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

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

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

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

modeling flow and transport in highly heterogeneous three dimensional aquifers ergodicity gaussianity and anomalous behavior 2 approximate semianalytical solution 1 flow and transport take place in a heterogeneous medium of lognormal distribution of the conductivity k flow is uniform in the mean and the system is defined by u mean velocity y2 log conductivity variance and integral scale i transport is analyzed in terms of the breakthrough curve of the solute identical to the traveltime distribution at control planes at distance x from the source the self consistent approximation is used where the traveltime is approximated by the sum of pertinent to the different separate inclusions and the neglected interaction between inclusions is accounted for by using the effective conductivity the pdf f x where x is the control plane distance is derived by a simple convolution it is found that f x has an early arrival time portion that captures most of the mass and a long tail which is related to the slow solute particles that are trapped in blocks of low k the macrodispersivity is very large and is independent of x the tail f x is highly skewed and only for extremely large x i depending on y2 the plume becomes gaussian comparison with numerical simulations shows very good agreement in spite of the absence of parameter fitting it is found that finite plumes are not ergodic and a cutoff of f x is needed in order to fit the mass flux of a finite plume depending on y2 and x i the bulk of f x can be approximated by a gaussian shape with fitted equivalent parameters the issue of anomalous behavior is examined with the aid of the model

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

a fast automatic algorithm for image denoising by a regularization method a new fast iterative algorithm is proposed to denoise images affected by gaussian additive noise by solving a constrained optimization problem a family of functionals parameterized by a few hyperparameters already proposed in the literature to solve denoising problems is considered the algorithm estimates jointly the image and the hyperparameters therefore providing an automatic method each member of the family is made up of a coherence with the data term and a term enforcing a roughness penalty and preserving jump discontinuities assuming we know the noise variance an adequacy constraint is also considered the algorithm computes a member of the family and a minimizer of it which satisfies the constraint a convergence proof is provided we then consider a heuristic version of the algorithm which gives restorations of comparable quality whose computational complexity is a linear function of the pixels number experimental results on synthetic and real data are presented moreover numerical comparisons with several fast denoising methods are provided

particle based transport model with markovian velocity processes for tracer dispersion in highly heterogeneous porous media 1 monte carlo mc studies of flow in heterogeneous porous formations in which the log conductivity field is multi gaussian have shown that as the log conductivity variance 2 y increases beyond about 0 5 the one point velocity probability density functions pdfs deviate significantly from gaussian behavior the velocity statistics become more complex due to the formation of preferential flow paths or channels as 2 y increases methods that employ low order approximations e g truncated perturbation expansions are limited to small 2 y and are unable to represent the complex velocity statistics associated with 2 y 1 here a stochastic transport model for highly heterogeneous domains i e 2 y 1 is proposed in the model the lagrangian velocity components of tracer particles are represented using continuous markovian stochastic processes in time the markovian velocity process mvp model is described using a set of first order ordinary and stochastic differential equations which are easy to solve using a particle based method once the mvp model is calibrated based on velocity statistics from mc simulations of the flow hydraulic head and velocity the mvp based model can be used to describe the evolution of the tracer concentration field accurately and efficiently specifically the mvp model is validated using mc simulations of longitudinal and transverse tracer spreading due to a point like injection in the domain we demonstrate that for 2 y 1 the ensemble averaged tracer cloud remains markedly non gaussian for relatively large travel distances from the point source the mvp transport model captures this behavior and reproduces the velocity statistics quite accurately

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

solute transport in highly heterogeneous aquifers the paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity y ln k available solutions obtained from perturbation methods are limited to first and second order solutions valid only for small values of the log conductivity variance 2y published numerical investigations give comparative results with finite values of 2y but some discrepancies among the results generate doubts about the capability of numerical methods to capture high order effects for media with large variance values when these large 2y values are encountered in natural formations the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process the nonlinearity consequences are here investigated in two dimensional isotropic porous media by the monte carlo technique coupled with a finite element analysis the analysis includes 2y values from 0 05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution to dissipate the doubts related to the numerical approach the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors the numerical results confirm the validity of the first order and second order analyses when 2y 0 the second order solution captures the nonlinear effects in a small range of log conductivity variance close to zero the nonlinear terms neglected in the first order formulation for higher 2y values give 1 late travel time longitudinal dispersion values greater than the linear solution 2 notably non gaussian distribution of the lagrangian velocity and particle displacements and 3 travel times to approach the asymptotic fickian regime longer than those obtained using the linear solution

is transverse macrodispersivity in three dimensional groundwater transport equal to zero a counterexample 1 in advective transport through weakly heterogeneous aquifers of random stationary and isotropic three dimensional permeability distribution transverse macrodispersivity t is found to be zero this was determined in the past by solving the transport equation at first order in the log conductivity variance 2 y however field findings indicate the presence of small but finite t the aim of the paper is to determine t for highly heterogeneous formations using a model that contains inclusions of conductivity k submerged in a matrix of conductivity k 0 for large k k 0 in the dilute medium approximation valid for small volume fraction n but arbitrary and for spherical inclusions it is found that t 0 because of the axisymmetry of flow past a sphere a medium made up of rotational ellipsoids of arbitrary random orientation macroscopically isotropic and of the same and n is devised as a counterexample it is found that because of the intertwining of streamlines t 0 being of order 1 4 for 1 these findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions for 10 and n 0 2 jamming limit the large value t l 0 15 is attained the numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon coined as advective mixing the two point covariance used in practice in order to characterize the aquifer structure is not able to detect the structures that produce advective mixing nevertheless the presence of high conductivity lenses inclined with respect to the mean flow may explain the occurrence of finite t in field applications

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

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

representation theory of u1 h in the symmetric tensors abstract the representations of the group of unitary operators which are trace class perturbations of the identity on an infinite dimensional separable hilbert space are classified according to factoriality quasi equivalence and semifiniteness by relating these representations to the quasi free representations of the weyl algebra this answers the problem posed by str til and d voiculescu lecture notes in mathematics vol 486 springer verlag berlin new york 1975

numerical assessment of 3 d macrodispersion in heterogeneous porous media 1 hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneous porous media that critically depends on dimensionality the transverse macrodispersion asymptotic dispersion transverse to the mean velocity direction is known to vanish only in 2 d and not in 3 d using classical gaussian correlated permeability fields with a lognormal distribution of variance y2 we determine numerically the longitudinal and transverse dispersivities as functions of heterogeneity and dimensionality we show that the transverse macrodispersion steeply increases with y2 underlying the essential role of flow lines braiding a mechanism specific to 3 d systems that we qualitatively characterize by the increasing expansion of the flow lines transversally to the flow direction the transverse macrodispersion remains however at least two orders of magnitude smaller than the longitudinal macrodispersion which increases even more steeply with y2 at moderate to high levels of heterogeneity the transverse dispersion also converges much faster to its asymptotic regime than do the longitudinal dispersion braiding cannot be thus taken as the sole mechanism responsible for the high longitudinal macrodispersions it could be either supplemented or superseded by stronger velocity correlations in 3 d than in 2 d this assumption is supported by the much larger longitudinal macrodispersions obtained in 3 d than in 2 d up to a factor of 5 for y2 6 25

stochastic analysis of unsaturated steady flows above the water table steady flowtakes place into a three dimensional partially saturated porous medium where due to their spatial variability the saturated conductivity ks and the relative conductivity kr are modeled as random space functions rsf s as a consequence the flow variables fvs i e pressure head and specific flux are also rsfs the focus of the present paper consists into quantifying the uncertainty of the fvs above the water table the simple expressions most of which in closed form of the second order moments pertaining to the fvs allow one to follow the transitional behavior from the zone close to the water table where the fvs are non stationary till to their far field limit where the fvs become stationary rsfs in particular it is shown how the stationary limits and the distance from the water table at which stationarity is attained depend upon the statistical structure of the rsfs ks kr and the infiltrating rate the mean pressure head k l has been also computed and it is expressed as k l 0 1 being a characteristic heterogeneity function which modifies the zero order approximation 0 of the pressure head valid for a vadose zone of uniform soil properties to account for the spatial variability of ks and kr two asymptotic limits i e close near field and away far field from the water table are derived into a very general manner whereas the transitional behavior of between the near far field can be determined after specifying the shape of the various input soil properties besides the theoretical interest results of the present paper are useful for practical purposes as well indeed the model is tested against to real data and in particular it is shown how it is possible for the specific case study to grasp the behavior of the fvs within an environment i e the vadose zone close to the water table which is generally very difficult to access by direct inspection

estimation of macrodispersion in 2 d highly heterogeneous porous media using the eulerian lagrangian localized adjoint method 1 the eulerian lagrangian localized adjoint method ellam formulation of younes et al 2006 is applied to solve the advection dispersion equation used to describe conservative solute transport in highly heterogeneous porous media heterogeneity is described by a correlated random field with an exponential covariance macrodispersion coefficients are calculated for a broad range of heterogeneities using monte carlo mc simulations on large size domains ellam circumvents some drawbacks of usual particle tracking and eulerian lagrangian methods when local dispersion diffusion is added ellam is also highly efficient and well adapted for advective dominant transport and for mc simulations for pure advection first order approximation provides good estimates of the duration of the preasymptotic regime and of the longitudinal macrodispersion coefficient for a variance of the log conductivity 2 2 25 higher order theories overestimated this coefficient for higher variances computed transverse macrodispersion is equal to 0 for each studied variance of log conductivity 2 0 25 1 0 2 25 4 0 6 25 9 0 local dispersion diffusion affects the macrodispersion for quite low peclet number 100 compared to previous work for a peclet number of 10 it leads to an increase of the longitudinal and transverse macrodispersion for low variances 2 0 25 for higher heterogeneity 2 9 0 for local dispersion and 2 4 0 for local diffusion the longitudinal macrodispersion decreases due to local transverse mixing

projective repesentations of the hilbert lie group u h 2 via quasifree statres on the car algebra the group u h 2 of unitary operators on a hilbert space h which differ from the identity by a hilbert schmidt operator may be imbedded in the group of bogoliubov automorphisms of the car algebra over h in such a way as to be weakly inner in any gauge invariant quasifree representation consequently each such quasifree representation determines a projective representation of u h 2 if 0 a i is the operator on h determining the quasifree representation a and a denotes the cyclic projective representation of u h 2 generated from the g n s cyclic vector a for a then the 2 cocycle in u h 2 determined by a can be given explicitly we prove that this 2 cocycle is a coboundary if any only if a or 1 a is hilbert schmidt the representations a on restriction to the group u h 1 consisting of unitaries which differ from the identity by a trace class operator always determine 2 cocycles which are coboundaries these representations of u h 1 have already been investigated by stratila and voiculescu lecture notes in mathematics no 486 springer new york 1975 math anal 235 1978 87 110 thus the stratila voiculescu representations of u h 1 always extend to projective representations of u h 2 and to ordinary representations when a or 1 a is hilbert schmidt this fact enables exploitation of the type analysis of stratila and voiculescu to determine the type of the von neumann algebra a u h 2 in the special case where 0 and 1 are not eigenvalues of a a is cyclic and separating for a u h 2 and hence determines a k m s state on this algebra it is shown that for special choices of a type iii 0 1 factors a u h 2 may be constructed

modeling three dimensional flow about ellipsoidal inhomogeneities with application to flow to a gravel packed well and flow through lens shaped inhomogeneities analytic functions are superimposed to model three dimensional steady groundwater flow in regions containing one or more inhomogeneities shaped like prolate or oblate ellipsoids of revolution each function and the sum of such functions are solutions of laplace s equation the governing differential equation for steady groundwater flow the functions are implemented in a manner that provides exact continuity of flow across the entire boundary of each inhomogeneity in general continuity of head is provided at specified control points on the boundary and is approximated between control points for the case of one inhomogeneity in a uniform flow field it turns out that there is exact continuity of head across the entire surface of the inhomogeneity the method is implemented in a computer program written by the author two applications are demonstrated 1 flow to a gravel packed well and 2 flow through a series of lens shaped inhomogeneities the examples demonstrate that the approximation of continuity of head can be made acceptable for many problems a possible application of the technique would be testing various theories regarding contaminant migration and dispersion by simulating flow and chemical diffusion through large numbers of lens shaped inhomogeneities

temporal behavior of a solute cloud in a heterogeneous porous medium 3 numerical simulations 1 the article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three dimensional heterogeneous medium the groundwater flow is derived from the linearized solution of the darcy equation with gauss distributed log hydraulic conductivity the transport of a passive solute is studied by a random walk method which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients the numerical results are compared to the second order perturbation theory expressions given in two companion papers dentz et al 2000a 2000b and to nonperturbative results which follow from corrsin s conjecture the low order perturbation theory is intrinsically based on the assumption of small heterogeneity while corrsin s conjecture does not take into account certain contributions due to higher order terms of the perturbation series the simulations yield for the first time systematic quantitative information on the validity and the limitations of these analytic approximations for increasing heterogeneities considerable deviations from the theoretically predicted transport behavior are observed

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

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

solute transport in heterogeneous formations of bimodal conductivity distribution 1 theory transport of a conservative solute takes place in a formation made up from a matrix of conductivity k0 and porosity 0 and inclusions of properties k for given inclusions shape the system is characterized by the two parameters k k0 and the inclusions volume fraction n in the past approximate solutions of the flow and transport problems were obtained under the limit of low variability i e 1 1 and arbitrary n rubin 1995 the present study aims at solving the problem under the opposite limit of a dilute system i e n 1 and arbitrary we are particularly interested in elongated inclusions high length thickness ratio of high permeability contrast to the matrix such configurations are related to applications in which lenses or cracks are present in a medium of highly different conductivity figure 1 the basic procedure was developed by eames and bush 1999 for cylindrical or spherical inclusions with no porosity contrast they compute the macrodispersion coefficient for advective transport past a large number of inclusions located at random it is based on the solution for the distortion of a material surface of marked particles moving past an individual inclusion in an unbounded domain and with uniform flow at infinity in the present study we extend the approach to inclusions of arbitrary porosity and elliptical shape characterized by the parameter e the ratio between the small and large axes with emphasis on e 1 we present the analytical solution of the flow problem and the procedure requiring two quadratures to calculate the macrodispersivity analytical solutions are obtained for two particular limits 1 and 1 the latter is compared with the limit n 1 of the solution of rubin 1995 the theoretical results are applied to a few cases of hydrological interest lessoff and dagan this issue

approximation properties for group algebras and group von neumann algebras let g be a locally compact group let c g resp vn g be the c algebra resp the von neumann algebra associated with the left regular representation i of g let a g be the fourier algebra of g and let moa g be the set of completely bounded multipliers of a g with the completely bounded norm moa g is a dual space and we say that g has the approximation property ap if there is a net ua of functions in a g with compact support such that ua a 1 in the associated weak topology in particular g has the ap if g is weakly amenable a g has an approximate identity that is bounded in the completely bounded norm for a discrete group f we show that f has the ap x c f has the slice map property for subspaces of any c algebra x vn jt has the slice map property for a weakly closed subspaces of any von neumann algebra property se the semidirect product of weakly amenable groups need not be weakly amenable we show that the larger class of groups with the ap is stable with respect to semidirect products and more generally this class is stable with respect to group extensions we also obtain some results concerning crossed products for example we show that the crossed product m a g of a von neumann algebra m with property s by a group g with the ap also has property s

asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations we determine the asymptotic dispersion coefficients in 2d exponentially correlated lognormally distributed permeability fields by using parallel computing fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker to obtain a well defined asymptotic regime under ergodic conditions initial plume size much larger than the correlation length of the permeability field the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length we determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 0 9 where s 2 is the lognormal permeability variance for purely advective transport the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

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

comparative analysis of multiscale gaussian random field simulation algorithms abstract we analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution we consider in particular the fourier wavelet method and three variants of the randomization method a without any stratified sampling of wavenumber space b with stratified sampling of wavenumbers with equal energy subdivision c with stratified sampling with a logarithmically uniform subdivision we focus primarily on fractal gaussian random fields with kolmogorov type spectra previous work has shown that variants a and b of the randomization method are only able to generate a self similar structure function over three to four decades with reasonable computational effort by contrast variant c along with the fourier wavelet method is able to reproduce accurate self similar scaling of the structure function over a number of decades increasing linearly with computational effort for our examples we will show that nine decades can be reproduced we provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method we find that when evaluating ensemble averaged quantities like the correlation and structure functions as well as some multi point statistical characteristics the randomization method can provide good accuracy with less cost than the fourier wavelet method the cost of the randomization method relative to the fourier wavelet method however appears to increase with the complexity of the random field statistics which are to be calculated accurately moreover the fourier wavelet method has better ergodic properties and hence becomes more efficient for the computation of spatial rather than ensemble averages which may be important in simulating the solutions to partial differential equations with random field coefficients

self consistent four point closure for transport in steady random flows ensemble averaging of advection dispersion equations describing transport of a passive scalar in incompressible random velocity fields requires a closure approximation commonly used approaches such as the direct interaction approximation and large eddy simulations as well as equivalent renormalization schemes employ so called two point or one loop closures these approaches have proven to be adequate for transport in zero mean unbiased time dependent random velocity fields with increasing accuracy for decreasing temporal coherence in the opposite limit of steady velocity fields with finite bias however these schemes fail to predict effective transport properties both quantitatively and qualitatively leading to an obvious inconsistency for transverse dispersion in two spatial dimensions for this case two point closures predict that macroscopic transverse dispersion increases as the square root of the disorder variance while it has been proven rigorously that there is no disorder induced contribution to macroscopic transverse dispersion for purely advective transport furthermore two point closures significantly underestimate the disorder induced contribution to longitudinal dispersion we derive a four point closure for stochastically averaged transport equations that goes beyond classical one loop schemes and demonstrate that it is exact for transverse dispersion and correctly predicts an increase of the longitudinal disorder induced dispersion coefficient with the square of the variance of the strong disorder the predicted values of asymptotic longitudinal dispersion coefficients are consistent with those obtained via monte carlo random walk simulations

a note on the theoretical foundations of particle tracking methods in heterogeneous porous media 1 in this technical note the stochastic process for the path of particles undergoing convection dispersion in a heterogeneous media with sharp interfaces is derived the derived process is shown to correspond to the numerical particle tracking scheme introduced on a more heuristic basis by hoteit et al 2002 who validated the accuracy of their numerical method in controlled laboratory experiments specifically it is shown that in the direction transverse to the flow u y and across an interface where the diffusion coefficient is discontinuous the stochastic particle path t y t is given by a rescaled version of ito and mckean s 1963 skew brownian motion for a uniquely determined value of a that depends on the values of the diffusion coefficient at either side of the interface these results help supply a theoretical foundation that supports the numerical schemes and experimental findings of hoteit et al 2002

reply to comment by a fiori et al on asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations the comment by a fiori g dagan and i jankoviae fiori et al 2008 compares numerical results de dreuzy et al 2007 on the longitudinal asymptotic dispersion coefficient to a self consistent solution fiori et al 2003

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

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

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

numerical simulations of preasymptotic transport in heterogeneous porous media departures from the gaussian limit 1 the objective of this work is to determine whether conventional solute macrodispersion theories adequately predict the behavior of individual subsurface plumes over time and length scales relevant to practical risk assessment and remediation activities this issue is studied with a set of high resolution numerical simulations of conservative tracers moving through saturated two dimensional heterogeneous conductivity fields the simulation experiments are designed to mimic long term field studies spatially correlated statistically stationary lognormal conductivity realizations are generated with a fourier transform procedure steady state velocity solutions are calculated for these fields using an accurate darcian solver velocity contour plots reveal the presence of disconnected networks of preferential pathways over a range of correlation lengths reverse flow cells are rare the velocity probability density functions have exponential tails and strong longitudinal asymmetries solute concentrations are derived from the simulated velocity fields with an accurate taut spline transport code that minimizes numerical dispersion the resulting plumes are tracked for travel distances of over one hundred spatial correlation lengths corresponding to scales of practical interest first moments and macrodispersivities which measure the location and extent of the plume are reasonably well approximated by conventional fickian theories but continue to vary after long travel distances this temporal variability highlights the slow convergence of the plume moments to gaussian limits calculations of the relative dilution index which is a measure of the plume mixing state indicate strongly non gaussian behavior other measures of plume structure suggested by anomalous dispersion theories also reveal the persistence of non gaussian behavior after long travel distances these measures appear to be more sensitive to non gaussian behavior than the spatial moments taken together the simulation results suggest that conservative solute plumes moving through statistically stationary random media may not converge to gaussian limits even after traveling hundreds of log conductivity correlation scales

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

flow and travel time statistics in highly heterogeneous porous media 1 in this paper we present flow and travel time ensemble statistics based on a new simulation methodology the adaptive fup monte carlo method afmcm as a benchmark case we considered two dimensional steady flow in a rectangular domain characterized by multi gaussian heterogeneity structure with an isotropic exponential correlation and lnk variance y2 up to 8 advective transport is investigated using the travel time framework where lagrangian variables e g velocity transverse displacement or travel time depend on space rather than on time we find that eulerian and lagrangian velocity distributions diverge for increasing lnk variance due to enhanced channeling transverse displacement is a nonnormal for all y2 and control planes close to the injection area but after xiy 20 was found to be nearly normal even for high y2 travel time distribution deviates from the fickian model for large lnk variance and exhibits increasing skewness and a power law tail for large lnk variance the slope of which decreases for increasing distance from the source no anomalous features are found second moment of advective transport is analyzed with respect to the covariance of two lagrangian velocity variables slowness and slope which are directly related to the travel time and transverse displacement variance which are subsequently related to the longitudinal and transverse dispersion we provide simple estimators for the eulerian velocity variance travel time variance slowness and longitudinal dispersivity as a practical contribution of this analysis both two parameter models considered the advection dispersion equation and the lognormal model provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media we identify the need for further theoretical and experimental scrutiny of early arrival times and the need for computing higher order moments for a more accurate characterization of the travel time probability density function a brief discussion is presented on the challenges and extensions for which afmcm is suggested as a suitable approach

flow and transport in highly heterogeneous formations 3 numerical simulations and comparison with theoretical results 1 in parts 1 dagan et al 2003 and 2 fiori et al 2003 a multi indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations the isotropic medium is made up from circular 2 d or spherical 3 d inclusions of different conductivities k submerged in a matrix of effective conductivity this structure is different from the multi gaussian one even for equal log conductivity distribution and integral scale a snapshot of a two dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture the present study was limited however to investigating the statistical moments of ergodic plumes two approximate semianalytical solutions based on a self consistent model sc and on a first order perturbation in the log conductivity variance fo are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf in this paper an efficient and accurate numerical procedure based on the analytic element method strack 1989 is used in order to validate the approximate results the solution satisfies exactly the continuity equation and at high accuracy the continuity of heads at inclusion boundaries the dimensionless dependent variables depend on two parameters the volume fraction n of inclusions in the medium and the log conductivity variance y2 for inclusions of uniform radius the largest n was 0 9 2 d and 0 7 3 d whereas the largest y2 was equal to 10 the sc approximation underestimates the longitudinal eulerian velocity variance for increasing n and increasing y2 in 2 d and to a lesser extent in 3 d as compared to numerical results the fo approximation overestimates these variances and these effects are larger in the transverse direction the longitudinal velocity pdf is highly skewed and negative velocities are present at high y2 especially in 2 d the main results are in the comparison of the macrodispersivities computed with the aid of the lagrangian velocity covariances as functions of travel time for the longitudinal macrodispersivity the sc approximation yields results close to the numerical ones in 2 d for n 0 4 but underestimates them for n 0 9 the asymptotic large travel time values of macrodispersivities in the sc and fo approximations are close for low to moderate y2 as shown and explained in part 1 however while the slow tendency to fickian behavior is well reproduced by sc it is much quicker in the fo approximation in 3 d the sc approximation is closer to numerical one for the highest n 0 7 and the different y2 2 4 8 and the comparison improves if molecular diffusion is taken into account transverse macrodispersivity for small travel times is underestimated by sc in 2 d and is closer to numerical results in 3 d whereas fo overestimates them transverse macrodispersivity asymptotically tends to zero in 2 d for large travel times in 3 d the numerical simulations lead to a small but persistent transverse macrodispersivity for large travel times whereas it tends to zero in the approximate solutions the results suggest that the self consistent semianalytical approximation provides a valuable tool to model transport in highly heterogeneous isotropic formations of a 3 d structure in terms of trajectories statistical moments it captures effects like slow transition to fickian behavior and to gaussian trajectory distribution which are neglected by the first order approximation

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

flow and transport in highly heterogeneous formations 2 semianalytical results for isotropic media 1 flow and transport of nonreactive solutes in heterogeneous porous media is studied by adopting a multi indicator model of permeability structure the porous formation is modeled as a collection of blocks of uniform permeability k implanted at random in a matrix of constant conductivity k0 the multi indicator model leads to simple semianalytical solutions based on the self consistent argument which are valid for a high degree of heterogeneity the methodology is applied to isotropic formations to derive a few statistical moments of the velocity field and of the solute particles trajectory as functions of time and of the log conductivity variance y2 along the common models of aquifer permeability distribution the distribution of y ln k is assumed to be normal all the semianalytical results degenerate in the well known first order results when y2 1 in particular it is shown that the first order longitudinal dispersivity l seems to hold for values of the log conductivity variance much larger than expected up to y2 4 this results from a compensation of errors associated with the first order approximation in contrast for y2 1 the asymptotic l grows exponentially with y2 the effect of molecular diffusion is considered in a simple manner by introducing a cutoff c for the conductivity contrast the time to reach the asymptotic l grows with the log conductivity variance it is thus observed that transport in highly heterogeneous formations can be characterized by a very prolonged non fickian stage with dispersivity l growing continuously with time further analysis of third order moment of trajectory indicates that for growing values of y2 the time needed for the plume to become gaussian can be quite large

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

asymptotic dispersion for two dimensional highly heterogeneous permeability fields under temporally fluctuating flow 1 temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media unfortunately most results have been obtained by linear stochastic theories on permeability fields of limited variability worse results are inconsistent regarding the impact of fluctuations on longitudinal dispersion which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion we perform numerical monte carlo simulations for highly variable permeability fields of up to 800 correlation lengths we find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines fluctuations transverse to the main flow direction not only increase transverse dispersion which is well known but also reduce the longitudinal macrodispersion in a significant and consistent way which contradicts previous findings the reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion most surprisingly for high heterogeneity temporal fluctuations cause total longitudinal plus transverse macrodispersion to drop with respect to the steady state one enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and lagrangian correlation reduction of the longitudinal macrodispersion results from the reduction of the lagrangian correlation of the longitudinal velocity that is temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low permeability zones that tend to block or trap the solute in steady state flow conditions

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

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

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

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

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

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

a bayesian approach to synthetic magnetic resonance imaging summary synthetic magnetic resonance imaging involves the estimation based on a set of measured images with noise of three basic physical quantities that are nonlinearly related to the observations the methods currently available for this ill conditoned inverse problem either do not provide sufficiently accurate estimates or require time consuming data collection we formulate this nonlinear problem in a bayesian framework taking into account knowledge about the physics of the magnetic resonance imaging experiment statistical properties of the experimental noise and prior information about the underlying physical quantities modelled by a suitable markov random field a new multilayer markov random field is proposed inference is drawn by means of markov chain monte carlo methods or iterated conditional modes some examples are included to demonstrate how synthetic magnetic resonance imaging by this approach can be performed in an accurate and reliable way

geological entropy and solute transport in heterogeneous porous media we propose a novel approach to link solute transport behavior to the physical heterogeneity of the aquifer which we fully characterize with two measurable parameters the variance of the log k values math formula and a new indicator hr that integrates multiple properties of the k field into a global measure of spatial disorder or geological entropy from the results of a detailed numerical experiment considering solute transport in k fields representing realistic distributions of hydrofacies in alluvial aquifers we identify empirical relationship between the two parameters and the first three central moments of the distributions of arrival times of solute particles at a selected control plane the analysis of experimental data indicates that the mean and the variance of the solutes arrival times tend to increase with spatial disorder i e hr increasing while highly skewed distributions are observed in more orderly structures i e hr decreasing or at higher math formula we found that simple closed form empirical expressions of the bivariate dependency of skewness on hr and math formula can be used to predict the emergence of non fickian transport in k fields considering a range of structures and heterogeneity levels some of which based on documented real aquifers the accuracy of these predictions and in general the results from this study indicate that a description of the global variability and structure of the k field in terms of variance and geological entropy offers a valid and broadly applicable approach for the interpretation and prediction of transport in heterogeneous porous media

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

behavior of indicator variograms and transition probabilities in relation to the variance in lengths of hydrofacies when using indicator geostatistics to represent the distribution of hydrofacies or lithofacies the range and curvature of the autotransition probabilities or the autovariogram are related to the variation in facies lengths as the coefficients of variation for the length of each facies increase toward unity the effective range increases while the indicator correlation structure evolves from a periodic linear structure to a somewhat periodic spherical structure to an aperiodic exponential structure multimodal distributions of facies lengths can give rise to autotransition probabilities or autovariograms that appear to have nested structures with the understanding of these relationships it is possible to choose a model form and model parameters for the autotransition probability or autovariogram based upon the facies proportions and the modality mean and variance in length of the facies this is illustrated with data from a case study

significance of higher moments for complete characterization of the travel time probability density function in heterogeneous porous media using the maximum entropy principle 1 the travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually e g for incorporating retention processes and in applications where typically the travel time peak early and late arrivals of contaminants are of major concern in a regulatory or remediation context furthermore the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers in view of this interest the travel time distribution has been studied in the literature however the link to the hydraulic conductivity statistics has been typically restricted to the first two moments here we investigate the influence of higher travel time moments on the travel time probability density function pdf in heterogeneous porous media combining monte carlo simulations with the maximum entropy principle the monte carlo experimental pdf is obtained by the adaptive fup monte carlo method afmcm for advective transport characterized by a multi gaussian structure with exponential covariance considering two injection modes in flux and resident and lnk variance up to 8 a maximum entropy maxent algorithm based on fup basis functions is used for the complete characterization of the travel time pdf all travel time moments become linear with distance initial nonlinearity is found mainly for the resident injection mode which exhibits a strong nonlinearity within first 5iy for high heterogeneity for the resident injection mode the form of variance and all higher moments changes from the familiar concave form predicted by the first order theory to a convex form for the in flux mode linearity is preserved even for high heterogeneity the number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level mean and variance completely describe travel time pdf for low and mild heterogeneity skewness is dominant for lnk variance around 4 while kurtosis and fifth moment are required for lnk variance higher than 4 including skewness seems sufficient for describing the peak and late arrivals linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and fickian transport however higher order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source

analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1 steady flow the statistical structures of the hydraulic conductivity head gradient specific discharge and head fields are analyzed with the aid of a simplified model consisting of a collection of spherical in three dimensions and cylindrical in two dimensions blocks of different conductivity and diameter set at random and independently in space results of a previous work for the effective conductivity and unbounded formations dagan 1979 are recovered and given a rational derivation the influence of the aquifer boundary and nonuniform average flows are examined for the first time the growth of the head variance with distance from boundary in two dimensional flows is compared with that of the monte carlo simulations of smith and freeze 1979

statistical formulation of generalized tracer retention in fractured rock we study tracer retention in fractured rock by combing lagrangian and time domain random walk frameworks as well as a statistical representation of the retention process mass transfer is quantified by the retention time distribution that follows from a lagrangian coupling between advective transport and mass exchange processes applicable for advection dominated transport a unifying parametrisation is presented for generalised diffusion using two rates denoted by k1 and k2 where k1 is a forward rate and k2 a reverse rate plus an exponent as an additional parameter for the fickian diffusion model k1 and k2 are related to measurable retention properties of the fracture matrix by the method of moments whereas for the non fickian case dimensional analysis is used the derived retention time distributions are exemplified for interpreting tracer tests as well as for predictive modelling of expected tracer breakthrough we show that non fickian effects can be notable when transport is upscaled based on a non fickian interpretation of a tracer test for which deviations from fickianity are relatively small the statistical representation of retention clearly shows the significance of the forward rate k1 which depends on the active specific surface area and is the most difficult parameter to characterise in the field

solute transport in low heterogeneity sandboxes the role of correlation length and permeability variance this work examines how heterogeneity structure in particular correlation length controls flow and solute transport we used two dimensional 2d sandboxes 21 9 cm 20 6 cm and four modeling approaches including 2d advection dispersion equation ade with explicit heterogeneity structure 1d ade with average properties and nonlocal continuous time random walk ctrw and fractional ade fade the goal is to answer two questions 1 how and to what extent does correlation length control effective permeability and breakthrough curves btc 2 which model can best reproduce data under what conditions sandboxes were packed with the same 20 v v fine and 80 v v coarse sands in three patterns that differ in correlation length the mixed cases contain uniformly distributed fine and coarse grains the four zone and one zone cases have four and one square fine zones respectively a total of seven experiments were carried out with permeability variance of 0 10 lc 0 22 mc and 0 43 hc experimental data show that the btc curves depend strongly on correlation length especially in the hc cases the hc one zone hco case shows distinct breakthrough steps arising from fast advection in the coarse zone slow advection in the fine zone and slow diffusion while the lco and mco btcs do not exhibit such behavior with explicit representation of heterogeneity structure 2d ade reproduces btcs well in all cases ctrw reproduces temporal moments with smaller deviation from data than fade in all cases except hco where fade has the lowest deviation

variational approach for edge preserving regularization using coupled pdes this paper deals with edge preserving regularization for inverse problems in image processing we first present a synthesis of the main results we have obtained in edge preserving regularization by using a variational approach we recall the model involving regularizing functions spl phi and we analyze the geometry driven diffusion process of this model in the three dimensional 3 d case then a half quadratic theorem is used to give a very simple reconstruction algorithm after a critical analysis of this model we propose another functional to minimize for edge preserving reconstruction purposes it results in solving two coupled partial differential equations pdes one processes the intensity the other the edges we study the relationship with similar pde systems in particular with the functional proposed by ambrosio tortorelli 1990 1992 in order to approach the mumford shah 1989 functional developed in the segmentation application experimental results on synthetic and real images are presented

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

impact of aquifer heterogeneity structure and local scale dispersion on solute concentration uncertainty in this paper we study the influence of high log conductivity variance sigma 2 y and local scale dispersion on the first two concentration moments as well as on higher order moments skewness

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

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

approximations for transport parameters and self averaging properties for point like injections in heterogeneous media we focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and gaussian random fields the parameters are determined by ensemble averages we study to what extent these averages represent the behaviour in a single realization we calculate the centre of mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation and on the iterative solution of the langevin equation compared with simulations the perturbation theory reproduces the numerical results only poorly whereas the iterative solution yields good results using these approximations we investigate the self averaging properties the ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles the dispersion coefficient is not self averaging in the ensemble of periodic fields for the gaussian ensemble the asymptotic dispersion coefficient is self averaging for finite times however the fluctuations are so large that the average does not represent the behaviour in a single realization

application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity selected groundwater flow scenarios are used in a two way comparison between the mixed hybrid finite element method and the standard finite element method also called the conforming finite element method the simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists the basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element by contrast the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential the conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity the mixed hybrid finite element method provides velocities everywhere in the field as well as potentials at the center of each element and each edge moreover the normal component of the velocity field is continuous between adjacent elements the results of the simulations are presented in the form of streamlines to avoid the problem of velocity discontinuity the method of cordes and kinzelbach 1992 is used it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method the comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy it is also superior to the conforming method in terms of computational effort the potential fields obtained by the mixed hybrid and the conforming finite element methods are the same