the oscillation and asymptotically monotone solutions of second order quasilinear differential equations abstract we offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation a t y 1 y q t y p t y y as well as for the existence of a positive monotone solution of the damped differential equation a t y 1 y b t y 1 y h t y 0 where 0 examples that dwell upon the importance of our results are also included

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

reversible simulation of bipartite product hamiltonians consider two quantum systems a and b interacting according to a product hamiltonian h h sub a spl ominus h sub b we show that any two such hamiltonians can be used to simulate each other reversibly i e without efficiency losses with the help of local unitary operations and local ancillas accordingly all nonlocal features of a product hamiltonian including the rate at which it can be used to produce entanglement transmit classical or quantum information or simulate other hamiltonians depend only upon a single parameter we identify this parameter and use it to obtain an explicit expression for the entanglement capacity of all product hamiltonians finally we show how the notion of simulation leads to a natural formulation of measures of the strength of a nonlocal hamiltonian

hierarchical preconditioning for the stochastic galerkin method one of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic galerkin method where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables related systems of linear equations are thus usually huge studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning for a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened cauchy bunyakowsky schwarz cbs constants with respect to the scalar product defined by the operator of the weak form of the underlying equation small values of the cbs constant indicate that certain additive or multiplicative two by two block preconditioners reduce enough the condition number of the system moreover we show that a recursive multiplicative two by two block preconditioning can be used resulting in the algebraic multilevel iterative amli method we present the conditions under which the amli preconditioning is of an optimal order numerical experiments confirm the introduced theoretical estimates

oscillation criteria for second order quasilinear functional differential equations abstract some oscillation criteria for the second order quasilinear functional differential equations are obtained our results generalize and improve some known results of both differential equations and delay differential equations

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

a new perspective on the solution of uncertainty quantification and reliability analysis of large scale problems abstract this work revisits the computational performance of non intrusive monte carlo versus intrusive galerkin methods of large scale stochastic systems in the framework of high performance computing environments the purpose of this work is to perform an assessment of the range of the relative superiority of these approaches with regard to a variety of stochastic parameters in both approaches the solution of the resulting algebraic equations is performed with a combination of primal and dual domain decomposition methods implementing specifically tailored preconditioners the solution of repeated simulations of the monte carlo method is accelerated with an a orthogonalization procedure aiming at reducing the iterations of subsequent simulations while the solution of the augmented equations of the stochastic galerkin method is enhanced with preconditioners which combine the block diagonal features of the resulting matrices as well as the sparsity pattern of the off block diagonal terms numerical results are presented demonstrating the efficiency of the proposed implementations on a large scale 3d problem with different stochastic characteristics and useful conclusions are derived regarding the ranges of stochastic parameters in which non intrusive solvers have a superior performance compared to intrusive ones and vice versa

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

equivalence of decoupling schemes and orthogonal arrays decoupling schemes are used in quantum information processing to selectively switch off unwanted interactions in a multipartite hamiltonian a decoupling scheme consists of a sequence of local unitary operations which are applied to the system s qudits and alternate with the natural time evolution of the hamiltonian several constructions of decoupling schemes have been given in the literature here we focus on two such schemes the first is based on certain triples of submatrices of hadamard matrices that are closed under pointwise multiplication see leung simulation and reversal of n qubit hamiltonians using hadamard matrices j mod opt vol 49 pp 1199 1217 2002 the second uses orthogonal arrays see stollsteimer and mahler suppression of arbitrary internal couplings in a quantum register phys rev a vol 64 p 052301 2001 we show that both methods lead to the same class of decoupling schemes we extend the first method to 2 local qudit hamiltonians where dges2 furthermore we extend the second method to t local qudit hamiltonians where tges2 and dges2 by using orthogonal arrays of strength t we also establish a characterization of orthogonal arrays of strength t by showing that they are equivalent to decoupling schemes for t local hamiltonians which have the property that they can be refined to have time slots of equal length the methods used to derive efficient decoupling schemes are based on classical error correcting codes

second order quasilinear oscillation with impulses abstract sufficient conditions for oscillation of all solutions of a class of second order quasilinear differential equations with fixed moments of impulse effect are found

oscillation theorems for certain nonlinear differential equations of second order abstract we present new oscillation criteria for the second order nonlinear differential equations these criteria involve the use of averaging functions our theorems are stated in general form they complement and extend related results known in the literature also the relevance of our results is to show that some of manojlovic s results contain the superfluous condition

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

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

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

existence theorems for second order discrete boundary value problems abstract this article is concerned with the existence of solutions of boundary value problems for nonlinear second order difference equations of the type p n x n 1 q n x n f n x n where 0 is the ratio of odd positive integers p n and q n are real sequences the authors apply the linking theorem and the mountain pass lemma in the critical point theory and give some new results for the existence of solutions

on oscillatory solutions of quasilinear differential equations abstract necessary and sufficient conditions for the existence of at least one oscillatory solution of a second order quasilinear differential equation are presented these results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero counting function

asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations mingshu peng department of applied mathematics beijing institute of technology beijing 100081 p r china qianli xu department of mathematics yiyang teachers college yiyang 413049 p r china lihong huang department of applied mathematics hunan university changsha 410082 p r china weigao ge

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

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

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

nonoscillation criteria for second order nonlinear differential equations necessary and sufficient conditions are obtained for the nonoscillation of all solutions of a class of nonlinear differential equations of second order

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

on the capacities of bipartite hamiltonians and unitary gates we consider interactions as bidirectional channels we investigate the capacities for interaction hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information we give analytic expressions for the entanglement generating capacity and entanglement assisted one way classical communication capacity of interactions and show that these quantities are additive so that the asymptotic capacities equal the corresponding 1 shot capacities we give general bounds on other capacities discuss some examples and conclude with some open questions

integral averaging technique and oscillation of certain even order delay differential equations abstract using the generalized riccati technique and the averaging technique new oscillation criteria for certain even order delay differential equation of the form x n 1 t 1 x n 1 t f t x g t 0 n even are established where g c t0 r f c t0 r r and 0 is a constant

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

preconditioning markov chain monte carlo simulations using coarse scale models we study the preconditioning of markov chain monte carlo mcmc methods using coarse scale models with applications to subsurface characterization the purpose of preconditioning is to reduce the fine scale computational cost and increase the acceptance rate in the mcmc sampling this goal is achieved by generating markov chains based on two stage computations in the first stage a new proposal is first tested by the coarse scale model based on multiscale finite volume methods the full fine scale computation will be conducted only if the proposal passes the coarse scale screening for more efficient simulations an approximation of the full fine scale computation using precomputed multiscale basis functions can also be used comparing with the regular mcmc method the preconditioned mcmc method generates a modified markov chain by incorporating the coarse scale information of the problem the conditions under which the modified markov chain will converge to the correct posterior distribution are stated in the paper the validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed we would like to note that coarse scale models used in the simulations need to be inexpensive but not necessarily very accurate as our analysis and numerical simulations demonstrate we present numerical examples for sampling permeability fields using two point geostatistics the karhunen loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data such as production data as well as some static data our numerical examples show that the acceptance rate can be increased by more than 10 times if mcmc simulations are preconditioned using coarse scale models

groundwater flow in heterogeneous composite aquifers 1 we introduce a stochastic model of flow through highly heterogeneous composite porous media that greatly improves estimates of pressure head statistics composite porous media consist of disjoint blocks of permeable materials each block comprising a single material type within a composite medium hydraulic conductivity can be represented through a pair of random processes 1 a boundary process that determines block arrangement and extent and 2 a stationary process that defines conductivity within a given block we obtain second order statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous next we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation in most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media we make this statement precise we illustrate the composite model with examples of one dimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions we also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in monte carlo simulations

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

acceleration of uncertainty propagation through lagrange multipliers in partitioned stochastic method abstract a partitioned stochastic method psm is proposed for the solution of static structural mechanics problems with uncertainties whose solution vectors are the displacements for each partition and lagrange multipliers along with the partition interfaces the proposed partitioned stochastic method employs three stochastic basis selection steps an arbitrary initial choice of displacement random bases a set of conjugate bases for the lagrange multipliers and finally modification of the displacement bases affected by those of the lagrange multipliers the present psm thus propagates the uncertainties instantly across partitioned substructures resulting in an improved rate of convergence numerical experiments illustrate the proposed psm outperforms a conventional partitioned solution method for structural mechanics 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

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

asymptotic behavior of nonoscillatory solutions of second order differential equations in this paper we establish some sufficient conditions for the asymptotic behavior of all nonoscillatory solutions of the differential equation p t f y t q t f y t 0 under suitable condition on email protected c t 0 0 email protected c t 0 r and f email protected c r r

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

oscillatory behavior of solutions of certain second order nonlinear differential equations abstract in this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation formula where 0 p q with p q odd integers or p even and q odd integers several examples which dwell upon the importance of our results are also included

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

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