domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

estimates for riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds let 2 and be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact n dimensional manifold m the riesz means of order of an integrable function on m are defined by r x m 1 2 2 x y y d y in this paper we study the kernels and the operator norms of he operators r on lp m 1 p 2 n 1 n 3 we also prove that if 1 p 2 n 1 n 3 and np n 1 2 then the operators r are of weak type p p uniformly with respect to

divergence of eigenfunction expansions we shall use a result of rellich on the convergence of spectral projections to analyze mean and almost everywhere convergence of eigenfunction expansions the basic result is a transplantation theorem which reduces these questions on manifolds to questions on the boundedness of fourier multiplier operators on the cotangent space results such as these were long known in the context of fourier series where they were introduced by stein and weiss 121 various authors gave generalizations but the most far reaching was obtained by mitjagin 8 who applied his results to elliptic problems in this paper we shall give several extensions of mitjagin s results in the main part of the paper we shall employ these extensions and the results of kenig and tomas s 61 to analyze certain nonelliptic problems this paper is divided into four sections in the first we shall give mitjagin s original proof that one may transplant problems on the convergence of eigenfunction expansions in lp mean to fourier multiplier problems on i we shall also extend this to convergence of other summability methods and also to joint eigenexpansions of commuting selfadjoint operators

coupling stokes darcy flow with transport a mathematical and numerical model describing chemical transport in a stokes darcy flow system is discussed the flow equations are solved through domain decomposition using classical finite element methods in the stokes region and mixed finite element methods in the darcy region the local discontinuous galerkin ldg method is used to solve the transport equation models dealing with coupling between stokes and darcy equations have been extensively discussed in the literature this paper focuses on the approximation of the transport equation stability of the ldg scheme is analyzed and an a priori error estimate is proved several numerical examples verifying the theory and illustrating the capabilities of the method are presented

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

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

hybrid uncertainty analysis of functionally graded plates via multiple imprecise random field modelling of uncertain material properties abstract this paper presents the mixed probabilistic non probabilistic uncertainty problem regarding the hybrid uncertainty propagation on physical model of functionally graded plates through dissimilarly uncertain characterisation of material properties in the presence of the multiple imprecise random field modelling the spectral stochastic finite element framework is extended to perform mixed uncertainty analysis of functionally graded material whereby the spatial variation of elastic modulus of the plate is uniquely characterised as the multiple imprecise random field modelling more than a purely homogeneous random field this characterisation has originated in heeding the construction of physical models and the significant influences of separately uncertain characteristics propagated on the physical model through different material ingredients are examined to make physical models more reliable and realistic mechanics and mathematics intrusive formulations are then established to explicitly produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation bounded surfaces of statistical moments and bounding distribution functions are vividly depicted for demonstration the feasibility and effectiveness of the presented work are illustrated via test cases of numerical examples

on the asymptotics of the bochner riesz kernel on compact lie groups suppose g is an n dimensional compact connected lie group and for 0 let dr r 0 be the bochner riesz kernel on g we prove that the lebesgue constants dr 1 are of the same order as r n 1 2 as r tends to infinity 0 n 1 2 while dr 1 log r if n 1 2 as a convergence we find best conditions for the norm convergence of the bochner riesz means on the spaces l1 g and c g

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

multigrid on the interface for mortar mixed finite element methods for elliptic problems we consider mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks we assume that the subdomain grids are locally defined and need not match across the block boundaries specially chosen mortar finite element spaces are introduced on the interfaces for approximating the scalar variable pressure the mortars also serve as lagrange multipliers for imposing flux matching conditions the method is implemented by reducing the algebraic system to a positive definite interface problem in the mortar spaces this problem is then solved using a multigrid on the interface with conjugate gradient smoothing the algorithm is very efficient in a distributed parallel computing environment as only subdomain solves are required on each conjugate gradient iteration the standard variational assumptions for the multigrid are not satisfied since the interface bilinear forms vary from level to level we present theoretical results for the convergence of the v cycle and the w cycle computational results in two and three dimensions are given to illustrate and confirm the theory

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

lyapunov exponents and the natural invariant density determination of chaotic maps an iterative maximum entropy ansatz we apply the maximum entropy principle to construct the natural invariant density and the lyapunov exponent of one dimensional chaotic maps using a novel function reconstruction technique that is based on the solution of the hausdorff moment problem via maximizing shannon entropy we estimate the invariant density and the lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments the accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available furthermore we also consider a very complex example for which no exact analytical result for the invariant density is available a comparison of our results to those available in the literature is also discussed

a fast monte carlo method with a reduced basis of control variates applied to uncertainty propagation and bayesian estimation abstract the reduced basis control variate monte carlo method was introduced recently in s boyaval t lelievre a variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm commun math sci 8 2010 735 762 special issue mathematical issues on complex fluids as an improved monte carlo method for the fast estimation of many parametrized expected values at many parameter values we provide here a more complete analysis of the method including precise error estimates and convergence results we also numerically demonstrate that it can be useful to some parametric frameworks in uncertainty quantification in particular i the case where the parametrized expectation is a scalar output of the solution to a partial differential equation pde with stochastic coefficients an uncertainty propagation problem and ii the case where the parametrized expectation is the bayesian estimator of a scalar output in a similar pde context moreover in each case a pde has to be solved many times for many values of its coefficients this is costly and we also use a reduced basis of pde solutions like in s boyaval c le bris y maday n nguyen a patera a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient comput methods appl mech eng 198 2009 3187 3206 to our knowledge this is the first combination of various reduced basis ideas with a view to reducing as much as possible the computational cost of a simple versatile monte carlo approach to uncertainty quantification

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

probabilistic nonconvex constrained optimization with fixed number of function evaluations summary a methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations a diffusion manifold over the initial set of data points is first identified and an associated basis computed the joint probability density function of this initial set is estimated using a kernel density model and an ito stochastic differential equation constructed with this model as its invariant measure this isde is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters design variables and function values are obtained as solutions of the isde the expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations the methodology brings together novel ideas from manifold learning and stochastic hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization three examples are presented to highlight different aspects of the proposed methodology this article is protected by copyright all rights reserved

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

nonparametric estimation in random coefficients binary choice models this paper considers random coefficients binary choice models the main goal is to estimate the density of the random coefficients nonparametrically this is an ill posed inverse problem characterized by an integral transform a new density estimator for the random coefficients is developed utilizing fourier laplace series on spheres this approach offers a clear insight on the identification problem more importantly it leads to a closed form estimator formula that yields a simple plug in procedure requiring no numerical optimization the new estimator therefore is easy to implement in empirical applications while being flexible about the treatment of unobserved heterogeneity extensions including treatments of nonrandom coefficients and models with endogeneity are discussed

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

polynomial chaos representation of databases on manifolds characterizing the polynomial chaos expansion pce of a vector valued random variable with probability distribution concentrated on a manifold is a relevant problem in data driven settings the probability distribution of such random vectors is multimodal in general leading to potentially very slow convergence of the pce in this paper we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold the proposed methodology constructs a pce of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold the method is robust and remains efficient for high dimension and large datasets the resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data driven queries

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

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

high order entropy based closures for linear transport in slab geometry ii a computational study of the optimization problem we present a numerical algorithm to implement entropy based m n moment models in the context of a simple linear kinetic equation for particles moving through a material slab the closure for these models as is the case for all entropy based models is derived through the solution of a constrained convex optimization problem the algorithm has two components the first component is a discretization of the moment equations which preserves the set of realizable moments thereby ensuring that the optimization problem has a solution in exact arithmetic the discretization is a second order kinetic scheme which uses muscl type limiting in space and a strong stability preserving runge kutta time integrator the second component of the algorithm is a newton based solver for the dual optimization problem which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives the accuracy of the numerical solution to the dual problem plays a key role in the time step restric

function reconstruction as a classical moment problem a maximum entropy approach we present a systematic study of the reconstruction of non negative functions via maximum entropy approach utilizing the information contained in a finite number of moments of the functions for testing the efficacy of the approach we reconstruct a set of functions using an iterative entropy optimization scheme and study the convergence profile as the number of moments is increased a wide variety of functions are considered that include a distribution with a sharp discontinuity an oscillatory function a distribution with singularities and finally a distribution with several spikes and fine structure the last example is important in the context of the determination of the natural density of the logistic map the convergence of the method is studied by comparing the moments of the approximated functions with the exact ones furthermore by varying the number of moments and iterations we examine to what extent the features of the functions such as the divergence behavior at singular points within the interval is reproduced the proximity of the reconstructed maximum entropy solution to the exact solution is examined via kullback leibler divergence and variation measures for different number of moments

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

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

maximum entropy in the hamburger moments problem the maximum entropy approach to the solution of the hamburger inverse problem of moments in which one seeks to recover a positive density function p x where x from the values of a finite n 1 of its moments is considered the obtained results show that unexpected upper bounds for the moments do not exist in the general hamburger finite moment problem unlike in the symmetric case previously considered some physical examples illustrating the use of partial information to determine the approximate function are presented

sharp estimates for lebesgue constants on compact lie groups suppose g is a n dimensional compact connected semisimple lie group and dr is the spherical dirichlet kernel on g we prove the existence of a positive constant k such that dr 1 kr n 1 2 this complements the known result dr 1 hr n 1 2 we also prove that for a polyhedral dirichlet kernel dn the above inequalities hold with np in place of r n 1 2 p is the number of positive roots of g

the influence of uncertain local subsoil conditions on the response of buildings to ground vibration abstract this paper examines the influence of imperfect knowledge of the local subsoil conditions on the prediction of building response to ground borne vibration the focus is on problems of environmental ground vibration in the wide frequency range between 1 hz and 80 hz a probabilistic finite element perfectly matched layers model is developed for the analysis of the dynamic soil structure interaction problem where the shear modulus of the soil is modeled as a conditional random field a subdomain formulation is employed to impose loading by an incident wave field in the model the uncertainty on the subsoil properties is propagated to the response of a building by means of monte carlo simulation a case study is considered to investigate the influence of the spatial correlation length of the random field representing the shear modulus of the subsoil and the foundation type of the building the structural response uncertainty varies over frequency bands but as a general trend increases with frequency the foundation type of the building is a crucial parameter determining the structural response and the associated uncertainty bounds

ito sde mcmc method for bayesian characterization of errors associated with data limitations in stochastic expansion methods for uncertainty quantification abstract this paper is concerned with the characterization and the propagation of errors associated with data limitations in polynomial chaos based stochastic methods for uncertainty quantification such an issue can arise in uncertainty quantification when only a limited amount of data is available when the available information does not suffice to accurately determine the probability distributions that must be assigned to the uncertain variables the bayesian method for assigning these probability distributions becomes attractive because it allows the stochastic model to account explicitly for insufficiency of the available information in previous work such applications of the bayesian method had already been implemented by using the metropolis hastings and gibbs markov chain monte carlo mcmc methods in this paper we present an alternative implementation which uses an alternative mcmc method built around an ito stochastic differential equation sde that is ergodic for the bayesian posterior we draw together from the mathematics literature a number of formal properties of this ito sde that lend support to its use in the implementation of the bayesian method and we describe its discretization including the choice of the free parameters by using the implicit euler method we demonstrate the proposed methodology on a problem of uncertainty quantification in a complex nonlinear engineering application relevant to metal forming

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

new developments of the extended quadrature method of moments to solve population balance equations abstract population balance models have a wide range of applications in many industrial fields as they allow accounting for heterogeneity among properties which are crucial for some system modelling they actually describe the evolution of a number density function ndf using a population balance equation pbe for instance they are applied to gas liquid columns or stirred reactors aerosol technology crystallisation processes fine particles or biological systems there is a significant interest for fast stable and accurate numerical methods in order to solve for pbes a class of such methods actually does not solve directly the ndf but resolves their moments these methods of moments and in particular quadrature based methods of moments have been successfully applied to a variety of systems point wise values of the ndf are sometimes required but are not directly accessible from the moments to address these issues the extended quadrature method of moments eqmom has been developed in the past few years and approximates the ndf from its moments as a convex mixture of kernel density functions kdfs of the same parametric family in the present work eqmom is further developed on two aspects the main one is a significant improvement of the core iterative procedure of that method the corresponding reduction of its computational cost is estimated to range from 60 up to 95 the second aspect is an extension of eqmom to two new kdfs used for the approximation the weibull and the laplace kernels all matlab source codes used for this article are provided with this article

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

fourth order pseudo maximum likelihood methods we extend pml theory to account for information on the conditional moments up to order four but without assuming a parametric model to avoid a risk of misspecification of the conditional distribution the key statistical tool is the quartic exponential family which allows us to generalize the pml2 and qgpml1 methods proposed in gourieroux et al 1984 to pml4 and qgpml2 methods respectively an asymptotic theory is developed the key numerical tool that we use is the gauss freud integration scheme that solves a computational problem that has previously been raised in several fields simulation exercises demonstrate the feasibility and robustness of the methods

identification of bayesian posteriors for coefficients of chaos expansions this article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data the data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized the random variables and fields are approximated by a polynomial chaos expansion and the coefficients of this expansion are viewed as unknown parameters to be identified it is shown how the bayesian paradigm can be applied to formulate and solve the inverse problem the estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the bayesian posterior this allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients as well as on subsequent predictions an illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

multiscale mortar expanded mixed discretization of nonlinear elliptic problems abstract we consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method this is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non overlapping subdomains an unknown lagrange multiplier is introduced on the interfaces which serve as pressure dirichlet boundary condition for local subdomain problems a finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface we demonstrated solvability of discretization and established a priori l2 error estimates for both vector and scalar approximations we proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation the uniqueness of discrete problem is shown for sufficiently small values of mesh size an error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient we also present the analysis of the linear second order elliptic problem and prove the similar results the computational experiments are presented to validate theory

hamburger moment problem and maximum entropy on the existence conditions the existence of maximum entropy solution for the reduced hamburger moment problem is reconsidered existence conditions previously appeared in literature are revisited allowing an easy way to identifying the existence of maximum entropy solution the obtained results suggest that except for special sequences of moments unknown a priori the maximum entropy solution for the non symmetric reduced hamburger moment problem exists for practical purposes the replacing of the support r with a large enough finite interval finds a theoretical warranty the symmetric case may be formulated as follows once assigned the first 2m moments if maxent density does not exist conclusion drawn uniquely from numerical evidence maxent density with the first 2m 2 moments exists in such a case even if the first 2m moments are known we have to settle for a density which carries less information theoretical results are illustrated through some numerical examples

stochastic reduced order models for uncertain geometrically nonlinear dynamical systems a general methodology is presented for the consideration of both parameter and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems the approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model the mean model uncertainty both from parameters and model is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model i e while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model this randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion affects all coefficients of the reduced order model and is computationally easily achieved the reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented these applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems

generalized stochastic approach for constitutive equation in linear elasticity a random matrix model this work is concerned with the construction of stochastic models for random elasticity matrices allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely integrating the statistical dependence between the random stiffness components or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations to this aim we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components having recourse to the maxent principle strategies for random generation and estimation were further reviewed and the approach was exemplified in the case of a material that was transversely isotropic almost surely in a second stage we made use of such derivations to propose a generalized model for random elasticity matrices that took into accountconstraints on both the level of stochastic anisotropy and the level of statistical fluctuations

generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions a new generalized probabilistic approach of uncertainties is proposed for computational model in structural linear dynamics and can be extended without difficulty to computational linear vibroacoustics and computational non linear structural dynamics this method allows the prior probability model of each type of uncertainties model parameter uncertainties and modeling errors to be separately constructed and identified the modeling errors are not taken into account with the usual output prediction error method but with the nonparametric probabilistic approach of modeling errors recently introduced and based on the use of the random matrix theory the theory an identification procedure and a numerical validation are presented then a chaos decomposition with random coefficients is proposed to represent the prior probabilistic model of random responses the random germ is related to the prior probability model of model parameter uncertainties the random coefficients arc related to the prior probability model of modeling errors and then depends on the random matrices introduced by the nonparametric probabilistic approach of modeling errors a validation is presented finally a future perspective is introduced when experimental data are available the prior probability model of the random coefficients can be improved in constructing a posterior probability model using the bayesian approach

time domain formulation in computational dynamics for linear viscoelastic media with model uncertainties and stochastic excitation the paper is devoted to the computational time domain formulation of linear viscoelastic systems submitted to a nonstationary stochastic excitation and in the presence of model uncertainties which are modeled in the framework of the probability theory the objective is to introduce and to develop an adapted and complete formulation of such a problem in the context of computational mechanics a reduced order model in the time domain with stochastic excitation is derived from the computational model for the reduced order model the stochastic modeling of both computational model parameter uncertainties and modeling errors is carried out using the nonparametric probabilistic approach and the random matrix theory we present a new formulation of model uncertainties to construct the random operators for viscoelastic media we then obtained a linear stochastic integro differential equation side with random operators and with a stochastic nonhomogeneous part stochastic excitation a time discretization of this side is proposed in a first step the side is transformed to a linear ito stochastic differential equation isde with random operators then the isde is discretized using an extension of the stormer verlet scheme which is a particularly well adapted algorithm for long time good behavior of the numerical solution finally for the stochastic solver and statistical estimations of the random responses we propose to use the monte carlo simulation for gaussian and non gaussian excitations

norms of characters and fourier series on compact lie groups abstract we prove new estimates for the p norms of irreducible characters of compact lie groups these estimates are applied to give negative results on p mean convergence of fourier series on compact lie groups

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

on the application of maximum entropy to the moments problem the use of the maximum entropy condition to obtain a solution to the moments problem in which one seeks to reconstruct a function from the values of a finite set of its moments is of interest in a variety of physical applications unfortunately the standard analysis which is available and on which these applications are based takes as its starting point precise values of the moments whereas the physical applications involve imprecise data in this paper it will be shown how to solve the moments problem using the maximum entropy condition starting from imprecise data values with errors specified by a 2 function

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

parametrization of random vectors in polynomial chaos expansions via optimal transportation polynomial chaos pc expansions are used for the propagation of uncertainty through dynamical systems as an alternative to monte carlo methods model parameters in a given dynamical system are assumed to have known expansions which correspond to simple standard distributions and one is usually interested in the polynomial expansion of the system solution we are concerned with the problem of estimating the pc expansion of a parameter vector when only realizations from its distribution are given to this end we apply ideas from optimal transportation theory and network optimization

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

maximum entropy autoregressive conditional heteroskedasticity model abstract in many applications it has been found that the autoregressive conditional heteroskedasticity arch model under the conditional normal or student s t distributions are not general enough to account for the excess kurtosis in the data moreover asymmetry in the financial data is rarely modeled in a systematic way in this paper we suggest a general density function based on the maximum entropy me approach that takes account of asymmetry excess kurtosis and also of high peakedness the me principle is based on the efficient use of available information and as is well known many of the standard family of distributions can be derived from the me approach we demonstrate how we can extract information functional from the data in the form of moment functions we also propose a test procedure for selecting appropriate moment functions our procedure is illustrated with an application to the nyse stock returns the empirical results reveal that the me approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

two dimensional distribution of streamwise velocity in open channel flow using maximum entropy principle incorporation of additional constraints based on conservation laws abstract this study derived the vertical and transverse distribution of streamwise velocity in open channels using the information theoretic concept involving the maximum entropy maxent principle moment constraints expressed by the conservation of mass momentum and energy and a hypothesis invoking the connection between probability and space domains through a generalized coordinate system explicit analytical solutions were obtained for velocity profiles using a non perturbation approach along with pade approximation the convergence of the series solution was shown both theoretically and numerically the lagrange multipliers arising through the application of maxent principle were obtained by minimizing a convex potential and the quasi newton method along with bfgs scheme was applied for solving the unconstrained optimization problem for applying the conservation laws the momentum and energy coefficients were found to influence the velocity profile in a channel cross section with the expressions of these coefficients the derived velocity equations were validated for a wide range of experimental and field data

norms of zonal spherical functions and fourier series on compact symmetric spaces abstract new estimates are given for the p norms of zonal spherical functions on compact symmetric spaces these estimates are applied to give a cohen type inequality for convolutor norms which leads to negative results on p mean convergence of fourier series on compact symmetric spaces of arbitrary rank

weak type estimates for bochner riesz spherical summation multipliers we consider the bochner riesz multiplier tofw 1 1 12 tjw 8 0 where denotes the fourier transform it is shown that the multiplier operator to is weak type po po acting on lpo rn radial functions where po is the critical value 2n n 1 28

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

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

analysis of multiscale mortar mixed approximation of nonlinear elliptic equations abstract a multiscale mortar mixed finite element method is established to approximate non linear second order elliptic equations the method is based on non overlapping domain decomposition and mortar finite element methods the existence and uniqueness of the approximation are demonstrated and a priori l 2 error estimates for the velocity and pressure are derived an error bound for mortar pressure is proved convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete pressure dependent coefficient optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation the quadratic convergence of the newton raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem numerical experiments are performed to support theoretic results

maximum entropy approximation for lyapunov exponents of chaotic maps let s 0 1 0 1 be a nonsingular chaotic map that preserves an integrable density f that describes the statistics of the orbits in this article we use the maximum entropy approach to approximate the density f and the corresponding lyapunov exponent

efficient numerical approximation of maximum entropy estimates a class of algorithms for approximation of the maximum entropy estimate of probability density functions on the basis of a finite number of sampled data is introduced the algorithms are presented as a finite sequence in order of increasing accuracy and decreasing computational efficiency in which the last element of the sequence is the exact maximum entropy estimate numerical and applicative examples are reported

observer participant models of neural processing a model is proposed in which the neuron serves as an information channel channel distortion occurs through the channel since the mapping from input boolean codes to output codes are many to one in that neuron outputs consist of just two distinguished states within the described model the neuron performs a decision making function decisions are made regarding the validity of a question passively posed by the neuron this question becomes defined through learning hence learning is viewed as the process of determining an appropriate question based on supplied input ensembles an application of the shannon information measures of entropy and mutual information taken together in the context of the proposed model lead to the hopfield neuron model with conditionalized hebbian learning rules neural decisions are shown to be based on a sigmoidal transfer characteristic or in the limit as computational temperature tends to zero a maximum likelihood decision rule the described work is contrasted with the information theoretic approach of linsker

a computational inverse method for identification of non gaussian random fields using the bayesian approach in very high dimension this paper is devoted to the identification of bayesian posteriors for the random coefficients of the high dimension polynomial chaos expansions of non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem must be solved to perform the identification of the random field a complete methodology is proposed to solve this very challenging problem in high dimension which consists in using the first four steps introduced in a previous paper followed by the identification of the posterior model the steps of the methodology are the following 1 introduction of a family of prior algebraic stochastic model pasm 2 identification of an optimal pasm in the constructed family using the partial experimental data 3 construction of a statistical reduced order optimal pasm 4 construction in high dimension of the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal pasm 5 substitution of these deterministic vector valued coefficients by random vector valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified 6 construction of the prior probability model of these random vector valued coefficients and finally 7 identification of the posterior probability model of these random vector valued coefficients using partial and limited experimental data through the stochastic boundary value problem two methods are proposed to carry out the identification of the posterior model the first one is based on the use of the classical bayesian method the second one is a new approach derived from the bayesian method which is more efficient in high dimension an application is presented for which several millions of random coefficients are identified

higher order moments and the maximum entropy inference the thermodynamic limit approach according to jaynes 1963 statistical mechanics can be interpreted as a special type of statistical inference based on the principle of maximum entropy the result of such an inference depends on the available information about a given physical system but the principle itself does not decide what kind of information is essential and what is not the well known gibbs canonical state results from the principle when the statistical mean value of energy is supposed to be known the maximum entropy inference employing higher order moments of energy is examined from the point of view of the thermodynamic limit it is shown that the contribution to the total entropy arising from the extra information corresponding to the higher order moments is o n when n to infinity n v constant n is the number of particles and v is the volume consequently from a purely thermodynamic point of view this extra information is non essential and can be neglected in the maximum entropy procedure these conclusions also characterise the maximum entropy inference employing higher order moments of almost arbitrary extensive quantities one can say that the maximum entropy inference has a certain stability property with respect to information corresponding to higher order moments of extensive physical quantities

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

l p norms of certain kernels on the n dimensional torus in this paper we study a class of kernels fr which generalize the bochner riesz kernels on the n dimensional torus our main result consists in upper estimates for the lp norms of fr as r tends to infinity as a consequence we prove a convergence theorem for means of functions belonging to suitable besov spaces 1 throughout this paper we identify the n dimensional torus tn n 2 with the n dimensional cube qn x e r n 1 2 0 we form the means fr g t f r m g m exp 2 7imt 1 m where t e tn m ranges on the integer lattice zn and denotes the convolution in tn in several cases one is able to estimate the l 1 tn norm of the kernel fr t e f r 1m exp 2 srimt 2 m and if 0 e s and f o 1 deduce convergence results for the means 1 as r x oc when g belongs to certain classes of periodic functions see e g 1 3 7 8 12 for instance if f x 1 i x12 8 when lxl 1 then fr k a is the familiar bochner riesz kernel in this case k i babenko cf 1 has shown that iikriil is of the same order as r n 1 2 as r tends to infinity n 2 0 8 n 1 2 while stein proved the estimate iikr il log r if 8 n 1 2 9 recently yudin 12 obtained the estimate lf o r n1 2 if f is the characteristic function of a closed balanced set c whose boundary has finite upper minkowski measure the method of yudin which also yields estimates for the lp tn norms of fr uses jackson approximation theorem which in tum involves estimates for the l2 r n modulus of continuity of f such a method can be adapted to a more general situation as we show in this paper in 2 4 we consider kernels of the type 2 associated with functionsf whose derivatives of certain orders are controlled by not necessarily positive received by the editors may 13 1980 1980 mathematics subject classification primary 42b99 secondary 46e35

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

data driven probability concentration and sampling on manifold a new methodology is proposed for generating realizations of a random vector with values in a finite dimensional euclidean space that are statistically consistent with a dataset of observations of this vector the probability distribution of this random vector while a priori not known is presumed to be concentrated on an unknown subset of the euclidean space a random matrix is introduced whose columns are independent copies of the random vector and for which the number of columns is the number of data points in the dataset the approach is based on the use of i the multidimensional kernel density estimation method for estimating the probability distribution of the random matrix ii a mcmc method for generating realizations for the random matrix iii the diffusion maps approach for discovering and characterizing the geometry and the structure of the dataset and iv a reduced order representation of the random matrix which is constructed using the diffusion maps vectors associated with the first eigenvalues of the transition matrix relative to the given dataset the convergence aspects of the proposed methodology are analyzed and a numerical validation is explored through three applications of increasing complexity the proposed method is found to be robust to noise levels and data complexity as well as to the intrinsic dimension of data and the size of experimental datasets both the methodology and the underlying mathematical framework presented in this paper contribute new capabilities and perspectives at the interface of uncertainty quantification statistical data analysis stochastic modeling and associated statistical inverse problems