stochastic upscaling in solid mechanics an excercise in machine learning this paper presents a consistent theoretical and computational framework for upscaling in random microstructures we adopt an information theoretic approach in order to quantify the informational content of the microstructural details and find ways to condense it while assessing quantitatively the approximation introduced in particular we substitute the high dimensional microscale description by a lower dimensional representation corresponding for example to an equivalent homogeneous medium the probabilistic characteristics of the latter are determined by minimizing the distortion between actual macroscale predictions and the predictions made using the coarse model a machine learning framework is essentially adopted in which a vector quantizer is trained using data generated computationally or collected experimentally several parallels and differences with similar problems in source coding theory are pointed out and an efficient computational tool is employed various applications in linear and non linear problems in solid mechanics are examined

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

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

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

accurate multiscale finite element methods for two phase flow simulations in this paper we propose a modified multiscale finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the global fine scale solution at initial time to determine the boundary conditions of the basis functions this method provides a significant improvement in two phase flow simulations in porous media where the long range effects are important this is typical for some recent benchmark tests such as the spe comparative solution project m christie m blunt tenth spe comparative solution project a comparison of upscaling techniques spe reser eval eng 4 2001 308 317 where porous media have a channelized structure the use of global information allows us to capture the long range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions we present some analysis of the proposed method to illustrate that the method can indeed capture the long range effect in channelized media

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

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

noise sensitivity of band limited signal derivative interpolation the sensitivity of interpolation of the pth derivative of a band limited signal directly from the signal s samples in the presence of additive stationary noise is considered oversampling and filtering generally decrease the interpolation noise level when the data noise is not band limited a lower bound on the interpolation noise level can be approached arbitrarily closely by increasing the sampling rate the lower bound is equivalent to the noise level obtained by low pass filtering and pth order differentiation of the unsampled additive input noise

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

generalized variability response functions for two dimensional elasticity problems the ability to determine probabilistic characteristics of response quantities in structural mechanics e g displacements stresses as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters the concept of the variability response function vrf has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response the key property of the vrf in its classical sense is its independence from the marginal probability distribution function pdf and the spectral density function sdf of the uncertain system parameters it depends only on the deterministic structural configuration and boundary conditions proofs have been provided for the existence of vrfs for linear and some nonlinear statically determinate beams for statically indeterminate structures the monte carlo based generalized variability response function gvrf methodology has been proposed recently as a generalization of the vrf concept to indeterminate linear and some nonlinear beams the methodology computes gvrfs which are analogous to vrfs for statically determinate structures and evaluates their dependence or lack thereof on the pdf and sdf of the random field thereby providing an estimate of the accuracy of the gvrf in this paper the gvrf methodology is extended to problems involving two dimensional linear continua whose stochasticity is characterized by statistically homogeneous random fields after detailing the gvrf methodology for two dimensional random fields two numerical examples are provided gvrfs are computed for the displacement response and for the effective compliance of linear plane stress systems

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

identifiability of the deconvolution problem abstract a deconvolution problem is stated and its identifiability properties are analysed the information assumed available consists of the spectral density of the output measurements and a dynamical description of the system the input to the system and the measurement noise are modelled as independent arma processes in order to obtain an optimal estimate in a mean square sense of the input signal the parameters of the input model and measurement noise are required either to be known a priori or correctly estimated since the normal situation requires estimation of the parameters a successful estimate of the input signal depends crucially on the identifiability properties hence necessary and sufficient conditions for parameter identifiability are derived in terms of model orders with the aid of spectral factorization an example shows that when these conditions are violated only generic results can be obtained a sufficient condition for generic parameter identifiability is given

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

optimal differentiation based on stochastic signal models the problem of estimating the time derivative of a signal from sampled measurements is addressed the measurements may be corrupted by colored noise a key idea is to use stochastic models of the signal to be differentiated and of the measurement noise two approaches are suggested the first is based on a continuous time stochastic process as a model of the signal the second uses a discrete time arma model of the signal and a discrete time approximation of the derivative operator digital differentiators are presented in a shift operator polynomial form they minimize the mean square estimation error and are calculated from a linear polynomial equation and a polynomial spectral factorization the three obstacles to perfect differentiation namely a finite smoothing lag measurement noise and aliasing effects due to sampling are discussed

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

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

numerical material representation using proper orthogonal decomposition and diffuse approximation from numerical point of view analysis and optimization in computational material engineering require efficient approaches for microstructure representation this paper develops an approach to establish an image based interpolation model in order to efficiently parameterize microstructures of a representative volume element rve based on proper orthogonal decomposition pod reduction of density maps snapshots when the parameters of the rve snapshot are known a priori the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters otherwise a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients we showcase the approach and its potential applications by considering a set of two phase composite snapshots the choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization

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

a stochastic ansatz and its relationship with the dichotomic markov process abstract it is shown that a simple stochastic ansatz for the solution of a stochastic differential equation leads to exact results for the dichotomic markov process this has implications for more complex problems where the ansatz leads to a simplified approach to the solution of stochastic equations in engineering and applied physics

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

development of optimal models of porous media by combining static and dynamic data the permeability and porosity distributions we describe a method for the development of the optimal spatial distributions of the porosity phi and permeability k of a large scale porous medium the optimal distributions are constrained by static and dynamic data the static data that we utilize are limited data for phi and k which the method honors in the optimal model and utilizes their correlation functions in the optimization process the dynamic data include the first arrival fa times at a number of receivers of seismic waves that have propagated in the porous medium and the time dependent production rates of a fluid that flows in the medium the method combines the simulated annealing method with a simulator that solves numerically the three dimensional 3d acoustic wave equation and computes the fa times and a second simulator that solves the 3d governing equation for the fluid s pressure as a function of time to our knowledge this is the first time that an optimization method has been developed to determine simultaneously the global minima of two distinct total energy functions as a stringent test of the method s accuracy we solve for flow of two immiscible fluids in the same porous medium without using any data for the two phase flow problem in the optimization process we show that the optimal model in addition to honoring the data also yields accurate spatial distributions of phi and k as well as providing accurate quantitative predictions for the single and two phase flow problems the efficiency of the computations is discussed in detail

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

microstructure reconstruction using entropic descriptors a multi scale approach to the inverse reconstruction of a pattern s microstructure is reported instead of a correlation function a pair of entropic descriptors eds is proposed for a stochastic optimization method the first of them measures a spatial inhomogeneity for a binary pattern or compositional one for a greyscale image the second one quantifies a spatial or compositional statistical complexity the eds reveal structural information that is dissimilar at least in part to that given by correlation functions at almost all discrete length scales the method is tested on a few digitized binary and greyscale images in each of the cases the persuasive reconstruction of the microstructure is found

a hybrid robust non homogeneous finite time differentiator a variant of super twisting differentiator is proposed lyapunov function is designed and an estimate on finite time of derivatives estimation is given the differentiator is equipped with hybrid adaptation algorithm that ensures global differentiation ability independently on amplitude of the differentiated signal and measurement noise

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

numerical homogenization and correctors for nonlinear elliptic equations in this paper we consider numerical homogenization and correctors for nonlinear elliptic equations the numerical correctors are constructed for operators with homogeneous random coefficients the construction employs two scales one a physical scale and the other a numerical scale a numerical homogenization technique is proposed and analyzed this procedure is developed within finite element formulation the convergence ofthe numerical procedure is presented for the case of general heterogeneities using g convergence theory the proposed numerical homogenization procedure for elliptic equations can be considered as a generalization of multiscale finite element methods to nonlinear equations using corrector results we construct an approximation of oscillatory solutions numerical examples are presented

effect of the image resolution on the statistical descriptors of heterogeneous media the characterization and reconstruction of heterogeneous materials such as porous media and electrode materials involve the application of image processing methods to data acquired by scanning electron microscopy or other microscopy techniques among them binarization and decimation are critical in order to compute the correlation functions that characterize the microstructure of the above mentioned materials in this study we present a theoretical analysis of the effects of the image size reduction due to the progressive and sequential decimation of the original image three different decimation procedures random bilinear and bicubic were implemented and their consequences on the discrete correlation functions two point line path and pore size distribution and the coarseness derived from the local volume fraction are reported and analyzed the chosen statistical descriptors correlation functions and coarseness are typically employed to characterize and reconstruct heterogeneous materials a normalization for each of the correlation functions has been performed when the loss of statistical information has not been significant for a decimated image its normalized correlation function is forecast by the trend of the original image reference function in contrast when the decimated image does not hold statistical evidence of the original one the normalized correlation function diverts from the reference function moreover the equally weighted sum of the average of the squared difference between the discrete correlation functions of the decimated images and the reference functions leads to a definition of an overall error during the first stages of the gradual decimation the error remains relatively small and independent of the decimation procedure above a threshold defined by the correlation length of the reference function the error becomes a function of the number of decimation steps at this stage some statistical information is lost and the error becomes dependent on the decimation procedure these results may help us to restrict the amount of information that one can afford to lose during a decimation process in order to reduce the computational and memory cost when one aims to diminish the time consumed by a characterization or reconstruction technique yet maintaining the statistical quality of the digitized sample

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

proper orthogonal decomposition method for multiscale elliptic pdes with random coefficients abstract in this paper we develop an efficient multiscale reduced basis method to solve elliptic pdes with multiscale and random coefficients in a multi query setting our method consists of offline and online stages in the offline stage a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition pod method moreover local tensor spaces are defined to approximate the solution space of the multiscale random pdes in the online stage a weak formulation is derived and discretized using the galerkin method to compute the solution since the multiscale reduced basis functions can efficiently approximate the high dimensional solution space our method is very efficient in solving multiscale elliptic pdes with random coefficients convergence analysis of the proposed method is presented which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the pod method finally numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space

a fuzzy stochastic multiscale model for fiber composites a one dimensional study abstract we study mathematical and computational models for computing the deformation of fiber reinforced polymers due to external forces a thorough study requires an understanding of both micro structural effects and uncertainty variability in the manufacturing process such as uncertainty in the size and distribution of fibers and variability in material properties and fracture parameters we first show that the present uncertainties variabilities which are of both random and non random types cannot be accurately characterized by current stochastic multiscale models based on precise probability theory such as stationary random fields next we present a new hybrid fuzzy stochastic model which can more accurately describe uncertainties variabilities in fiber composites the new model which is referred to as a fuzzy stationary random field consists of a random field with fuzzy moments we then construct a global local multiscale algorithm in a fuzzy stochastic framework for efficiently computing output quantities of interest such as displacements and stresses in regions of relatively small size e g hot spots the algorithm utilizes the concept of representative volume elements and homogenization and constructs a global solution to compute a local approximation that captures the microscale features of the problem the results are based on and backed by real experimental data through a calibration validation approach

a newton raphson method for moving average spectral factorization using the euclid algorithm an implementation of the newton raphson approach to compute the minimum phase moving average spectral factor of a finite positive definite correlation sequence is presented each step in the successive approximation method involves a system of linear equations that is solved using either the levinson algorithm backwards the jury stability test or a symmetrized version of the euclid algorithm various properties of the newton raphson map are studied the algorithm is generalized to other symmetries other than with respect to the unit circle the special case of the symmetry with respect to the imaginary axis is presented and related to the routh hurwitz stability test for continuous time transfer function

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

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

percolation phase diagrams for multi phase models built on the overlapping sphere model abstract the overlapping sphere os percolation model gives a two phase microstructure matrix plus inclusions that is useful for testing composite material ideas and other applications as well as serving as a paradigm of overlapping object percolation and phase transitions real materials often have more than two phases so it is of interest to extend the applicability of the os model a flexible variant of the os model can be constructed by randomly assigning the spheres different phase labels according to a uniform probability distribution as they are inserted one by one into the matrix the resulting three or more phase models can have different amounts of percolating and non percolating phases depending on the volume fraction of each phase and the total os volume fraction a three dimensional digital image approach is used to approximately map out the percolation phase diagram of such models explicitly up to four phases one matrix plus three spherical inclusion phases and implicitly for n 4 phases for the three phase model it was found that a single os sub phase has a percolation threshold that ranges from about a volume fraction of 0 16 when the matrix volume fraction is about 0 01 to about 0 30 at a matrix volume fraction of about 0 7 the approximate analytical dependence of this sub phase percolation threshold on the defining model parameters serves to guide the building of the percolation phase diagram for the n phase model and is used to determine the maximum value of n n 6 at which all n phases can be simultaneously percolated

optimal deconvolution based on polynomial methods the problem of estimating the input to a known linear system is treated in a shift operator polynomial formulation the mean square estimation error is minimized the input and a colored measurement noise are described by independent arma autoregressive moving average processes the filter is calculated by performing a spectral factorization and solving a polynomial equation the approach can be applied to input prediction filtering and smoothing problems as well as to the use of prefilters in the quadratic criterion it applies to nonminimum phase as well as unstable systems as illustrated by two examples

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

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

towards multiscale functions enriching finite element spaces with local but not bubble like functions in this paper we propose a novel way via finite elements to treat problems that can be singular perturbed a reaction diffusion equation in our case we enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem as in the residual free bubble rfb setting but do not require these functions to vanish on each element edge a departure from the rfb paradigm such multiscale functions have an analytic expression for triangles and rectangles bubbles are the choice for the test functions allowing static condensation thus our method is of petrov galerkin type we perform several numerical validations which confirm the good performance of the method 2004 elsevier b v all rights reserved

on precision limits for derivatives numerically calculated from noisy data abstract this paper has three purposes 1 1 to verify an error formula from which the maximal precision in derivatives obtained from noisy measurement data can be calculated the formula is verified by comparison with the resulting noise in derivatives obtained by local least squares polynomial fitting it is also verified that differentiation by fourier series expansion gives noise exceeding the minimum value according to the error formula 2 2 an index called the relative noise amplification rna is introduced for an arbitrary differentiating filter it is defined as the noise transmission of the filter divided by the minimal noise transmission according to the above mentioned error formula when the filter produces unbiased estimates the value of the rna always exceeds one the rna can be used as a quality index for differentiating filters the filter with the smallest value on the rna also has the smallest amount of noise superimposed on the calculated derivatives provided the input noise is white and additive 3 3 the bandwidth and the relative noise amplification of differentiating filters obtained by local least squares polynomial fitting are presented the 0th 1st and 2nd order derivatives have been investigated for polynomials up to 8th order these results can be used for the determination of suitable filter parameters in various applications

a mapping approach to rate distortion computation and analysis in rate distortion theory results are often derived and stated in terms of the optimizing density over the reproduction space in this paper the problem is reformulated in terms of the optimal mapping from the unit interval with lebesgue measure that would induce the desired reproduction probability density this results in optimality conditions that are random relatives of the known lloyd 1982 optimality conditions for deterministic quantizers the validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces we show that for the squared error distortion the optimal reproduction random variable is purely discrete at supercritical distortion where the shannon 1948 lower bound is not tight the gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate to analyze the evolution of the optimal reproduction distribution we use the mapping formulation and establish an analogy to statistical mechanics the solutions are given by the distribution at isothermal statistical equilibrium and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate distortion theory the analysis of an annealing process shows how the number of symbols grows as the system undergoes phase transitions thus an algorithm based on the mapping approach often needs but a few variables to find the exact solution while the blahut 1972 algorithm would only approach it at the limit of infinite resolution finally a quick deterministic annealing algorithm to generate the rate distortion curve is suggested the resulting curve is exact as long as continuous phase transitions in the process are accurately followed

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

analytical results for random line networks applications to fracture networks and disordered fiber composites abstract we study the geometrical properties of a continuum model consisting of an ensemble of intersecting lines placed within a finite region in two dimensions we consider the model in the high density regime where it emulates fiber composites filtering elements and fracture networks we report exact results for the probability distribution of segment lengths produced by line intersections segment loop perimeter and area distributions are estimated numerically for the probability distribution of perimeters we found the interesting phenomenon of small pore perimeter power law repulsion for pores of four or more sides two subsidiary models with infinite line length are solved exactly and reveal the salient features of the finite line model important applications to fiber filter selectivity and optimal intersection trajectories in fracture networks are discussed

digital differentiation of noisy data measured through a dynamic system the design of discrete time differentiating filters in the presence of colored noise and nonneglectable transducer dynamics is investigated the signal and noise are described by autoregressive moving average arma models possibly with poles on mod z mod 1 the mean square error mse optimal filter based on a discrete time approximation of the derivative operator is given by a spectral factorization and a linear polynomial equation

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

a comparison of the squared energy and teager kaiser operators for short term energy estimation in additive noise time frequency distributions that evaluate the signal s energy content both in the time and frequency domains are indispensable signal processing tools especially for nonstationary signals various short time energy computation schemes are used in practice including the mean squared amplitude and teager kaiser energy approaches herein we focus primarily on the short and medium term properties of these two energy estimation schemes as well as on their performance in the presence of additive noise to facilitate this analysis and generalize the approach we use a harmonic noise model to approximate the noise component the error analysis is conducted both in the continuous and discrete time domains deriving similar conclusions the estimation errors are measured in terms of normalized deviations from the expected signal energy and are shown to greatly depend on both the signals spectral content and the analysis window length when medium and long term analysis windows are employed the teager kaiser energy operator is proven superior to the common squared energy operator provided that the spectral content of the noise is more lowpass than the corresponding signal content and vice versa however for shorter window lengths the teager kaiser operator always outperforms the squared energy operator the theoretical results are experimentally verified for synthetic signals finally the performance of the proposed energy operators is evaluated for short term analysis of noisy speech signals and the implications for speech processing applications are outlined

an analysis of polynomial chaos approximations for modeling single fluid phase flow in porous medium systems we examine a variety of polynomial chaos motivated approximations to a stochastic form of a steady state groundwater flow model we consider approaches for truncating the infinite dimensional problem and producing decoupled systems we discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature it would be necessary to find new multivariate cubature rules finally we use the acceleration of monte carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than monte carlo for the relatively small domains considered in this work a curse of dimensionality in the series expansion of the log normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work other than the monte carlo method

a stabilized stochastic finite element second order projection method for modeling natural convection in random porous media we consider natural convection in flow saturated porous media with random porosity the porosity is treated as a random field and a stochastic finite element method is developed the stochastic projection method is considered for the solution of the high dimensional stochastic navier stokes equations since it leads to the uncoupling of the velocity and pressure degrees of freedom because of the porosity dependence of the pressure gradient term in the governing flow equations one cannot use the first order projection method a stabilized stochastic finite element second order projection method is presented based on a pressure gradient projection a two dimensional stochastic problem with moderate and large variation in the random porosity field is examined and the results are compared with monte carlo and sparse grid smolyak collocation approaches excellent agreement between these results indicates the effectiveness and accuracy of the proposed methodology

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

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

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

error bounds in differentiation of noisy signals by high gain observers the error in estimating the derivative s of a noisy signal by using a high gain observer is studied and quantified the error is bounded in terms of the infinity norms of the noise and a derivative of the signal the error bound is independent of the frequency or derivatives of the noise guidelines are presented for the observer gain design when it is used for on line differentiation analytical and simulation results are presented

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

modeling heterogeneous materials via two point correlation functions ii algorithmic details and applications in the first part of this series of two papers we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials media via two point correlation functions s2 and introduced an efficient heterogeneous medium reconstruction algorithm called the lattice point algorithm here we discuss the algorithmic details of the lattice point procedure and an algorithm modification using surface optimization to further speed up the reconstruction process the importance of the error tolerance which indicates to what accuracy the media are reconstructed is also emphasized and discussed we apply the algorithm to generate three dimensional digitized realizations of a fontainebleau sandstone and a boron carbide aluminum composite from the two dimensional tomographic images of their slices through the materials to ascertain whether the information contained in s2 is sufficient to capture the salient structural features we compute the two point cluster functions of the media which are superior signatures of the microstructure because they incorporate topological connectedness information we also study the reconstruction of a binary laser speckle pattern in two dimensions in which the algorithm fails to reproduce the pattern accurately we conclude that in general reconstructions using s2 only work well for heterogeneous materials with single scale structures however two point information via s2 is not sufficient to accurately model multiscale random media moreover we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two point correlation functions

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

enforcing positivity in intrusive pc uq methods for reactive ode systems we explore the relation between the development of a non negligible probability of negative states and the instability of numerical integration of the intrusive galerkin ordinary differential equation system describing uncertain chemical ignition to prevent this instability without resorting to either multi element local polynomial chaos pc methods or increasing the order of the pc representation in time we propose a procedure aimed at modifying the amplitude of the pc modes to bring the probability of negative state values below a user defined threshold this modification can be effectively described as a filtering procedure of the spectral pc coefficients which is applied on the fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold we demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor this is carried out by comparing different observables and error measures as obtained by non intrusive monte carlo and gauss quadrature integration and the filtered intrusive procedure the filtering procedure has been shown to effectively stabilize divergent intrusive solutions and also to improve the accuracy of stable intrusive solutions which are close to the stability limits

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

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

prediction of permeability for porous media reconstructed using multiple point statistics to predict multiphase flow through geologically realistic porous media it is necessary to have a three dimensional 3d representation of the pore space we use multiple point statistics based on two dimensional 2d thin sections as training images to generate geologically realistic 3d pore space representations thin section images can provide multiple point statistics which describe the statistical relation between multiple spatial locations and use the probability of occurrence of particular patterns assuming that the medium is isotropic a 3d image can be generated that preserves typical patterns of the void space seen in the thin sections the method is tested on berea sandstone for which a 3d image from micro ct computerized tomography scanning is available and shows that the use of multiple point statistics allows the long range connectivity of the structure to be preserved in contrast to two point statistics methods that tend to underestimate the connectivity furthermore a high resolution 2d thin section image of a carbonate reservoir rock is used to reconstruct 3d structures by the proposed method the permeabilities of the statistical images are computed using the lattice boltzmann method lbm the results are similar to the measured values to the permeability directly computed on the micro ct image for berea and to predictions using analysis of the 2d images and the effective medium approximation

illumination a new method for studying 3d percolation fronts in a concentration gradient a new method of studying percolation fronts is introduced in situations where the concentration is not constant gradient percolation approach it consists in an efficient way of determining subsets of the percolation cluster which are exactly at the percolation threshold the method is applied to both lattice and continuum percolation in order to determine the percolation threshold in 3d more accurately two variations of this method illumination and projection seem to give a similar scaling behaviour at low gradients for the cubic lattice case leading to a new estimate for the percolation threshold pc 0 31173 or 0 00007 for continuum percolation of overlapping spheres the new estimate for the critical volume fraction is 0 291 or 0 002 and the value of the critical exponent alpha sigma for the width of the front scaled with the concentration gradient is very close to the one for the lattice case

computational experience with the spectral smoothing method for differentiating noisy data abstract when applied to non exact noisy data numerical methods for calculating derivatives in particular derivatives of order higher than the first based on model functions fitted to exact data become unsatisfactory the spectral smoothing method of anderssen and bloomfield developed to solve this problem entails calculation of a smoothing parameter and the choice of an optimal order sobolev norm that is used as regularizer this method is used to differentiate smooth and integrate noisy data a likelihood function is minimized to determine the smoothing parameter we present numerical results suggesting that this function can be jointly minimized with respect to the smoothing parameter and the order of the regularizing norm thus yielding a fully automatic numerical differentiation procedure

the use of optimally regularized fourier series for estimating higher order derivatives of noisy biomechanical data abstract optimally regularized filtered fourier series can be used most effectively for estimating higher order derivatives of noisy data sequences such as occur in biomechanical investigations as the optimal filtering process takes place in the derivative domain use being made of the statistical information contained in the periodogram of the data sequence no numerical differentiation as such takes place and all the problems known to be associated with such a procedure are avoided the optimally filtered approximation to the function itself is obtained by the same technique the algorithm is based on known results from time series and approximation theory and offers several advantages when compared with other techniques the optimally filtered approximations of the function and all its required derivatives are obtained in a single run compact analytical fourier representations of the filtered function and the derivatives are available allowing interpolation between data points and further analytical treatment of the function all the desirable properties of fourier approximations such as periodicity end point continuity etc can be utilized if fast fourier transforms are used the algorithm is computationally far more efficient than other techniques

polynomial approach to wiener filtering a rigorous derivation is given of the polynomial method for finding wiener filters in both discrete time and continuous time in each case the method involves one spectral factorization and the solution of one diophantine equation the theory is applicable in multivariable situations but simple scalar examples are included here to compare the polynomial approach with the basic wiener method

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

fractal heterogeneous media a method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm the obtained structures are invasive stochastic fractals with the hurst exponent varying as a continuous parameter as opposed to lacunar deterministic fractals such as the menger sponge by employing the detrending moving average algorithm a carbone phys rev e 76 056703 2007 the hurst exponent of the generated structure can be subsequently checked the fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself consequently in this framework the hurst exponent should be intended as an estimator of compactness rather than of roughness applications can be envisaged for simulating and quantifying complex systems characterized by self similar heterogeneity across space for example exploitation areas range from the design and control of multifunctional self assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology environmental sciences geomorphological sciences etc

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 multi resolution non parametric bayesian framework for identification of spatially varying model parameters this paper proposes a hierarchical multi resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output such parameters are frequently encountered in pde based models and correspond to quantities such as density or pressure fields elasto plastic moduli and internal variables in solid mechanics conductivity fields in heat diffusion problems permeability fields in fluid flow through porous media etc the proposed model has all the advantages of traditional bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty in contrast to existing approaches it utilizes a parsimonious non parametric formulation that favors sparse representations and whose complexity can be determined from the data the proposed framework in non intrusive and makes use of a sequence of forward solvers operating at various resolutions as a result inexpensive coarse solvers are used to identify the most salient features of the unknown field s which are subsequently enriched by invoking solvers operating at finer resolutions this leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy it is based on a novel adaptive scheme based on sequential monte carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in markov chain monte carlo schemes the capabilities of the proposed methodology are illustrated in problems from nonlinear solid mechanics with special attention to cases where the data is contaminated with random noise and the scale of variability of the unknown field is smaller than the scale of the grid where observations are collected

polynomial systems approach to optimal linear filtering and prediction the solution of the optimal linear estimation problem is considered using a polynomial matrix description for the discrete system the filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and new insights into the disturbance rejection properties of the filter are obtained the plant or signal model can be stable or unstable and allowance is made for both control and disturbance input subsystems and white and coloured measurement noise or an output disturbance subsystem the model structure was determined by the needs of several industrial filtering problems the polynomial form of filter may easily be included in a self tuning algorithm and a simple adaptive estimator is described

geometrical ambiguity of pair statistics ii heterogeneous media in the first part of this series of two papers y jiao f h stillinger and s torquato phys rev e 81 011105 2010 we considered the geometrical ambiguity of pair statistics associated with point configurations here we focus on the analogous problem for heterogeneous media materials heterogeneous media are ubiquitous in a host of contexts including composites and granular media biological tissues ecological patterns and astrophysical structures the complex structures of heterogeneous media are usually characterized via statistical descriptors such as the n point correlation function s n an intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two point correlation function s 2 of a target medium recently general claims of the uniqueness of reconstructions using s 2 have been made based on numerical studies which implies that s 2 suffices to uniquely determine the structure of a medium within certain numerical accuracy in this paper we provide a systematic approach to characterize the geometrical ambiguity of s 2 for both continuous two phase heterogeneous media and their digitized representations in a mathematically precise way in particular we derive the exact conditions for the case where two distinct media possess identical s 2 i e they form a degenerate pair the degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations respectively by examining these equations and constructing their rigorous solutions for specific examples we conclusively show that in general s 2 is indeed not sufficient information to uniquely determine the structure of the medium which is consistent with the results of our recent study on heterogeneous media reconstruction y jiao f h stillinger and s torquato proc natl acad sci u s a 106 17634 2009 the analytical examples include complex patterns composed of building blocks bearing the letter t and the word water as well as degenerate stacking variants of the densest sphere packing in three dimensions barlow films several numerical examples of degeneracy e g reconstructions of polycrystal microstructures laser speckle patterns and sphere packings are also given which are virtually exact solutions of the degeneracy equations the uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed including two point quantities that contain topological connectedness information about the phases

differentiation of discrete multidimensional signals we describe the design of finite size linear phase separable kernels for differentiation of discrete multidimensional signals the problem is formulated as an optimization of the rotation invariance of the gradient operator which results in a simultaneous constraint on a set of one dimensional low pass prefilter and differentiator filters up to the desired order we also develop extensions of this formulation to both higher dimensions and higher order directional derivatives we develop a numerical procedure for optimizing the constraint and demonstrate its use in constructing a set of example filters the resulting filters are significantly more accurate than those commonly used in the image and multidimensional signal processing literature

accurate uncertainty quantification using inaccurate computational models this paper proposes a novel uncertainty quantification framework for computationally demanding systems characterized by a large vector of non gaussian uncertainties it combines state of the art techniques in advanced monte carlo sampling with bayesian formulations the key departure from existing works is the use of inexpensive approximate computational models in a rigorous manner such models can readily be derived by coarsening the discretization size in the solution of the governing pdes increasing the time step when integration of odes is performed using fewer iterations if a nonlinear solver is employed or making use of lower order models it is shown that even in cases where the inaccurate models provide very poor approximations of the high fidelity response statistics of the latter can be quantified accurately with significant reductions in the computational effort multiple approximate models can be used and rigorous confidence bounds of the estimates produced are provided at all stages