transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a bayesian approach to multiscale inverse problems with on the fly scale determination abstract a bayesian computational approach is presented to provide a multi resolution estimate of an unknown spatially varying parameter from indirect measurement data in particular we are interested in spatially varying parameters with multiscale characteristics in our work we consider the challenge of not knowing the characteristic length scale s of the unknown a priori and present an algorithm for on the fly scale determination our approach is based on representing the spatial field with a wavelet expansion wavelet basis functions are hierarchically structured localized in both spatial and frequency domains and tend to provide sparse representations in that a large number of wavelet coefficients are approximately zero for these reasons wavelet bases are suitable for representing permeability fields with non trivial correlation structures moreover the intra scale correlations between wavelet coefficients form a quadtree and this structure is exploited to identify additional basis functions to refine the model bayesian inference is performed using a sequential monte carlo smc sampler with a markov chain monte carlo mcmc transition kernel the smc sampler is used to move between posterior densities defined on different scales thereby providing a computationally efficient method for adaptive refinement of the wavelet representation we gain insight from the marginal likelihoods by computing bayes factors for model comparison and model selection the marginal likelihoods provide a termination criterion for our scale determination algorithm the bayesian computational approach is rather general and applicable to several inverse problems concerning the estimation of a spatially varying parameter the approach is demonstrated with permeability estimation for groundwater flow using pressure sensor measurements

relationship between juxtaposed overlapping and fractal representations of multimodal spatial variability 1 it has been shown by neuman 1990 di federico and neuman 1997 and di federico et al 1999 that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or gaussian variograms of overlapping homogeneous random fields modes having mutually uncorrelated increments and variance proportional to a power 2h of the integral spatial correlation scale where h is the hurst coefficient low and high frequency cutoffs are related to length scales of the sampling window domain and data support sample volume respectively intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy created by a hiatus of modes associated with discrete ranges of scales alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed mutually exclusive materials or categories each having its own architecture and attributes the spatial distribution of categories is characterized by indicator random variables and their attributes by random fields this paper focuses on expressions developed by lu and zhang 2002 in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories upon rewriting their expressions for statistically homogeneous and anisotropic media it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space overlap in a way that preserves their local architecture their attributes become mutually uncorrelated categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space the idea is analogous to the well known dual continuum concept in which two categories most commonly fractures and porous blocks are considered to overlap in reality the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space the variogram of an attribute sampled jointly over overlapping statistically homogeneous and anisotropic categories is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories weighted by their volumetric proportions it is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or gaussian variograms such that the product of category probability density function pdf and attribute variance is proportional to a power 2h 1 of the attribute integral scale they represent a fractal field characterized by a power or truncated power variogram as in the work of di federico and neuman any category pdf that satisfies this condition and any attribute pdf having a variance that satisfies the same condition are admissible neither of them need to be gaussian to form fractional brownian motion or levy like to approximate fractional levy motion for the attribute to form a random fractal intermediate cutoffs render the latter model discrete as is that of lu and zhang hence the two multimodal models are mutually consistent and complementary

stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

maps preserving the local spectrum of jordan product of matrices abstract let m n c be the algebra of all n n complex matrices and fix a nonzero vector x 0 c n we show that a map from m n c into itself satisfies t s s t x 0 t s s t x 0 t s m n c if and only if there exists an invertible matrix a m n c such that a x 0 x 0 and t a t a 1 for all t m n c

reconstructing random media ii three dimensional media from two dimensional cuts we report on an investigation concerning the utilization of morphological information obtained from a two dimensional 2d slice thin section of a random medium to reconstruct the full three dimensional 3d medium we apply a procedure that we developed in an earlier paper that incorporates any set of statistical correlation functions to reconstruct a fontainebleau sandstone in three dimensions since we have available the experimentally determined 3d representation of the sandstone we can probe the extent to which intrinsically 3d information such as connectedness is captured in the reconstruction we considered reconstructing the sandstone using the two point probability function and lineal path function as obtained from 2d cuts cross sections of the sample the reconstructions are able to reproduce accurately certain 3d properties of the pore space such as the pore size distribution the mean survival time of a brownian particle and the fluid permeability the degree of connectedness of the pore space also compares remarkably well with the actual sandstone however not surprisingly visualization of the 3d pore structures reveals that the reconstructions are not perfect a more refined reconstruction can be produced by incorporating higher order information at the expense of greater computational cost finally we remark that our reconstruction study sheds lightmore on the nature of information contained in the employed correlation functions thinsp copyright ital 1998 ital the american physical society less

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

identification of large scale spatial trends in hydrologic data it is often useful to distinguish different scales of variability in hydrologic properties such as hydraulic conductivity in the simplest two scale case large scale variations can be viewed as a trend while small scale fluctuations about this trend can be viewed as a random residual this paper describes a method for estimating spatial trends from scattered field measurements the basic concept is to treat both the trend and the residual as stationary random functions these functions are distinguished by their spatial spectral or covariance properties which may be estimated from available data or simply hypothesized we present two versions of a general algorithm for estimating spatial trends 1 a discrete version which is useful in practical applications where data are limited and irregularly spaced and 2 a continuous version which can be used to study the effects of using incorrect spectral parameters applications of the discrete algorithm to both synthetically generated data and field measurements yield satisfactory trend estimates an analysis based on the continuous algorithm shows that the estimation error lower bound for these applications depends on two dimensionless ratios that is the scale disparity ratio of the trend and residual correlation scales and the signal to noise ratio ratio of the trend and residual variances these ratios may be used to evaluate the feasibility of trend estimation before field samples are actually collected

scalable uncertainty estimation for nonlinear inverse problems using parameter reduction constraint mapping and geometric sampling marine controlled source electromagnetic examples we have developed a new uncertainty estimation method that accounts for nonlinearity inherent in most geophysical problems allows for the explicit search of model posterior space is scalable and maintains computational efficiencies on the order of deterministic inverse solutions we accomplish this by combining an efficient parameter reduction technique a parameter constraint mapping routine a sparse geometric sampling scheme and an efficient forward solver in order to reduce our model domain and determine an independent basis we implement both a typical principal component analysis which factorizes the model covariance matrix and an alternative compression method based on singular value decomposition which acts on training models directly and is storage efficient once we have a reduced base we map parameter constraints from our original model domain to this reduced domain to define a bounded geometric region of feasible model space we utilize an optimal scheme to sample this reduced mo

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

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

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

a level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods a method combining features of front tracking methods and fixed domain methods is presented to model dendritic solidification of pure materials to explicitly track the interface growth and shape of the solidifying crystals a front tracking approach based on the level set method is implemented to easily model the heat and momentum transport a fixed domain method is implemented assuming a diffused freezing front where the liquid fraction is defined in terms of the level set function the fixed domain approach by avoiding the explicit application of essential boundary conditions on the freezing front leads to an energy conserving methodology that is not sensitive to the mesh size to compute the freezing front morphology an extended stefan condition is considered applications to several classical stefan problems and two and three dimensional crystal growth of pure materials in an undercooled melt including the effects of melt flow are considered the computed results agree very well with available analytical solutions as well as with results obtained using front tracking techniques and the phase field method

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

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

on joint convexity of trace functions abstract we give the necessary and sufficient condition of the trace function f a b tr a p b q is jointly convex in a b

transient effective hydraulic conductivities under slowly and rapidly varying mean gradients in bounded three dimensional random media we have shown elsewhere tartakovsky and neuman this issue a that in randomly heterogeneous media the ensemble mean transient flux is generally nonlocal in space time and therefore non darcian we have also shown tartakovsky and neuman this issue b that there are special situations in which this flux can be localized so as to render it darcian in real or transformed domains each such situation gives rise to an effective hydraulic conductivity which relates mean gradient to mean flux at any point in real or transformed space time in this paper we develop first order analytical expressions for effective hydraulic conductivity under three dimensional transient flow through a boxshaped domain due to a mean hydraulic gradient that varies slowly in space and time when the mean gradient varies rapidly in time the laplace transform of the mean flux is local but its real time equivalent includes a temporal convolution integral we develop analytical expressions for the real time kernel of this convolution integral the box is embedded within a statistically homogeneous natural log hydraulic conductivity field that is gaussian and exhibits an anisotropic exponential spatial correlation structure by the effective hydraulic conductivity of a finite box in such a field we imply the ensemble mean expected value of all random equivalent conductivities that one could associate with the box under these conditions we explore the influence of domain size time and statistical anisotropy on effective conductivity and include a simple new formula for its variation with statistical anisotropy ratio in an infinite domain under steady state appendixes c e are available with entire article on microfiche order by mail from american geophysical union 2000 florida avenue n w washington dc 20009 or by phone at 800 966 2481 2 50 document w97 002 payment must accompany order

a probabilistic graphical model based stochastic input model construction abstract model reduction techniques have been widely used in modeling of high dimensional stochastic input in uncertainty quantification tasks however the probabilistic modeling of random variables projected into reduced order spaces presents a number of computational challenges due to the curse of dimensionality the underlying dependence relationships between these random variables are difficult to capture in this work a probabilistic graphical model based approach is employed to learn the dependence by running a number of conditional independence tests using observation data thus a probabilistic model of the joint pdf is obtained and the pdf is factorized into a set of conditional distributions based on the dependence structure of the variables the estimation of the joint pdf from data is then transformed to estimating conditional distributions under reduced dimensions to improve the computational efficiency a polynomial chaos expansion is further applied to represent the random field in terms of a set of standard random variables this technique is combined with both linear and nonlinear model reduction methods numerical examples are presented to demonstrate the accuracy and efficiency of the probabilistic graphical model based stochastic input models

preservers of unitary similarity functions on lie products of matrices abstract denote by m n the set of n n complex matrices let f m n 0 be a continuous map such that f u a u f a for any complex unit a m n and unitary u m n f x 0 if and only if x 0 and the induced map t f t x is monotonically increasing on 0 for any rank one nilpotent x m n characterization is given for surjective maps on m n satisfying f a b b a f a b b a the general theorem is then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius

modeling diffusion in random heterogeneous media data driven models stochastic collocation and the variational multiscale method in recent years there has been intense interest in understanding various physical phenomena in random heterogeneous media any accurate description simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties an accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response we propose a general methodology to construct a data driven reduced order model to describe property variations in realistic heterogeneous media this reduced order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media a decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties a sparse grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems a variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems an illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology

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

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

supersensitivity due to uncertain boundary conditions summary we study the viscous burgers equation subject to perturbations on the boundary conditions two kinds of perturbations are considered deterministic and random for deterministic perturbations we show that small perturbations can result in o 1 changes in the location of the transition layer for random perturbations we solve the stochastic burgers equation using different approaches first we employ the jacobi polynomial chaos which is a subset of the generalized polynomial chaos for stochastic modeling converged numerical results are reported up to seven significant digits and we observe similar stochastic supersensitivity for the mean location of the transition layer subsequently we employ up to fourth order perturbation expansions we show that even with small random inputs the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output two types of distributions are considered uniform distribution and a truncated gaussian distribution with no tails various solution statistics including the spatial evolution of probability density function at steady state are studied copyright 2004 john wiley sons ltd

reduced dimensional gaussian process emulators of parametrized partial differential equations based on isomap in this paper isomap and kernel isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a gaussian process emulator of parametrized partial differential equations the output space consists of spatial or spatio temporal fields that are functions of multiple input variables for such problems standard multi output gaussian process emulation strategies are computationally impractical and or make restrictive assumptions regarding the correlation structure the method we develop can be applied without modification to any problem involving vector valued targets and vector valued inputs it also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

geodesic entropic graphs for dimension and entropy estimation in manifold learning in the manifold learning problem one seeks to discover a smooth low dimensional surface i e a manifold embedded in a higher dimensional linear vector space based on a set of measured sample points on the surface in this paper we consider the closely related problem of estimating the manifold s intrinsic dimension and the intrinsic entropy of the sample points specifically we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold we introduce a novel geometric approach based on entropic graph methods although the theory presented applies to this general class of graphs we focus on the geodesic minimal spanning tree gmst to obtaining asymptotically consistent estimates of the manifold dimension and the re spl acute nyi spl alpha entropy of the sample density on the manifold the gmst approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples the gmst method simply constructs a minimal spanning tree mst sequence using a geodesic edge matrix and uses the overall lengths of the msts to simultaneously estimate manifold dimension and entropy we illustrate the gmst approach on standard synthetic manifolds as well as on real data sets consisting of images of faces

kernel principal component analysis for stochastic input model generation stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used these input models are often constructed from a set of experimental samples of the underlying random field to this end the karhunen loeve k l expansion also known as principal component analysis pca is the most popular model reduction method due to its uniform mean square convergence however it only projects the samples onto an optimal linear subspace which results in an unreasonable representation of the original data if they are non linearly related to each other in other words it only preserves the first order mean and second order statistics covariance of a random field which is insufficient for reproducing complex structures this paper applies kernel principal component analysis kpca to construct a reduced order stochastic input model for the material property variation in heterogeneous media kpca can be considered as a nonlinear version of pca through use of kernel functions kpca further enables the preservation of higher order statistics of the random field instead of just two point statistics as in the standard karhunen loeve k l expansion thus this method can model non gaussian non stationary random fields in this work we also propose a new approach to solve the pre image problem involved in kpca in addition polynomial chaos pc expansion is used to represent the random coefficients in kpca which provides a parametric stochastic input model thus realizations which are statistically consistent with the experimental data can be generated in an efficient way we showcase the methodology by constructing a low dimensional stochastic input model to represent channelized permeability in porous media

maps preserving the local spectrum of skew product of operators abstract let h and k be infinite dimensional complex hilbert spaces and let b h be the algebra of all bounded linear operators on h let t h denote the local spectrum of an operator t b h at any vector h h and fix two nonzero vectors h 0 h and k 0 k we show that if a map b h b k has a range containing all operators of rank at most two and satisfies t s h 0 t s k 0 for all t s b h then there exist two unitary operators u and v in b h k such that u h 0 k 0 for some nonzero c and t u t v for all t b h we also described maps b h b k satisfying t s t h 0 t s t k 0 for all t s b h and with the range containing all operators of rank at most four

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 non linear dimension reduction methodology for generating data driven stochastic input models stochastic analysis of random heterogeneous media polycrystalline materials porous media functionally graded materials provides information of significance only if realistic input models of the topology and property variations are used this paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data driven strategy given a set of microstructure realizations input samples generated from given statistical information about the medium topology the framework constructs a reduced order stochastic representation of the thermal diffusivity this problem of constructing a low dimensional stochastic representation of property variations is analogous to the problem of manifold learning and parametric fitting of hyper surfaces encountered in image processing and psychology denote by m the set of microstructures that satisfy the given experimental statistics a non linear dimension reduction strategy is utilized to map m to a low dimensional region a we first show that m is a compact manifold embedded in a high dimensional input space r n an isometric mapping f from m to a low dimensional compact connected set email protected r d email protected n is constructed given only a finite set of samples of the data the methodology uses arguments from graph theory and differential geometry to construct the isometric transformation f m a asymptotic convergence of the representation of m by a is shown this mapping f serves as an accurate low dimensional data driven representation of the property variations the reduced order model of the material topology and thermal diffusivity variations is subsequently used as an input in the solution of stochastic partial differential equations that describe the evolution of dependant variables a sparse grid collocation strategy smolyak algorithm is utilized to solve these stochastic equations efficiently we showcase the methodology by constructing low dimensional input stochastic models to represent thermal diffusivity in two phase microstructures this model is used in analyzing the effect of topological variations of two phase microstructures on the evolution of temperature in heat conduction processes

a fully symmetric nonlinear biorthogonal decomposition theory for random fields abstract we present a general approach for nonlinear biorthogonal decomposition of random fields the mathematical theory is developed based on a fully symmetric operator framework that unifies different types of expansions and allows for a simple formulation of necessary and sufficient conditions for their completeness the key idea of the method relies on an equivalence between nonlinear mappings of hilbert spaces and local inner products i e inner products that may be functionals of the random field being decomposed this extends previous work on the subject and allows for an effective formulation of field dependent and field independent representations the proposed new methodology can be applied in many areas of mathematical physics for stochastic low dimensional modelling of partial differential equations and dimensionality reduction of complex nonlinear phenomena an application to a transient stochastic heat conduction problem in a one dimensional infinite medium is presented and discussed

flow and transport in highly heterogeneous formations 1 conceptual framework and validity of first order approximations 1 flow of uniform mean velocity u takes place in a formation of spatially variable random conductivity k x advective transport of a plume of an inert solute is investigated by the lagrangean approach the aim of the study is to determine the spatial moments of the plume i e of fluid particle trajectories for highly heterogeneous aquifers for which y 1 where y ln k a multi indicator model of the permeability structure which is different from the common multi gaussian one is proposed the formation is modeled as a collection of n blocks of different k j the structure is defined by the distribution of k j the blocks shape and the coordinates of their centroids the following simplifications are adopted the blocks are inclusions of a regular shape circles spheres for isotropic media investigated here defined by the radius a and the inclusions are not overlapping and their centroids are distributed uniformly and independently in space at the continuous limit the model is characterized by the joint pdf f y a the model is shown to be quite general and to comprise binary bimodal indicator variograms and unimodal distributions of y as particular cases the study is focused on the latter case with y normal n y y2 and stationary covariance of given integral scale iy these are the parameters commonly estimated for sedimentary formations this leaves still freedom in selecting the pdf f a the simple model selected for semianalytical and numerical analysis is that of inclusions of radius r and volume fraction n submerged in a matrix of effective conductivity kef the latter represents the effect of inclusions of much smaller radius which appear as a nugget in the log conductivity two point covariance an approximate analytical solution of the flow is obtained by using a self consistent approximation while a fully numerical one is derived in part 3 jankovic et al 2003a transport is solved by particle tracking and the time dependent spatial moments trajectories variance skewness kurtosis are presented in part 2 fiori et al 2003 in the self consistent approximation the asymptotic longitudinal macrodispersivity l which is a function of y shows strong nonlinear effects inclusions of large positive y lead to a finite l whereas l grows unbounded for those of negative y this effect is not captured by the common first order approximation in y which is symmetrical and overestimates l for y 0 and underestimates it for y 0 as a result the second spatial moment is predicted accurately by the first order approximation by cancellation of errors provided that f y is symmetrical however the transient regime and higher order moments are not captured by the first order approximation

lie product and local spectrum preservers abstract let x and y be two infinite dimensional complex banach spaces and fix two nonzero vectors x 0 x and y 0 y let b x resp b y denote the algebra of all bounded linear operators on x resp on y and let e x 0 x be the collection of all operators t b x for which x 0 is an eigenvector and t r 1 x 2 is a nonzero scalar operator for some scalar r c we show that a map from b x onto b y satisfies t s s t y 0 t s s t x 0 t s b x if and only if there are two functions b x c and b x 1 1 and a bijective bounded linear mapping a x y such that a x 0 y 0 the function is constant on b x e x 0 x and t t a t a 1 t 1 y for all t b x

a stochastic mortar mixed finite element method for flow in porous media with multiple rock types this paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types the governing equations are based on darcy s law with nonstationary stochastic permeability represented as a sum of local karhunen loeve expansions the approximation uses stochastic collocation on either a tensor product or a sparse grid coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method the latter method requires solving a coarse scale mortar interface problem via an iterative procedure the traditional implementation requires the solution of local fine scale linear systems on each iteration we employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration in the stochastic setting the basis is further reused over multiple realizations leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude error analysis and numerical experiments are presented

some operator and trace function convexity theorems we consider trace functions a b tr a q 2 b pa q 2 s where a and b are positive n n matrices and ask when these functions are convex or concave we also consider operator convexity concavity of a q 2 b pa q 2 and convexity concavity of the closely related trace functional tr a q 2 b pa q 2 c r the concavity questions are completely resolved thereby settling cases left open by hiai the convexity questions are settled in many cases as a consequence the audenaert datta renyi entropy conjectures are proved for some cases

flow in heterogeneous media displaying a linear trend in the log conductivity the purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationary manifests itself in the form of a spatial trend in the mean log conductivity a stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity for a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector as well as a relation between the expected values of the head and the fluxes these expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean and they are correct to the first order in the variance of the log conductivity the expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space a general methodology is outlined for derivation of these expressions for any type of spatial covariance and for demonstration explicit results are obtained for a gaussian isotropic covariance 16 refs 6 figs

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

inequalities for quantum divergences and the audenaert datta conjecture given two density matrices rho and sigma there are a number of different expressions that reduce to the alpha renyi relative entropy of rho with respect to sigma in the classical case i e when rho and sigma commute only those expressions for which the data processing inequality dpi is valid are of potential interest as quantum divergences in quantum information theory audenaert and datta have made a conjecture on the validity of the dpi for an interesting family of quantum generalizations of the alpha renyi relative entropies the alpha z renyi relative entropies they and others have contributed to the partial solution of this conjecture we review the problem its context and the methods that have been used to obtain the results that are known at present presenting a unified treatment of developments that have unfolded in a number of different papers

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

an indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity solute transport takes place in heterogeneous porous formations with the log conductivity y ln k modeled as a stationary random space function of given univariate normal probability density fu

mid frequency structural dynamics with parameter uncertainty abstract in this paper a number of frequency domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range are integrated into the stochastic finite element method in all cases frequency domain model reduction strategies are used to minimize the computational effort in the mid frequency range firstly an energy operator approach eoa is investigated in this procedure an energy operator adapted to a fixed medium frequency band is defined whose dominant eigensubspace is used to construct a reduced model using a ritz galerkin method secondly the proper orthogonal decomposition method is used to extract the spatial dominant coherent structures in the vibration wave field in the mid frequency band the coherent structures are the eigenvectors corresponding to the dominant eigenvalues of the spatial autocorrelation function of the system response consequently a close relationship between the energy operator approach and the proper orthogonal decomposition method is identified although these two approaches are not mathematically identical the proper orthogonal decomposition method appears to be more straightforward compared to the energy operator method from the viewpoint of numerical implementation thirdly another approach namely the dynamic element method based on frequency dependent finite element shape functions is considered a stochastic reduction method is then utilized to represent the uncertain parameters modelled as stochastic processes in terms of their dominant scales of fluctuation the karhunen loeve and the polynomial chaos decompositions are used to that effect in the context of a stochastic finite element formalism the methodology adopted in the paper thus integrates efficient dynamical reduction techniques with a reduction scheme of stochastic processes for the mid frequency vibration of linear random systems the formulation is exemplified by its application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation the example is also used to highlight some of the relative features of the three dynamic reduction strategies

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

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

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

maps preserving the spectrum of polynomial products of matrices abstract let m p c be the algebra of all p p complex matrices and let t denote the spectrum of any matrix t m p c for an integer n 2 let f 1 n be a 1 linear polynomial with complex coefficients and n non commuting indeterminates 1 n under minor natural conditions on f we show that if a map on m p c satisfies 0 1 f t 1 t n f t 1 t n for all t 1 t n m p c then there exist a nonzero scalar and an invertible matrix a m p c such that has one of the following forms 0 2 t a t a 1 for all t m p c or 0 3 t a t t a 1 for all t m p c here t t denotes the transpose of t in general not all of mappings of the form 0 2 or the form 0 3 satisfy 0 1 when n 2 we describe the set of all scalars for which the mapping 0 2 or 0 3 satisfies 0 1 we also obtain analogue results of spectrum preserving maps on the real linear space h p of all self adjoint matrices in m p c our results extend and unify several results obtained earlier on maps on m p c preserving the spectrum of generalized product or generalized jordan product of matrices

uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data we present a computational approach to quantify predictive uncertainty in complex phenomena which is modeled by partial differential equations with uncertain parameters exhibiting multi scale variability the approach is motivated by flow in random composites whose internal architecture spatial arrangement of constitutive materials and spatial variability of properties of each material are both uncertain the proposed two scale framework combines a random domain decomposition rdd and a probabilistic collocation method pcm on sparse grids to quantify these two sources of uncertainty respectively the use of sparse grid points significantly reduces the overall computational cost especially for random processes with small correlation lengths a series of one two and three dimensional computational examples demonstrate that the combined rdd pcm approach yields efficient robust and non intrusive approximations for the statistics of diffusion in random composites

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

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

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

lagrangian models of reactive transport in heterogeneous porous media with uncertain properties we consider multi component reactive transport in heterogeneous porous media with uncertain hydraulic and chemical properties this parametric uncertainty is quantified by treating relevant flow and transport parameters as random fields which renders the governing equations stochastic we adopt a stochastic lagrangian framework to replace a three dimensional advection reaction transport equation with a one dimensional equation for solute travel times we derive approximate expressions for breakthrough curves and their temporal moments to illustrate our general theory we consider advective transport of dissolved species undergoing an irreversible bimolecular reaction

generalizing lieb s concavity theorem via operator interpolation abstract we introduce the notion of k trace and use interpolation of operators to prove the joint concavity of the function a b tr k b q 2 k a p k b q 2 s 1 k which generalizes lieb s concavity theorem and its following results from trace to a class of homogeneous functions tr k 1 k here tr k a denotes the k th elementary symmetric polynomial of the eigenvalues of a this result gives an alternative proof for the concavity of a tr k exp h log a 1 k that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of 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 treats easily a wide 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 of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

multivariate sensitivity analysis of saturated flow through simulated highly heterogeneous groundwater aquifers a multivariate analysis of variance anova is used to measure the relative sensitivity of groundwater flow to two factors that indicate different dimensions of aquifer heterogeneity an aquifer is modeled as the union of disjoint volumes or blocks composed of different materials with different hydraulic conductivities the factors are correlation between the hydraulic conductivities of the different materials and the contrast between mean conductivities in the different materials the precise values of aquifer properties are usually uncertain because they are only sparsely sampled yet are highly heterogeneous hence the spatial distribution of blocks and the distribution of materials in blocks are uncertain and are modeled as stochastic processes the anova is performed on a large sample of monte carlo simulations of a simple model flow system composed of two materials distributed within three disjoint blocks our key finding is that simulated flow is much more sensitive to the contrast between mean conductivities of the blocks than it is to the intensity of correlation although both factors are statistically significant the methodology of the experiment anova performed on monte carlo simulations of a multi material flow system constitutes the basis of additional studies of more complicated interactions between factors that define flow and transport in aquifers with uncertain properties

monotonicity of a relative renyi entropy we show that a recent definition of relative renyi entropy is monotone under completely positive trace preserving maps this proves a recent conjecture of muller lennert et al on quantum renyi entropies a new definition some properties j math phys 54 122203 2013 e print arxiv 1306 3142v1 see also e print arxiv 1306 3142

from wigner yanase dyson conjecture to carlen frank lieb conjecture abstract in this paper we study the joint convexity concavity of the trace functions p q s a b tr b q 2 k a p k b q 2 s p q s r where a and b are positive definite matrices and k is any fixed invertible matrix we will give full range of p q s r 3 for p q s to be jointly convex concave for all k as a consequence we confirm a conjecture of carlen frank and lieb in particular we confirm a weaker conjecture of audenaert and datta and obtain the full range of z for z renyi relative entropies to be monotone under completely positive trace preserving maps we also give simpler proofs of many known results including the concavity of p 0 1 p for 0 p 1 which was first proved by epstein using complex analysis the key is to reduce the problem to the joint convexity concavity of the trace functions p 1 p 1 a b tr k a p k b 1 p 1 p 1 using a variational method

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

concavity of certain matrix trace and norm functions ii abstract we refine epstein s method to prove joint concavity convexity of matrix trace functions of lieb type tr f a p 1 2 b q a p 1 2 and symmetric anti norm functions of the form f a p b q where and are positive linear maps is an operator mean and f x with a certain power is an operator monotone function on 0 moreover the variational method of carlen frank and lieb is extended to general non decreasing convex concave functions on 0 so that we prove joint concavity convexity of more trace functions of lieb type

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

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 nonlinear manifold based reduced order model for multiscale analysis of heterogeneous hyperelastic materials a new manifold based reduced order model for nonlinear problems in multiscale modeling of heterogeneous hyperelastic materials is presented the model relies on a global geometric framework for nonlinear dimensionality reduction isomap and the macroscopic loading parameters are linked to the reduced space using a neural network the proposed model provides both homogenization and localization of the multiscale solution in the context of computational homogenization to construct the manifold we perform a number of large three dimensional simulations of a statistically representative unit cell using a parallel finite strain finite element solver the manifold based reduced order model is verified using common principles from the machine learning community both homogenization and localization of the multiscale solution are demonstrated on a large three dimensional example and the local microscopic fields as well as the homogenized macroscopic potential are obtained with acceptable engineering accuracy

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

using stochastic analysis to capture unstable equilibrium in natural convection a stabilized stochastic finite element implementation for the natural convection system of equations under boussinesq assumptions with uncertainty in inputs is considered the stabilized formulations are derived using the variational multiscale framework assuming a one step trapezoidal time integration rule the stabilization parameters are shown to be functions of the time step size provision is made for explicit tracking of the subgrid scale solution through time a support space stochastic galerkin approach and the generalized polynomial chaos expansion gpce approach are considered for input output uncertainty representation stochastic versions of standard rayleigh benard convection problems are used to evaluate the approach it is shown that for simulations around critical points the gpce approach fails to capture the highly non linear input uncertainty propagation whereas the support space approach gives fairly accurate results a summary of the results and findings is provided

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach