendum to convexity and concavity constants in lorentz and marcinkiewicz spaces j math anal appl 343 2008 337 351 abstract in this note we state the accurate formulas of convexity and concavity constants in lorentz spaces correcting the formulas from kaminska and parrish 2008 2

weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators abstract in this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal as a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function in particular we extend some results given in carro et al 2005 7 to the multilinear context on the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator m b associated to a young function b and the multilinear maximal operators m m 0 t b t 1 n m n m n m as an application of these estimate we obtain a direct proof of the l p l q boundedness results of m b for the case b t t and b k t t 1 log t k when 1 q 1 p n we also give sufficient conditions on the weights involved in the boundedness results of m b that generalizes those given in moen 2009 22 for b t t finally we prove some boundedness results in banach function spaces for a generalized version of the multilinear fractional maximal operator

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

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd