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

application of polynomial chaos in stability and control the polynomial chaos of wiener provides a framework for the statistical analysis of dynamical systems with computational cost far superior to monte carlo simulations it is a useful tool for control systems analysis because it allows probabilistic description of the effects of uncertainty especially in systems having nonlinearities and where other techniques such as lyapunov s method may fail we show that stability of a system can be inferred from the evolution of modal amplitudes covering nearly the full support of the uncertain parameters with a finite series by casting uncertain parameters as unknown gains we show that the separation of stochastic from deterministic elements in the response points to fast iterative design methods for nonlinear control

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 unified framework for the comparison of treatments with ordinal responses different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable in this paper we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses the latent variable model is built upon the location scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables including the proportional odds model the ordered probit type model and the proportional hazards model a flexible estimation procedure is proposed for the identification and estimation of the general latent variable model which allows for the location and scale parameters to be freely estimated the framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data thus allowing for important statistical inferences such as location and or dispersion comparisons among treatments to be conveniently drawn analysis on real data sets is used to illustrate the proposed methods

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

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

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

generalised polynomial chaos expansion approaches to approximate stochastic model predictive control this paper considers the model predictive control of dynamic systems subject to stochastic uncertainties due to parametric uncertainties and exogenous disturbance the effects of uncertainties are quantified using generalised polynomial chaos expansions with an additive gaussian random process as the exogenous disturbance with gaussian approximation of the resulting solution trajectory of a stochastic differential equation using generalised polynomial chaos expansion convex finite horizon model predictive control problems are solved that are amenable to online computation of a stochastically robust control policy over the time horizon using generalised polynomial chaos expansions combined with convex relaxation methods the probabilistic constraints are replaced by convex deterministic constraints that approximate the probabilistic violations this approach to chance constrained model predictive control provides an explicit way to handle a stochastic system model in the presence of both model uncertain

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 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

model order reduction of random parameter dependent linear systems this paper proposes a new method to perform model reduction of linear time invariant lti systems where parameters are random variables governed by probabilistic laws it combines the well known truncation balanced realization tbr technique together with the generalized polynomial chaos gpc formalism a powerful tool for uncertainty propagation gpc formalism is used to represent and compute a random parameter dependent balancing transformation rpd bt which puts the random lti system in a balanced form almost surely within the probabilistic range of the uncertain parameters model reduction is then performed by truncating almost surely weakly controllable and observable states yielding a random parameter dependent truncated balanced realization rpd tbr the truncation error s moments are shown to be bounded by hankel singular values moments which are also estimated using gpc formalism as an illustrative example the proposed method is applied to a simple mechanical model of a two degrees of freedom mass spring system with uncertain stiffness and damping

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

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

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 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

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

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

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

brief paper gradient dynamic optimization with legendre chaos the polynomial chaos approach for stochastic simulation is applied to trajectory optimization by conceptually replacing random variables with free variables using the gradient method we generate with low computational cost an accurate parametrization of optimal trajectories

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

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

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

non intrusive generalized polynomial chaos for the robust stability analysis of uncertain nonlinear dynamic friction systems this paper is devoted to the stability analysis of uncertain nonlinear dynamic dry friction systems the stability property of dry friction systems is known to be very sensitive to the variations of friction laws moreover the friction coefficient admits dispersions due to the manufacturing processes therefore it becomes necessary to take this uncertainty into account in the stability analysis of dry friction systems to ensure robust predictions of stable and instable behaviours the generalized polynomial chaos formalism is proposed to deal with this challenging problem treated in most cases with the prohibitive monte carlo based techniques two equivalent methods presented here combine the non intrusive generalized polynomial chaos with the indirect lyapunov method both methods are shown to be efficient in the estimation of the stability and instability regions with high accuracy and high confidence levels and at lower cost compared with the classic monte carlo based method

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

region of attraction analysis of nonlinear stochastic systems using polynomial chaos expansion abstract a method is presented to estimate the region of attraction roa of stochastic systems with finite second moment and uncertainty dependent equilibria the approach employs polynomial chaos pc expansions to represent the stochastic system by a higher dimensional set of deterministic equations we first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty dependent equilibrium point of the stochastic system a connection between the boundedness of the moments of the stochastic system and the lyapunov stability of its pc expansion is then derived defining corresponding notions of a roa for both system representations we show how this connection can be leveraged to recover an estimate of the roa of the stochastic system from the roa of the pc expanded system two optimization programs obtained from sum of squares programming techniques are provided to compute inner estimates of the roa the first optimization program uses the lyapunov stability arguments to return an estimate of the roa of the pc expansion based on this result and user specifications on the moments for the initial conditions the second one employs the shown connection to provide the corresponding roa of the stochastic system the method is demonstrated by two examples

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

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

a new non intrusive polynomial chaos using higher order sensitivities abstract this paper proposes a new non intrusive method for uncertainty quantification called polynomial chaos decomposition with differentiation pcdd that uses higher order sensitivities of the response in pcdd the polynomial chaos expansion pce of the response is differentiated with respect to the basis random variables using multi indices this differentiation results in a system of linear equations which can then be solved to determine the expansion coefficients here the higher accuracy modified forward finite difference modffd that involves representation of the response using taylor expansion of order equal to the chaos order is used in combination with pce therefore the total number of samples required with this method is equal to the number of terms in the pce to verify the validity of this new technique two analytical problems and two stochastic composite laminate problems were studied the results of the analytical problems showed that the accuracy of pcdd using modffd is similar to that of pcdd using analytical sensitivities which in addition is comparable to the exact results for composite laminate problems the pcdd displayed very high accuracy comparable to 50 000 latin hypercube samples which underlines the computational efficiency of this proposed method

sample size calculations for ordered categorical data many clinical trials yield data on an ordered categorical scale such as very good good moderate poor under the assumption of proportional odds such data can be analysed using techniques of logistic regression in simple comparisons of two treatments this approach becomes equivalent to the mann whitney test in this paper sample size formulae consistent with an eventual logistic regression analysis are derived the influence on efficiency of the number and breadth of categories will be examined effects of misclassification and of stratification are discussed and examples of the calculations are given

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

polynomial chaos expansions for optimal control of nonlinear random oscillators the polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems the procedure is demonstrated on a base excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its karhunen loeve expansion an adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement velocity norm and is applied to the identification of phase orbits of nonlinear oscillators this approximation is then integrated in the design of an optimal polynomial controller allowing for the efficient estimation of statistics and probability density functions of quantities of interest numerical investigations are carried out employing the polynomial chaos expansions and the lyapunov asymptotic stability condition based control policy the results reveal that the performance as gaged by probabilistic quantities of interest of the controlled oscillators is greatly improved a comparative study is also presented against the classical stochastic optimal control whereby statistical linearization based lqg is employed to design the optimal controller it is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators

choice of test for comparing two groups with particular application to skewed outcomes we consider a clinical trial where a skewed outcome variable is to be compared between two groups while comparison of sample means may lack power we show that power also depends on the nature of the anticipated treatment effect for any given distribution in the control arm there is a family of true distributions in the intervention arm for which the most powerful test is a comparison of arithmetic means similar results hold for a comparison of geometric means and approximately for the wilcoxon rank sum test and a comparison of medians we discuss how these methods could be used in planning the analysis of a clinical trial in which the intervention effect alters the shape of the distribution these ideas are illustrated by a trial in community psychiatry where the primary outcome days in hospital was highly skewed but the intervention was mainly expected to reduce the frequency of values in the tail we show that a comparison of sample means is a reasonable choice in this case despite the skewness

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

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

comparison of two treatments with skewed ordinal responses in clinical studies the proportional odds model is widely used to compare treatment efficacies when the responses are categorically ordered however this model has been shown to be inappropriate when the proportional odds assumption is invalid mainly because it is unable to control the type i error rate in such circumstances to remedy this problem the latent normal model was recently promoted and has been demonstrated to be superior to the proportional odds model however the application of the latent normal model is limited to compare treatments with similar underlying distributions except possibly their means and variances when the underlying distributions are very different in skewness both of the aforementioned procedures suffer from the undesirable inflation of the type i error rate to solve the problem for clinical studies with ordinal responses we provide a viable solution that relies on the use of the latent weibull distribution which is a member of the log location scale family the proposed model is able to control the type i error rate regardless of the degree of skewness of the treatment responses in addition the power of the test also outperforms that of the latent normal model the testing procedure draws on newly developed theoretical results related to latent distributions from the location scale family the testing procedure is illustrated with two clinical examples

pairwise comparisons with ordered categorical data clinical trials frequently involve pairwise comparisons of different treatments to evaluate their relative efficacy in this study we examine methods for conducting pairwise tests of treatments with ordered categorical responses a modified version of the wilcoxon mann whitney test based on a logistic regression model assuming proportional odds is a popular choice for comparing two treatments this paper discusses the extension of this test to pairwise comparisons involving more than two treatments however when the proportional odds assumption is not valid the wilcoxon mann whitney type test procedure cannot control the overall type i error rate at the prespecified level of significance we therefore propose a better strategy in which a latent normal model is employed we presented a simulated comparative study of power and the overall type i error rate to illustrate the superiority of the latent normal model examples are also given for illustrative purposes copyright 2013 john wiley sons ltd

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

sample size calculation for the wilcoxon mann whitney test adjusting for ties in this paper we study sample size calculation methods for the asymptotic wilcoxon mann whitney test for data with or without ties the existing methods are applicable either to data with ties or to data without ties but not to both cases while the existing methods developed for data without ties perform well the methods developed for data with ties have limitations in that they are either applicable to proportional odds alternatives or have computational difficulties we propose a new method which has a closed form formula and therefore is very easy to calculate in addition the new method can be applied to both data with or without ties simulations have demonstrated that the new sample size formula performs very well as the corresponding actual powers are close to the nominal powers

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

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

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

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

assessing toxicities in a clinical trial bayesian inference for ordinal data nested within categories this article addresses modeling and inference for ordinal outcomes nested within categorical responses we propose a mixture of normal distributions for latent variables associated with the ordinal data this mixture model allows us to fix without loss of generality the cutpoint parameters that link the latent variable with the observed ordinal outcome moreover the mixture model is shown to be more flexible in estimating cell probabilities when compared to the traditional bayesian ordinal probit regression model with random cutpoint parameters we extend our model to take into account possible dependence among the outcomes in different categories we apply the model to a randomized phase iii study to compare treatments on the basis of toxicities recorded by type of toxicity and grade within type the data include the different categorical toxicity types exhibited in each patient each type of toxicity has an ordinal grade associated to it the dependence among the different types of toxicity exhibited by the same patient is modeled by introducing patient specific random effects

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

prediction of uncertain frequency response function bounds using polynomial chaos expansion a probabilistic method is developed to predict the uncertainty bounds on frequency response functions frfs developed from finite element models a non intrusive polynomial chaos expansion pce method is used to predict uncertainty regression models of the various parameters that make up a curvefit of the frf natural frequencies damping ratios complex amplitudes mass and stiffness residuals by making use of an efficient latin hypercube technique these uncertainty models are then combined to efficiently determine pdfs of the parameters and also the uncertainty bounds of the frfs the approach is demonstrated using two examples a simple beam containing uncertainty in young s modulus and a full scale aircraft composite wing model containing uncertainties in both young s modulus and the shear modulus the results were compared with monte carlo simulation mcs and it was found that the parameter pdfs and frf error bounds obtained using a 2nd order pce model agreed very well whilst requiring significantly less computation

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

simulation of stochastic systems via polynomial chaos expansions and convex optimization polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems yet deriving the expansion s coefficients for complex systems might require a significant and nontrivial manipulation of the model or the computation of large numbers of simulation runs rendering the approach too time consuming and impracticable for applications with more than a handful of random variables we introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions the approach exploits a regularization technique with a particular choice of weighting matrices which allows to take into account the specific features of polynomial chaos expansions the method completely based on convex optimization can be applied to problems with a large number of random variables and uses a modest number of monte carlo simulations while avoiding model manipulations additional information on the stochastic process when available can be also incorporated in the approach by means of convex constraints we show the effectiveness of the proposed technique in three applications in diverse fields including the analysis of a nonlinear electric circuit a chaotic model of organizational behavior and finally a chemical oscillator

finite differences on minimal grids conventional approximations to space derivatives by finite differences use orthogonal grids to compute second order space derivatives in a given direction two points are used thus 2n points are required in a space of dimension n however a centered finite difference approximation to a second order derivative may be obtained using only three points in 2 d the vertices of a triangle four points in 3 d the vertices of a tetrahedron and in general n 1 points in a space of dimension n a grid using n 1 points to compute derivatives is called minimal the use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids for instance the well known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids because all the components of a given tensor e g displacement or stress are known at the same points some numerical tests in 2 d show that the propagation of waves is as accurate as when performed with conventional grids although this method may have less intrinsic anisotropies than the conventional method no attempt has yet been made to obtain a quantitative estimation

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed