simplex stochastic collocation method with improved scalability the simplex stochastic collocation ssc method is a robust tool used to propagate uncertain input distributions through a computer code however it becomes prohibitively expensive for problems with dimensions higher than 5 the main purpose of this paper is to identify bottlenecks and to improve upon this bad scalability in order to do so we propose an alternative interpolation stencil technique based upon the set covering problem and we integrate the ssc method in the high dimensional model reduction framework in addition we address the issue of ill conditioned sample matrices and we present an analytical map to facilitate uniformly distributed simplex sampling

onset of cooperative entrainment in limit cycle oscillators with uniform all to all interactions bifurcation of the order function abstract the concept of order function was previously proposed as the key to a general theory of macroscopic mutual or cooperative entrainment in large populations of limit cycle oscillators with weak interactions such that every element is linked to all the other as well as with weakly dispersed intrinsic frequencies that is limit cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies following previous work a bifurcation theory of the order function is developed on the basis of its self consistent functional equation to elucidate in particular generic scaling behavior of such systems at the onset of cooperative entrainment among other results when the onset is not abrupt the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well studied sinusoidal coupling model as well as by typical mean fields models of thermodynamic phase transitions to which coupled oscillator models investigated here are analogous the theory also reveals what happens in nongeneric cases moreover a criterion is found of whether the bifurcation is normal or inverted all these analytical results and predictions are verified not only by numerically solving the equation of the order function but also by numerical simulations although this paper is mainly concerned with the critical behaviors noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average frequency spectra the theory however keeps some room to be further generalized a finding which suggests this is put forth

variety and generality of clustering in globally coupled oscillators abstract clustering behavior in globally coupled identical oscillators is studied by using the phase model any pair of oscillators is assumed to interact through their phase difference without specifying the coupling function the stability of symmetric cluster states in analyzed where symmetric means that the clusters contain the same number of oscillators it is found that higher harmonics in the coupling function are essential to clustering examples show that various cluster states appear depending on the coupling form including those with oscillating intervals anti phase states and chaotic long transient apart from the phase model we also study a general model for globally coupled limit cycle oscillators in the vicinity of the hopf bifurcation and find that such a system does not exhibit clustering within the lowest order approximation

simplex stochastic collocation with eno type stencil selection for robust uncertainty quantification multi element uncertainty quantification approaches can robustly resolve the high sensitivities caused by discontinuities in parametric space by reducing the polynomial degree locally to a piecewise linear approximation it is important to extend the higher degree interpolation in the smooth regions up to a thin layer of linear elements that contain the discontinuity to maintain a highly accurate solution this is achieved here by introducing essentially non oscillatory eno type stencil selection into the simplex stochastic collocation ssc method for each simplex in the discretization of the parametric space the stencil with the highest polynomial degree is selected from the set of candidate stencils to construct the local response surface approximation the application of the resulting ssc eno method to a discontinuous test function shows a sharper resolution of the jumps and a higher order approximation of the percentiles near the singularity ssc eno is also applied to a chemical model problem and a shock tube problem to study the impact of uncertainty both on the formation of discontinuities in time and on the location of discontinuities in space

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

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

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

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

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

reconstruction of two dimensional phase dynamics from experiments on coupled oscillators phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data we use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators based on the phases of the two oscillators independently and the phase difference respectively we discuss the benefits of the two dimensional approach relative to the one dimensional approach using phase difference we quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay we show differences in synchronization predictions of the two models using a toy model we show that the two dimensional approach reveals behavior not detected by the one dimensional model in a driven experimental oscillator this approach is broadly applicable to quantify interactions between nonlinear oscillators especially where intrinsic oscillator sensitivity and coupling evolve with time

simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces stochastic collocation sc methods for uncertainty quantification uq in computational problems are usually limited to hypercube probability spaces due to the structured grid of their quadrature rules nonhypercube probability spaces with an irregular shape of the parameter domain do however occur in practical engineering problems for example production tolerances and other geometrical uncertainties can lead to correlated random inputs on nonhypercube domains in this paper a simplex stochastic collocation ssc method is introduced as a multielement uq method based on simplex elements that can efficiently discretize nonhypercube probability spaces it combines the delaunay triangulation of randomized sampling at adaptive element refinements with polynomial extrapolation to the boundaries of the probability domain the robustness of the extrapolation is quantified by the definition of the essentially extremum diminishing eed robustness principle numerical examples show that the resulting ssc eed method achieves superlinear convergence and a linear increase of the initial number of samples with increasing dimensionality these properties are demonstrated for uniform and nonuniform distributions and correlated and uncorrelated parameters in problems with 15 dimensions and discontinuous responses

refinement criteria for simplex stochastic collocation with local extremum diminishing robustness the simplex stochastic collocation method is developed for adaptive uncertainty quantification in computational problems with random inputs the effectiveness of adaptive formulations is determined to a large extent by both the h and p refinement measures and the stopping criterion an improved stopping criterion and different h refinement measures are derived from an error estimate based on the root mean square of the hierarchical surpluses a p criterion for the polynomial degree p is established that achieves in principle a constant order of convergence with dimensionality the robustness of the adaptive method is guaranteed by the local extremum diminishing concept extended to probability space using the introduced local extremum conserving limiter for p numerical results show both that the stopping criterion is reliable owing to the accurate and conservative error estimate and that the refinement measures outperform uniform refinement

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

homogeneous weak solenoids a generalized weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps if the covering maps are regular then we call the inverse limit space a strong solenoid by a theorem of m c mccord strong solenoids are homogeneous we show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids we also give an example of a weak solenoid that has simply connected path components but which is not homogeneous

uncertainty quantification for systems of conservation laws uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non linear stochastic pdes in this paper we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with polynomial chaos pc methods the idea is to introduce a new variable the entropic variable in bijection with our vector of unknowns which we develop on the polynomial basis by performing a galerkin projection we obtain a deterministic system of conservation laws we state several properties of this deterministic system in the case of a general uncertain system of conservation laws we then apply the method to the case of the inviscid burgers equation with random initial conditions and we present some preliminary results for the euler system we systematically compare results from our new approach to results from the stochastic galerkin method in the vicinity of discontinuities the new method bounds the oscillations due to gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations it is found to be more precise than the stochastic galerkin method for smooth cases but above all for discontinuous cases

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

second order front tracking for the euler equations a second order front tracking method is developed for solving the hyperbolic system of euler equations of inviscid fluid dynamics numerically meshless front tracking methods are usually limited to first order accuracy since they are based on a piecewise constant approximation of the solution here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution the linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately in order to construct a physically correct linearization the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model this front interaction model is also extended to include front numbers used in the wave decomposition it is illustrated numerically for sod s riemann problem the two interacting blast waves problem and a two dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions

numerical analysis of the burgers equation in the presence of uncertainty the burgers equation with uncertain initial and boundary conditions is investigated using a polynomial chaos pc expansion approach where the solution is represented as a truncated series of stochastic orthogonal polynomials the analysis of well posedness for the system resulting after galerkin projection is presented and follows the pattern of the corresponding deterministic burgers equation the numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability similarly to the deterministic case the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix the time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients an estimate of the eigenvalues is provided a characteristic analysis of the truncated pc system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions it is shown that a precise statistical characterization of the input uncertainty is required and partial information e g the expected values and the variance are not sufficient to obtain a solution an analytical solution is derived and the coefficients of the infinite pc expansion are shown to be smooth while the corresponding coefficients of the truncated expansion are discontinuous

subcell resolution in simplex stochastic collocation for spatial discontinuities subcell resolution has been used in the finite volume method fvm to obtain accurate approximations of discontinuities in the physical space stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions however the adaptive refinement in the probability space is ineffective in the non intrusive uncertainty quantification framework if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location the dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics which leads to first order error convergence and an underprediction of the maximum standard deviation to avoid these problems we introduce subcell resolution into the simplex stochastic collocation ssc method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space the presented ssc sr method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location the applications to a linear advection problem the inviscid burgers equation a shock tube problem and the transonic flow over the rae 2822 airfoil show that ssc sr resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples

pade legendre approximants for uncertainty analysis with discontinuous response surfaces a novel uncertainty propagation method for problems characterized by highly non linear or discontinuous system responses is presented the approach is based on a pade legendre pl formalism which does not require modifications to existing computational tools non intrusive approach and it is a global method the paper presents a novel pl method for problems in multiple dimensions which is non trivial in the pade literature in addition a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities the numerical examples include fluid dynamic problems characterized by shock waves a simple dual throat nozzle problem with uncertain initial state and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain results are presented in terms of statistics of both shock position and strength and are compared to monte carlo simulations

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

equicontinuous foliated spaces riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a riemannian manifold their main structural features are well understood since the work of molina in this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms which gives rise to the notion of equicontinuous foliated space we show that equicontinuous foliated spaces have structural properties similar to those known for riemannian foliations the universal covers of their leaves are in the same quasi isometry class leaf closures are homogeneous spaces and the holonomy pseudogroup is indeed given by local isometries

amplitude death through a hopf bifurcation in coupled electrochemical oscillators experiments and simulations amplitude death was observed in experiments with two coupled periodic electrochemical oscillators without time delay simulation results confirmed that the observed amplitude death was obtained via a hopf bifurcation the two oscillators must have a minimum discrepancy and both be near their individual hopf bifurcations for amplitude death to occur phase drift coexisting with unstable asymmetric phase locked solutions amplitude death and in phase synchronization were observed in both the experiments and simulations as coupling strength was increased in addition the simulations showed that a stable asymmetric phase locked solution exists between the phase drift and amplitude death regions

stochastic investigation of flows about airfoils at transonic speeds aerodynamics nonlinearities are reported in the uncertain probabilistic space the efficiency of the present methodology are evaluated for the propagation of random disturbances associated with the angle of attack and the freestreammachnumber anerroranalysisiscarriedoutinordertodetermineappropriatephysicalandstochastic discretization levels different stochastic flow regimes are analyzed in details by means of various postprocessing procedures including error bars probabilistic density function of the aerodynamic field and sobol s coefficients

a domain adaptive stochastic collocation approach for analysis of mems under uncertainties this work proposes a domain adaptive stochastic collocation approach for uncertainty quantification suitable for effective handling of discontinuities or sharp variations in the random domain the basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains within each subdomain a sparse grid interpolant is constructed using the classical smolyak construction s smolyak quadrature and interpolation formulas for tensor products of certain classes of functions soviet math dokl 4 1963 240 243 to approximate the stochastic solution locally the adaptive strategy is governed by the hierarchical surpluses which are computed as part of the interpolation procedure these hierarchical surpluses then serve as an error indicator for each subdomain and lead to subdivision whenever it becomes greater than a threshold value the hierarchical surpluses also provide information about the more important dimensions and accordingly the random elements can be split along those dimensions the proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro electromechanical systems mems specifically we study the effect of uncertain material properties and geometrical parameters on the pull in behavior and actuation properties of a mems switch using the adaptive approach we resolve the pull in instability in mems switches the results from the proposed approach are verified using monte carlo simulations and it is demonstrated that it computes the required statistics effectively

non intrusive iterative stochastic spectral representation with application to compressible gas dynamics in this paper we propose a new iterative formulation improving the convergence of standard non intrusive stochastic spectral method for uncertainty quantification we demonstrate that the method is more accurate than the classical approach with the same level of approximation and at no significant additional computational or memory cost since it is deployed in a post processing stage moreover the accuracy of the representation improves no matter the regularity of the random quantity of interest therefore the method is particularly well suited when nonlinear transformations of random variables are in play and can be viewed as a new way of tackling the gibbs phenomenon we apply the method to several test cases with different levels of regularity dimensionality and complexity including the case of compressible gas dynamics and long time integration problems the new and the classical approaches are compared for the resolution of a stochastic riemann problem governed by an euler system

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws this paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters a finite volume scheme is used for the deterministic discretization while a piecewise polynomial representation is used at the stochastic level the methodology is designed in the context of intrusive galerkin projection methods with a roe type solver the adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain in addition the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time the dynamically evolving stochastic discretization uses a tree structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial

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

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a gpc based approach to uncertain transonic aerodynamics abstract the present paper focus on the stochastic response of a two dimensional transonic airfoil to parametric uncertainties both the freestream mach number and the angle of attack are considered as random parameters and the generalized polynomial chaos gpc theory is coupled with standard deterministic numerical simulations through a spectral collocation projection methodology the results allow for a better understanding of the flow sensitivity to such uncertainties and underline the coupling process between the stochastic parameters two kinds of non linearities are critical with respect to the skin friction uncertainties on one hand the leeward shock movement characteristic of the supercritical profile and on the other hand the boundary layer separation on the aft part of the airfoil downstream the shock the sensitivity analysis thanks to the sobol decomposition shows that a strong non linear coupling exists between the uncertain parameters comparisons with the one dimensional cases demonstrate that the multi dimensional parametric study is required to get the correct shape and magnitude of the standard deviation distributions of the flow quantities such as pressure and skin friction

uncertainty quantification for chaotic computational fluid dynamics we seek error models for simulations that model chaotic flow stable statistics for the solution and for the error are obtained after suitable averaging procedures