computational modeling of uncertainty in time domain electromagnetics we discuss computationally efficient ways of accounting for the impact of uncertainty e g lack of detailed knowledge about sources materials shapes etc in computational time domain electromagnetics in contrast to classic statistical monte carlo based methods we explore a probabilistic approach based on high order accurate expansions of general stochastic processes we show this to be highly efficient and accurate on both one and two dimensional examples enabling the computation of global sensitivities of measures of interest e g radar cross sections rcs in scattering applications for a variety of types of uncertainties

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

a stochastic framework for the variability analysis of textile antennas a novel framework to accurately quantify the effect of stochastic variations of design parameters on the performance of textile antennas is developed and tested first a sensitivity analysis is applied to get a rough idea about the effect of these random variations on the textile antenna s performance next a more detailed view is obtained by a generalized polynomial chaos technique that accurately quantifies the statistical distribution of the textile antenna s figures of merit for a given range over which geometry and material parameters vary statistically according to a given distribution the method is validated both for a simple inset fed patch textile microstrip antenna and for a dual polarized textile antenna for the latter the probability density function corresponding to its most sensitive design parameter being the width is experimentally estimated by means of measurements performed on 100 patches a kolmogorov smirnoff test proves that for all considered examples the results are as accurate as those obtained via monte carlo analysis while the new technique is much more efficient indeed speedups up to a factor 1667 are demonstrated

distributed model predictive control of linear systems with stochastic parametric uncertainties and coupled probabilistic constraints this paper investigates the distributed stochastic model predictive control dsmpc for multiple constrained dynamically decoupled subsystems subject to stochastic uncertainties in the parameters to obtain a computationally tractable formulation for real control applications a spectral method called generalized polynomial chaos expansions gpces is utilized to propagate the stochastic parametric uncertainties through the system model by using gpces combined with the probabilistic information on the uncertainties both local probabilistic constraints and coupled probabilistic constraints are converted explicitly into deterministic convex second order cone constraints the constraints can achieve satisfaction of coupled probabilistic constraints in a distributed way by permitting a single subsystem to optimize a local cost function at each time step while freezing the plans of others the proposed gpces based dsmpc algorithm guarantees recursive feasibility with respect to both local and coupled prob

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations we consider a hyperbolic system with uncertainty in the boundary and initial data our aim is to show that dierent boundary conditions gives dierent convergence rates of the variance of the solution this means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution a variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented as applications we study the eect of this technique on maxwell s equations as well as on a subsonic outow boundary for the euler equations

stochastic analysis of the impact of substrate compression on the performance of textile antennas one of the many adverse effects modifying the performance of textile antennas in real operating conditions is substrate compression therefore this communication presents a stochastic collocation method scm that either relies on the generalized polynomial chaos gpc expansion or on a novel hermite pade approximant the method is introduced to rigorously quantify the effect of random variations in the height and the permittivity of the substrate on the figures of merit of a textile antenna next the joint height and permittivity probability distribution of a compressible substrate are characterized by means of a new measurement setup based on a resonant perturbation technique finally the method is validated for a probe fed gps textile antenna it is shown that hermite pade approximants model the highly nonlinear relationship between these substrate random variables and the figures of merit of the antenna more efficiently than the gpc moreover a kolmogorov smirnoff test proves that the resulting distributions of the antenna s figures of merit are as accurate as those obtained by means of a monte carlo mc analysis with demonstrated speedup factors up to 123

energy conserving galerkin approximation of two dimensional wave equations with random coefficients abstract wave propagation problems for heterogeneous media are known to have many applications in physics and engineering recently there has been an increasing interest in stochastic effects due to the uncertainty which may arise from impurities of the media this work considers a two dimensional wave equation with random coefficients which may be discontinuous in space generalized polynomial chaos method is used in conjunction with stochastic galerkin approximation and local discontinuous galerkin method is used for spatial discretization our method is shown to be energy preserving in semi discrete form as well as in fully discrete form when leap frog time discretization is used its convergence rate is proved to be optimal and the error grows linearly in time the theoretical properties of the proposed scheme are validated by numerical tests

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

a flexible numerical approach for quantification of epistemic uncertainty in the field of uncertainty quantification uq epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available largely due to our lack of knowledge about the uncertainty quantification of the impacts of epistemic uncertainty is naturally difficult because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty and there have been few studies and methods to deal with epistemic uncertainty a recent work can be found in j jakeman m eldred d xiu numerical approach for quantification of epistemic uncertainty j comput phys 229 2010 4648 4663 where a framework for numerical treatment of epistemic uncertainty was proposed the method is based on solving an encapsulation problem without using any probability information in a hypercube that encapsulates the unknown epistemic probability space if more probabilistic information about the epistemic variables is known a posteriori the solution statistics can then be evaluated at post process steps in this paper we present a new method similar to that of jakeman et al but significantly extending its capabilities most notably the new method 1 does not require the encapsulation problem to be in a bounded domain such as a hypercube 2 does not require the solution of the encapsulation problem to converge point wise in the current formulation the encapsulation problem could reside in an unbounded domain and more importantly its numerical approximation could be sought in l p norm these features thus make the new approach more flexible and amicable to practical implementation both the mathematical framework and numerical analysis are presented to demonstrate the effectiveness of the new approach

efficient computation of rcs from scatterers of uncertain shapes we propose a way of accounting for the lack of detailed knowledge about material shapes in computational time domain electromagnetics we use legendre gauss lobatto stroud 2 and stroud 3 quadrature formulas to solve the resulting stochastic equation and we show the efficiency of the proposed method over statistical monte carlo simulations we also show how the radar cross section rcs in scattering is affected by the uncertainty in shape of the objects and by the direction of the incident field

a new surrogate modeling technique combining kriging and polynomial chaos expansions application to uncertainty analysis in computational dosimetry in numerical dosimetry the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate sar characterizing the human exposure to electromagnetic waves however this procedure remains time consuming and a single simulation can request several hours as a consequence the influence of uncertain input parameters on the sar cannot be analyzed using crude monte carlo simulation the solution presented here to perform such an analysis is surrogate modeling this paper proposes a novel approach to build such a surrogate model from a design of experiments considering a sparse representation of the polynomial chaos expansions using least angle regression as a selection algorithm to retain the most influential polynomials this paper proposes to use the selected polynomials as regression functions for the universal kriging model the leave one out cross validation is used to select the optimal number of polynomials in the deterministic part of the kriging model the proposed approach called lars kriging pc modeling is applied to three benchmark examples and then to a full scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device the performances of the lars kriging pc are compared to an ordinary kriging model and to a classical sparse polynomial chaos expansion the lars kriging pc appears to have better performances than the two other approaches a significant accuracy improvement is observed compared to the ordinary kriging or to the sparse polynomial chaos depending on the studied case this approach seems to be an optimal solution between the two other classical approaches a global sensitivity analysis is finally performed on the lars kriging pc model of the fetus exposure problem

modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions this study proposes a framework for uncertainty analysis by incorporating explicit numerical solutions of governing equations for flood wave propagation with the expectation operator it aims at effectively evaluating the effect of variations in initial and boundary conditions on the estimation of flood waves spatiotemporal semi variogram models are employed to quantify the correlation of the variables in time and space the 1 d nonlinear kinematic wave equation for the overland flow named evo ns kwe is applied in the model development model validation is made by comparison with the monte carlo simulation mcs model in the calculation of statistical properties of model outputs i e flow depths that is the mean standard deviation and coefficient of variation cv the results from the model validation show that the evo ns kwe model can produce excellent approximations of the mean and less satisfactory approximations of the standard deviation and cv compared with those obtained using the mcs model it concludes that the uncertainties of flow depths in the domain are significantly affected by variations in the boundary condition future application of the evo ns kwe model enable the evaluation of uncertainty in model outputs induced by the initial and boundary condition subject to uncertainty and will also provide corresponding probabilistic information for risk quantification method

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos abstract polynomial chaos pc is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems in the case of a structural system under dynamic loading the pc method is combined with a time integration scheme the pc method with galerkin projection converts the stochastic pde to a set of simultaneous deterministic pde which can be solved using any standard time integration scheme however the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response the probability density functions of the responses are not of constant type as the time progresses however in the case of generalized pc gpc these are approximated using initially considered gpc thus loses its efficiency time dependent general polynomial chaos tdgpc can be considered to overcome these difficulties where polynomials are generated on the fly to match with the pdf of the responses the present study considered an investigation of structural mechanics responses with both gaussian and non gaussian random system properties under dynamic loading using tdgpc an adaptive updation scheme based on the rms value of the coefficients of pc expansion is investigated for a proper updation of pc expansion the curse of dimensionality of pc expansion is addressed by considering only the dominant components of the responses evaluated using kl expansion further starting with lower order pc the order of pc expansion is increased only after the first updation thus reducing computational complexities

stable recovery of time dependent coefficient from arbitrary measurements for wave equation abstract this paper concerns the stability in the inverse problem of determining the time dependent coefficient in the wave equation from partial knowledge of the dirichlet to neumann map at the boundary of a bounded domain of interest we first establish a stability estimate in the determination of the time dependent coefficient from partial boundary observations then we prove that by enlarging the set of data the unknown coefficient can be stably determined in larger regions

on transformations into normal form abstract we discuss a certain class of transformations of an ordinary differential equation into normal form near a stationary point these transformations preserve certain special structures and one can derive from them a universally applicable algorithm to determine a normal form up to any desired degree the algorithm can be programmed with the help of any computer algebra system in the same manner transformations into normal form on an invariant manifold are discussed

sequential subspace reliability method with univariate revolving integration a new computational framework called the sequential subspace reliability method ssrm is presented this method decomposes the multidimensional random space into multiple two dimensional subspaces in this manner ssrm is able to approximate bivariate interaction effects when the reliability estimate contribution is calculated subspace by subspace the final assessment is updated in a progressive manner the iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest in a decision making situation the flexibility of the proposed ssrm to provide iterative updates on reliability estimation becomes especially valuable in dealing with large scale and complex problems under the constraints of limited time and resources to calculate the individual subspace contributions a novel univariate revolving integration uri method is proposed the uri method takes advantage of the axisymmetric nature of a joint pr

multi fidelity non intrusive polynomial chaos based on regression abstract in this paper we present a multi fidelity mf extension of non intrusive polynomial chaos based on regression point collocation for uncertainty quantification purposes the proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients due to its usage of regression to estimate the coefficients the present method offers high flexibility in the sampling and generation of the polynomial basis the method takes advantage of a nested sampling plan to create the samples for the low fidelity lf and correction expansions where the high fidelity hf samples are a subset of the lf ones to build the polynomial basis a total order or hyperbolic truncation strategy is used with a highly flexible combination of the lf and correction polynomial expansions the method is demonstrated on some artificial test problems and aerodynamic problems of the euler flow around an airfoil and common three dimensional research models in order to derive the strategies for successful mf approximation the effect of the correlation and the errors between the lf and hf functions is also studied the results show that high correlation and moderately low errors are important to improve the mf approximation s accuracy on a common research model problem the mf approach with partially converged simulations as the lf samples can successfully reduce the computational cost to about 40 for similar accuracy compared to an approach using a single hf expansion

polynomial meta models with canonical low rank approximations the growing need for uncertainty analysis of complex computational models has led to an expanding use of meta models across engineering and sciences the efficiency of meta modeling techniques relies on their ability to provide statistically equivalent analytical representations based on relatively few evaluations of the original model polynomial chaos expansions pce have proven a powerful tool for developing meta models in a wide range of applications the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials the classical pce approach nevertheless faces the curse of dimensionality namely the exponential increase of the basis size with increasing input dimension to address this limitation the sparse pce technique has been proposed in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm an alternative for developing meta models with polynomial functions in high dimensional problems is offered by the newly emerged low rank approximations lra approach by exploiting the tensor product structure of the multivariate basis lra can provide polynomial representations in highly compressed formats through extensive numerical investigations we herein first shed light on issues relating to the construction of canonical lra with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions specifically we examine the selection of optimal rank stopping criteria in the updating of the polynomial coefficients and error estimation in the sequel we confront canonical lra to sparse pce in structural mechanics and heat conduction applications based on finite element solutions canonical lra exhibit smaller errors than sparse pce in cases when the number of available model evaluations is small with respect to the input dimension a situation that is often encountered in real life problems by introducing the conditional generalization error we further demonstrate that canonical lra tend to outperform sparse pce in the prediction of extreme model responses which is critical in reliability analysis

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

boundedness and unboundedness in polynomial differential systems on relie le caractere fini ou infini d un systeme differentiel polynomial autonome aux signes des invariants du systeme differentiel correspondant des polynomes homogenes de plus haut degre

an efficient technique based on polynomial chaos to model the uncertainty in the resonance frequency of textile antennas due to bending the generalized polynomial chaos theory is combined with a dedicated cavity model for curved textile antennas to statistically quantify variations in the antenna s resonance frequency under randomly varying bending conditions the nonintrusive stochastic method solves the dispersion relation for the resonance frequencies of a set of radius of curvature realizations corresponding to the gauss quadrature points belonging to the orthogonal polynomials having the probability density function of the random variable as a weighting function the formalism is applied to different distributions for the radius of curvature either using a priori known or on the fly constructed sets of orthogonal polynomials numerical and experimental validation shows that the new approach is at least as accurate as monte carlo simulations while being at least 100 times faster this makes the method especially suited as a design tool to account for performance variability when textile antennas are deployed on persons with varying body morphology

multidimensional inverse problems and completeness of the products of solutions to pde a method is given for proving uniqueness theorems for some inverse problems the method is based on a result on completeness of the products of solutions to pde as an example the following uniqueness theorems are proved 1 the scattering amplitude a k known for all s2 and a fixed k 0 determines the compactly supported q x l2 d uniquely 2 the surface data u x y k known for all x y p x x3 0 and a fixed k 0 determine the compactly supported x l2 d d r 3 x x3 0 is arbitrarily small determine aj x j 1 2 uniquely here 2u k2u k2a1x a2 x u x y in r3 a1 l2 d a2 h2 d the same conclusion holds if the surface data are known at two distinct frequencies 4 the surface data u x y k known for all x y p and all k 0 determine v x and h k uniquely here 2 k2 k2 x u x y h k x l2 d h k is fourier transform of a wavelet of compact support 5 the conductivity x w2 2 d x c 0 is uniquely determined by the measurements of u and un on d here n is the outward normal to d d r3 is a bounded domain with a smooth boundary d x u 0 in d 6 necessary and sufficient conditions are given for a function a k s2 k 0 is fixed to be the scattering amplitude corresponding to a local potential from a certain class

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

stability in the determination of a time dependent coefficient for wave equations from partial data let be a c2 bounded domain of rn n 2 and fix q 0 t with t 0 we consider the stability in the inverse problem of determining a time dependent coefficient of order zero q appearing in a dirichlet initial boundary value problem for a wave equation t2u xu q t x u 0 in q from partial observations on q the observation is given by a boundary operator associated to the wave equation using suitable geometric optics solutions and carleman estimates we prove a stability estimate in the determination of q from the boundary operator

scattering from two dimensional objects of varying shape combining the method of moments with the stochastic galerkin method in this communication the combined field integral equation for perfect electrically conducting scatterers is combined with the stochastic galerkin method sgm to model the impact of stochastic variations of the shape of the scatterer on the radar cross section and on the induced current distribution the sgm is compared to the stochastic collocation method scm and it is shown that for a modest number of random variables the sgm is a good alternative to the scm

hyperbolicity preserving and well balanced stochastic galerkin method for shallow water equations a stochastic galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve the hyperbolicity of the original system in this work we develop a hyperbolicity pr

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

data based importance sampling estimates for extreme events abstract an accurate efficient and conceptually simple method is developed to estimate distributions of maxima of solutions of stochastic equations i e ordinary or partial differential equations with random entries the method is data based it constructs importance sampling is or biasing measures from samples of surrogates of full model solutions of stochastic equations and uses these measures and mixtures of surrogate and full model samples to estimate probabilities of extreme events numerical examples are presented to illustrate the implementations of the proposed method and demonstrate numerically its performance

necessary and sufficient condition for l to have property c abstract let l m j 0aj j be a formal differential operator with constant coefficients in r n n 2 j is a multi index we say that l has property c iff for any bounded domain d r n the set of products uw u w nd l u l u 0 in d u hm d is complete in l2 d here hm d is the sobolev space a necessary and sufficient condition is given for l to have property c

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

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

projections of polynomial vector fields and the poincare sphere abstract we investigate projections of homogeneous polynomial vector fields to level sets of homogeneous polynomials this allows a systematic study of stationary points at infinity for polynomial differential equations in n dimensional real space results include some general criteria for the existence of unbounded solutions and a fairly complete discussion of boundedness of solutions for second order equations in one dependent variable

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

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

a polynomial expansion approach for response analysis of periodical composite structural acoustic problems with multi scale mixed aleatory and epistemic uncertainties abstract the response analysis of periodical composite structural acoustic problem with multi scale mixed aleatory and epistemic uncertainties is investigated based on homogenization method in this paper the aleatory uncertainties are presented by bounded random variables whereas the epistemic uncertainties are presented by interval variables and evidence variables when dealing with the combination of bounded random variables interval variables and evidence variables enormous computation is needed to estimate the output probability bounds of the sound pressure response of the periodical composite structural acoustic system to reduce the involved computational cost but without losing accuracy by transforming all of the bounded random variables and interval variables into evidence variables appropriately an evidence theory based polynomial expansion method epem is developed in which the gegenbauer series expansion is employed to approximate the variation range of the response with respect to evidence variables by using epem the probability bounds of the response can be obtained efficiently a numerical example is used to validate the proposed method and two engineering examples are given to demonstrate its efficiency

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

fourier mode analysis of multigrid methods for partial differential equations with random coefficients partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models various approaches exist for computing the stochastic characteristics of the solution of such a differential equation in this paper we consider the spectral expansion approach this method transforms the continuous model into a large discrete algebraic system we study the convergence properties of iterative methods for solving this discretized system we consider one level and multi level methods the classical fourier mode analysis technique is extended towards the stochastic case this is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem we show how the convergence properties depend on the particulars of the algorithm on the discretization parameters and on the stochastic characteristics of the model numerical results are added to illustrate some of our theoretical findings

sequential experimental design based generalised anova over the last decade surrogate modelling technique has gained wide popularity in the field of uncertainty quantification optimization model exploration and sensitivity analysis this approach relies on experimental design to generate training points and regression interpolation for generating the surrogate in this work it is argued that conventional experimental design may render a surrogate model inefficient in order to address this issue this paper presents a novel distribution adaptive sequential experimental design da sed the proposed da sed has been coupled with a variant of generalised analysis of variance g anova developed by representing the component function using the generalised polynomial chaos expansion moreover generalised analytical expressions for calculating the first two statistical moments of the response which are utilized in predicting the probability of failure have also been developed the proposed approach has been utilized in predicting probability of failure of three structural mechanics problems it is observed that the proposed approach yields accurate and computationally efficient estimate of the failure probability

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

boundary control wave field continuation and inverse problems for the wave equation abstract we consider the problem of the wave field continuation and recovering of coefficients for the wave equation in a bounded domain in r n n 1 the inverse data is a response operator mapping neumann boundary data into dirichlet ones the reconstruction procedure is local this means that observing boundary response for larger times we may recover coefficients deeper in the domain the approach is based upon ideas and results of the boundary control theory yielding some natural multidimensional analogs of the classical gel fand levitan krein equations

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

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

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

uniqueness theorems for multidimensional inverse problems with unbounded coefficients abstract in a g ramm inverse problems 3 1987 l77 l82 a method to prove uniqueness theorems is given the method uses an estimate of the solution to a partial differential equation with l d coefficients here the estimate is established for l2 d coefficients and the resulting uniqueness theorems are formulated

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

backward uncertainty propagation method in flow problems application to the prediction of rarefaction shock waves abstract a computational method for taking into account backward uncertainty propagation in flow problems is presented and applied to the study of rarefaction shock waves rsw in a dense gas shock tube previous theoretical and numerical studies have shown that a rsw is relatively weak and that the prediction of its occurrence and intensity are highly sensitive to uncertainties on the initial flow conditions and on the fluid thermodynamic model the objective of this work is to introduce an innovative flexible and efficient algorithm combining computational fluid dynamics cfd uncertainty quantification uq tools and metamodel based optimization in order to obtain a reliable estimate for the rsw probability of occurrence and to prescribe the experimental accuracy requirements ensuring the reproducibility of the measurements with sufficient confidence

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

uncertainty analysis for the steady state flows in a dual throat nozzle it is well known that the steady state of an isentropic flow in a dual throat nozzle with equal throat areas is not unique in particular there is a possibility that the flow contains a shock wave whose location is determined solely by the initial condition in this paper we consider cases with uncertainty in this initial condition and use generalized polynomial chaos methods to study the steady state solutions for stochastic initial conditions special interest is given to the statistics of the shock location the polynomial chaos pc expansion modes are shown to be smooth functions of the spatial variable x although each solution realization is discontinuous in the spatial variable x when the variance of the initial condition is small the probability density function of the shock location is computed with high accuracy otherwise many terms are needed in the pc expansion to produce reasonable results due to the slow convergence of the pc expansion caused by non smoothness in random space

parallel domain decomposition methods for stochastic elliptic equations we present parallel schwarz type domain decomposition preconditioned recycling krylov subspace methods for the numerical solution of stochastic elliptic problems whose coefficients are assumed to be a random field with finite variance karhunen loeve kl expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations the key to an efficient algorithm lies in recycling computed subspaces based on a careful analysis of the kl expansion we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation we show theoretically and experimentally that the schwarz preconditioned recycling gmres method is optimal for the entire family of linear systems a fully parallel implementation is provided and scalability results are reported in the paper

monte carlo simulation of electromagnetic scattering from two dimensional random rough surfaces the fast multipole method fast fourier transform fmm fft method is developed to compute the scattering of an electromagnetic wave from a two dimensional 2 d rough surface the resulting algorithm computes a matrix vector multiply in o n log n operations this algorithm is shown to be more efficient than another o n log n algorithm the multilevel fast multipole algorithm mlfma for surfaces of small height for surfaces with larger roughness the mlfma is found to be more efficient using the mlfma monte carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from 2 d random rough surfaces using a workstation for the rougher surface backscattering enhancement is clearly observable as a pronounced peak in the backscattering direction of the computed bistatic scattering coefficient for the smoother surface the monte carlo results compare well with the results of the approximate kirchhoff theory

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

bayesian sparse polynomial chaos expansion for global sensitivity analysis polynomial chaos expansions are frequently used by engineers and modellers for uncertainty and sensitivity analyses of computer models they allow representing the input output relations of computer models usually only a few terms are really relevant in such a representation it is a challenge to infer the best sparse polynomial chaos expansion of a given model input output data set in the present article sparse polynomial chaos expansions are investigated for global sensitivity analysis of computer model responses a new bayesian approach is proposed to perform this task based on the kashyap information criterion for model selection the efficiency of the proposed algorithm is assessed on several benchmarks before applying the algorithm to identify the most relevant inputs of a double diffusive convection model

property c and an inverse problem for a hyperbolic equation let utt u q x t u 0 in d 0 t where d r3 is a bounded domain with a smooth boundary d t d d diam d q x t c 0 t l d suppose that for every u d f x t c1 d 0 t the value un d h s t is known where n is the outer normal to d u solves and and satisfies the initial conditions u ut 0 at t 0 then q x t is uniquely determined by the data f h f c1 d 0 t in the subset s of d 0 t consisting of the lines which make 45 with the t axis and which meet the planes t 0 and t t outside d 0 t provided that q x t is known outside s here d is the closure of d

use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory scale porous media in this work we show how the use of global sensitivity analysis gsa in conjunction with the polynomial chaos expansion pce methodology can provide relevant information for the interpretation of transport experiments in laboratory scale heterogeneous porous media we perform gsa by calculating the sobol indices which provide a variance based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model the choice of pce has the following two benefits 1 it provides the global sensitivity indices in a straightforward manner and 2 pce can serve as a surrogate model for the calibration of parameters the coefficients of the pce are computed by probabilistic collocation the methodology is applied to two nonreactive transport experiments available in the literature while considering both transient and pseudo steady state transport regimes this method allows a rigorous investigation of the relative effects and importance of different uncertain quantities which include boundary conditions as well as porous medium hydraulic and dispersive parameters the parameters that are most relevant to depicting the system s behavior can then be evaluated in addition one can assess the space time distribution of measurement points which is the most influential factor for the identifiability of parameters our work indicates that these methods can be valuable tools in the proper design of model based transport experiments

reduced order models of finite element approximations of electromagnetic devices exhibiting statistical variability a methodology is proposed for the development of reduced order models of finite element approximations of electromagnetic devices exhibiting uncertainty or statistical variability in their input parameters in this approach the reduced order system matrices are represented in terms of their orthogonal polynomial chaos expansions on the probability space defined by the input random variables the coefficients of these polynomials which are matrices are obtained through the repeated deterministic model order reduction of finite element models generated for specific values of the input random parameters these values are chosen efficiently in a multi dimensional grid using a smolyak algorithm the generated stochastic reduced order model is represented in the form of an augmented system that lends itself to the direct generation of the desired statistics of the device response the accuracy and efficiency of the proposed method is demonstrated through its application to the reduced order finite element modeling of a terminated coaxial cable and a circular wire loop antenna

numerical studies of the stochastic korteweg de vries equation we present numerical solutions of the stochastic korteweg de vries equation for three cases corresponding to additive time dependent noise multiplicative space dependent noise and a combination of the two we employ polynomial chaos for discretization in random space and discontinuous galerkin and finite difference for discretization in physical space the accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and monte carlo simulation results of particular interest is the interplay of spatial discretization error with the stochastic approximation error which is examined for different orders of spatial and stochastic approximation

stable determination outside a cloaking region of two time dependent coefficients in an hyperbolic equation from dirichlet to neumann map abstract in this paper we treat the inverse problem of determining two time dependent coefficients appearing in a dissipative wave equation from measured neumann boundary observations we establish in dimension n 2 stability estimates with respect to the dirichlet to neumann map of these coefficients provided that they are known outside cloaking regions moreover we prove that they can be stably recovered in larger subsets of the domain by enlarging the set of data