numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

a synthesis of approaches to upscaling of hydraulic conductivities simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them the process of defining block conductivities from point values is termed upscaling a number of approaches to upscaling are available most of which consider the uncertainty associated with any natural property so that they cast the problem in a stochastic frame recently indelman and dagan 1993a b provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function unfortunately they did not provide any practical method to compute block values from point ones the objective of this work is twofold first we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values second we check that all approaches verify approximately a number of conditions stated by indelman and dagan 1993a specifically we show analytically that for regular blocks the methodologies of both rubin and gomez hernandez 1990 and desbarats 1992 which we call practical methodologies satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support we call this the actual formation is identical to that obtained from the same field with conductivities defined at a larger support upscaled formation the analysis is carried out by working with the logarithm of block conductivities and using a small perturbation expansion and thus is strictly valid for small variances on the other hand we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy that block averaged dissipation values computed are indeed very close to the true dissipation values in each block the agreement is even better if we consider statistical moments instead of point values as an important conclusion we should note that all practical methodologies considered in this work perform equally well and more important constitute a simple way to treat an otherwise very complex problem

scaling of constitutive relationships in unsaturated heterogeneous media a numerical investigation numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured this study seeks to identify simple scaling laws linking parameters of block scale constitutive relationships with the parameters of their measurement scale counterparts represented here using gardner russo models the two parameters of these models saturated conductivity and pore size distribution parameter are viewed as cross correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential gaussian method upscaled constitutive curves for steady gravity drainage through three dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials the experiments reveal clear log linear relationships between upscaled relative conductivities and average matric potentials the slopes of which are used to define an upscaled pore size distribution parameter the reciprocal of this quantity which represents an upscaled characteristic capillary length is very well approximated by a spatial power average of its measurement scale values furthermore the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains when average matric potentials are normalized using this upscaled characteristic capillary length relative conductivity and moisture saturation values for widely different flow field realizations collapse along well defined curves however for wet conditions the form of these curves differs from that of the models assumed at the measurement scale

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

polynomial matrix solution of the h infinity filtering problem and the relationship to riccati equation state space results a solution to the standard suboptimal h filtering problem is presented using a new polynomial systems ap proach the polynomial matrix solution is closely related to the recent h state space based riccati equation results and the links between the polynomial and state equation methodologies are demonstrated the h filter is derived by solving a special type of minimum variance filtering problem using a technique based on game theory the solution of this game problem in a stochastic setting by polynomial methods is novel the cal culation of the h filter involves a j spectral factorization and the solution of two coupled diophantine equations there are several computational advantages over previous polynomial methods for calculating h filters

optimal multivariable lqg control using a single diophantine equation the general solution of the optimal lqg problem using the polynomial equation approach involves the solution of a couple of two sided matrix polynomial equations the conditions under which a single one sided equation which is related to the original two the implied diophantine equation yields the unique optimal controller are derived in this paper the result has considerable relevance to aspects such as computation and self tuning control applications

modeling the large scale dynamics of saturated groundwater flow using spatial filtering theory 1 theoretical development we develop a model and a framework for understanding the explicitly resolved large scale dynamics of groundwater flow in fully saturated heterogeneous porous media the approach is based upon the large eddy simulation les methodology from geophysical fluid dynamics the idea behind the les approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales the technical foundation of the method rests on a spatial filtering operation which allows one to precisely define the scale of the model we spatially filter darcy s law and the continuity equation to produce a model for the resolved scale dynamics we develop les approximations for the resulting flow equations and a closure model for unresolved subgrid scale terms we model the subgrid terms using resolved scale quantities only and thus do not require subgrid scale information the accuracy of these closure models depends upon the scale at which hydraulic conductivity is measured by field tests we characterize the approximations used to model subgrid scale effects asymptotically with a small parameter the dimensionless filter width defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables

upscaling of permeability of anisotropic heterogeneous formations 3 applications the general results of indelman and dagan this issue a b are applied to some cases of interest in applications the exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first order analysis the explicit expressions of the two first moments of the upscaled permeability are derived for media of two dimensional isotropic and three dimensional axisymmetric actual log permeability of gaussian covariance and for parallelepipedic rectangular for two dimensional averaging elements the effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed the asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity the strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples in contrast the homogenization of the upscaled media by large blocks depends primarily on the volume area for two dimensions of the partition element relative to the one based on the correlation scale of the log permeability

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

polynomial equations for the linear mmse state estimation the linear minimum mean square error mmse state estimation problem is solved via spectral factorization and a pair of bilateral diophantine equations detectability and or stability requirements in the riccati based solution context are expressed in terms of the stability of the greatest common right and left divisors of polynomial matrices

mmse deconvolution via polynomial methods and its dual lqg regulation abstract the problem of linear minimum mean square error mmse multichannel deconvolution of sampled signals from noisy observations is approached via matrix polynomial equations the general solution is given in terms of a left spectral factorization and a pair of bilateral diophantine equations the first diophantine equation is obtained by imposing optimality of the deconvolution filter whereas the second ensures stability of the filter should the signal model be unstable the proposed solution encompasses classical wiener as well as stationary kalman filtering prediction and fixed lag smoothing the duality with the polynomial equations for lqg regulation is also discussed

a stochastic approach to the problem of upscaling of conductivity in disordered media theory and unconditional numerical simulations a general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model in this study a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size assuming a multilognormal distribution for the point conductivities the block conductivity appears as a random function whose distribution conditional to point measurements can be derived this conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data these realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known the analytical results are compared to results obtained through a monte carlo analysis and a good agreement was found even for variances of ln t larger than 1

continuous time lq regulator design by polynomial equations abstract the deterministic continuous time lq output regulation problem is solved by polynomial equations as an alternative to the usual riccati equation approach in particular it is shown that riccati based and polynomial methods are fully conceptually equivalent in steady state lq regulation

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

analysis of variance method for the equivalent conductivity of rectangular blocks we use an analysis of variance anova decomposition of the log hydraulic conductivity f x ln k x as the sum of an average value f plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in d dimensional space some of the anova components are dealt with exactly the effect of the other components is approximated through calibration to the effective conductivity keff under ergodic conditions our analysis applies to both isotropic and anisotropic lognormal k fields we evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations we find that the anova formula is nearly unbiased although its performance is generally inferior to numerical renormalization advantages of the anova formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity kb on the various components of fluctuation of f inside the block such dependence is more complex than previously assumed considering the anova formula to be unbiased we derive the bias of the geometric mean of the cardwell parsons bounds for three dimensional rectangular blocks

bayesian variable selection in structured high dimensional covariate spaces with applications in genomics we consider the problem of variable selection in regression modeling in high dimensional spaces where there is known structure among the covariates this is an unconventional variable selection problem for two reasons 1 the dimension of the covariate space is comparable and often much larger than the number of subjects in the study and 2 the covariate space is highly structured and in some cases it is desirable to incorporate this structural information in to the model building process we approach this problem through the bayesian variable selection framework where we assume that the covariates lie on an undirected graph and formulate an ising prior on the model space for incorporating structural information certain computational and statistical problems arise that are unique to such high dimensional structured settings the most interesting being the phenomenon of phase transitions we propose theoretical and computational schemes to mitigate these problems we illustrate our methods on two

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

on the polynomial equations for the mimo lq stochastic regulator the role played by three polynomial equations in the lq stochastic regulation problem is discussed the emphasis is on establishing conditions under which the lq regulator can be obtained via the solution of a single uncoupled diophantine equation nevertheless the authors do not claim that there is any advantage in solving a single uncoupled equation in place of two coupled diophantine equations

upscaling of permeability of anisotropic heterogeneous formations 2 general structure and small perturbation analysis the general methodology developed in part 1 indelman and dagan this issue of this study is applied to the detailed analysis of upscaled permeability first the structure of the dissipation function for the general conductivity a tensor of stationary random components is examined with the aid of dimensional analysis it is shown that for arbitrary shapes of the numerical elements the upscaled permeability has also this general structure and the numbers of unknown parameters and equations match this result suggests that the upscaling problem has an unique solution in the particular case of scalar permeability of isotropic covariance of the actual formation it is shown that a similar upscaled permeability is possible only for spherical circular numerical elements otherwise the upscaled permeability has to be a tensor of anisotropic covariance if the actual formation has a scalar permeability of axisymmetric covariance upscaling preserves the last property only for axisymmetric partition elements i e for a sphere cylinder and ellipsoid explicit expressions for the first moments of the upscaled permeability mean covariance are derived at first order in the log permeability variance the detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results the upscaled permeability expected values are components of an axisymmetric tensor whereas the fluctuations are determined by a scalar random space function of anisotropic covariance the general case of upscaling by a first order approximation is examined in appendix b these results will be applied to a few particular cases in part 3 indelman this issue

detecting differentially expressed genes in microarrays using bayesian model selection dna microarrays open up a broad new horizon for investigators interested in studying the genetic determinants of disease the high throughput nature of these arrays where differential expression for thousands of genes can be measured simultaneously creates an enormous wealth of information but also poses a challenge for data analysis because of the large multiple testing problem involved the solution has generally been to focus on optimizing false discovery rates while sacrificing power the drawback of this approach is that more subtle expression differences will be missed that might give investigators more insight into the genetic environment necessary for a disease process to take hold we introduce a new method for detecting differentially expressed genes based on a high dimensional model selection technique bayesian anova for microarrays bam which strikes a balance between false rejections and false nonrejections the basis of the new approach involves a weighted average of generalized ridge

representation of grid block permeability in coarse scale models of randomly heterogeneous porous media the two basic approaches for the assignment of coarse scale grid block permeabilities from highly detailed permeability descriptions are sampling where the permeability of the coarse scale grid block is simply specified as the fine scale permeability evaluated at the grid block center no averaging is performed and scaleup or homogenization where the coarse scale permeability is determined through some procedure which appropriately averages the fine scale data the intent of this paper is to quantitatively compare these two approaches through extensive numerical calculations for the modeling of single phase flow through two dimensional realizations of randomly heterogeneous porous media with heterogeneity quantified through the correlation length and the variance of the log of permeability 2 several different flow scenarios and two scaleup techniques local and global are considered over a range of l and it is seen that for flows driven by smoothly varying boundary conditions a sampling procedure is more accurate than global scaleup when the length scale of a typical grid block is less than l independent of for flows driven by source terms by contrast more resolution is required for a sampling approach to be preferable to global scaleup particularly at high variances local scaleup is shown to be preferable to either sampling or global scaleup in essentially all cases though its use for more general permeability fields requires additional numerical calculations

spike and slab gene selection for multigroup microarray data dna microarrays can provide insight into genetic changes that characterize different stages of a disease process accurate identification of these changes has significant therapeutic and diagnostic implications statistical analysis for multistage multigroup data is challenging however anova based extensions of two sample z tests a popular method for detecting differentially expressed genes in two groups do not work well in multigroup settings false detection rates are high because of variability of the ordinary least squares estimators and because of regression to the mean induced by correlated parameter estimates we develop a bayesian rescaled spike and slab hierarchical model specifically designed for the multigroup gene detection problem data preprocessing steps are introduced to deal with unique features of microarray data and to enhance selection performance we show theoretically that spike and slab models naturally encourage sparse solutions through a process called selective shrinkage t

minimax lq exact and inexact model matching problems abstract a mixed h 2 h optimal problem is presented for both disturbance attenuation and model mismatching minimization its solution is in the form of a two degrees of freedom controller whose feedback and feedforward paths can be independently designed if the conditions for exact model matching are not met an inexact model matching problem is addressed this solution provides an effective model matching design methodology for non minimum phase plants

spike and slab variable selection frequentist and bayesian strategies variable selection in the linear regression model takes many apparent faces from both frequentist and bayesian standpoints in this paper we introduce a variable selection method referred to as a rescaled spike and slab model we study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation specifically we study the usefulness of continuous bimodal priors to model hypervariance parameters and the effect scaling has on the posterior mean through its relationship to penalization several model selection strategies some frequentist and some bayesian in nature are developed and studied theoretically we demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification and show this is achieved using the posterior from a rescaled spike and slab model we also show how to verify a procedure s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy using this tool we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty

a polynomial solution to the generalized h control problem with measurement noise the paper deals with the solution of the generalized h gh control problem in the presence of stochastic measurement noise and output disturbance because of the presence of the measurement noise it is found that the problem cannot be solved as in the zero measurement case in a one shot but the so called iteration procedure is required to obtain the solution the problem is approached by means of polynomial methods and the solution is achieved by solving standard polynomial diophantine equations and a single well conditioned eigenvalue vector problem for whose solutions numerically reliable algorithms exist an example of an application is given in order to show the effectiveness of the g h strategy in the case of very highly coloured measurement noise with control and noise bandwidths overlapped

adapting geostatistical transmissivity simulations to finite difference flow simulators the validity of hydrogeological studies is often dependant on the correct estimation of the flux or equivalently the equivalent transmissivity of the field however depending on the level of fine scale heterogeneity of the porous media considered flow simulators based on the finite differences algorithm can be shown to underestimate the equivalent transmissivity this result is shown to depend on the value on the intermesh transmissivities and not the mesh transmissivities when working with geostatistically simulated point support transmissivities intermesh transmissivities can be directly calculated so that the finite differences result obtained is consistent with the spatial transmissivity distribution

high dimensional sparse factor modeling applications in gene expression genomics we describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data we discuss breast cancer applications and key aspects of the modeling and computational methodology our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways as well as links between aggregate patterns in gene expression profiles and clinical biomarkers based on the metaphor of statistically derived factors as representing biological subpathway structure we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity our methodology is based on sparsity modeling of multivariate regression anova and latent factor models as well as a class of models that combines all components hierarchical sparsity priors address questions of dimension reduction and multiple co

a fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media fast upscaling of hydraulic conductivity is a recurrent problem in modeling flow through heterogeneous porous media we propose a new renormalization technique it is based on the iterative application of the cardwell and parsons 1945 bounds on elementary groups of cells the combination of the bounds with a heuristic formula allows anisotropy to be taken into account the new technique is tested and compared with other fast techniques among the tested techniques the two most reliable ones are the tensorial renormalization and the new simplified renormalization the numerical efficiency of the simplified renormalization leads us to recommend it when a diagonal tensor of equivalent conductivity is sufficient

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

statistics of block conductivity through a simple bounded stochastic medium a monte carlo approach is employed to estimate the distribution of an equivalent conductivity measure the block conductivity which characterizes the total flow rate through a two dimensional bounded domain and which is itself a random variable chi square goodness of fit tests indicate that the lognormal distribution is an appropriate choice for the block conductivity distribution for square domains the mean and variance of the block conductivity are seen to be closely approximated using the statistics of local averages of log conductivity

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

upscaling transport with mass transfer models mean behavior and propagation of uncertainty 1 the choice of an adequate large scale conceptual transport model constitutes a major challenge associated with the upscaling of solute transport among the different alternatives to the classical advection dispersion model the multirate mass transfer model has been proposed as a valuable and convenient alternative to model the large scale behavior of solute transport this paper evaluates the use of mass transfer models as a constitutive equation for upscaling solute transport to achieve this we compare monte carlo simulations of solute transport at two different support scales transport simulations performed at the smallest scale represent a set of reference transport solutions described at a high resolution which are contrasted against transport simulations obtained using an upscaled model low resolution several formulations of the multirate mass transfer model which differ in the type of memory function single rate double rate and truncated power law are used as a constitutive transport equation the large scale scenario represents a simplified model obtained by partially homogenizing the reference solution results show that the double rate and the truncated power law mass transfer models are capable of properly describing the ensemble average behavior of the main features associated with the integrated breakthrough curves however the uncertainty associated with the upscaled mass transfer models was substantially smaller than that attributed to the reference solution importantly the cumulative distribution function of concentrations associated with the upscaled model follows a distribution similar to the reference solution but with smaller statistical dispersion the reason is that while appropriate memory functions can be used to preserve the residence time distribution of mass particles during upscaling the lack of memory in space prevents the model from reproducing mass fluxes in all directions specifically the reproduction of mass fluxes taking place at the interface between two homogenized blocks of the upscaled model is not satisfied thus providing a poor description of the spatial distribution of mass particles in a given realization

transmissivity upscaling in numerical models of steady aquifer flow conditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t for spatially varying t and by the stochastic approach t is modeled as a random field typically both the numerical element scale r and the log transmissivity integral scale i are of order of hundreds to thousands meters consequently the upscaled block transmissivity is random and its statistical moments depend on those of the fine scale t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we have derived dagan and lessoff water resources research 43 w05431 doi 10 1029 2006wr005235 2007 the unconditional statistics of the upscaled ln accurate to the first order in y2 the log transmissivity variance both cases of mean uniform flow solved previously by indelman and dagan transport in porous media 12 161 183 1993 and of strongly nonuniform flow in a circular block of radius r centered at a well of radius rw were considered such upscaling applies either to elements in regions of natural gradient flow or those surrounding a projected well respectively which are sufficiently far from points of measurement of t the present article extends the analysis to account for the presence of a measurement removed in the center of the element e g result of a pumping test the fine scale conditional log transmissivity yc is modeled as multi gaussian of nonstationary mean and of two point covariance which depend on the measured value and distance from the measurement location upscaling in mean uniform flow is related for instance to the case of a pumping well which is not operative but served to determine t while natural gradient flow conditions prevail in the area it was found that upscaling has a significant effect upon the mean upscaled log transmissivity that changes from the measured value for r i 1 to the effective one y ln tg for r i 6 similarly the conditional log transmissivity variance is greatly reduced compared to the unconditional case it tends to zero for r i 0 or r i and it reaches a maximal value of around 0 15 y2 for r i 1 the second case of upscaling in well flow applies to an element with an operative well at its center which was used to determine the transmissivity tw unlike mean uniform flow it is found that upscaling has a very small effect for the realistic values of rw i 0 10 3 and r i 5 in this case the upscaled transmissivity is practically deterministic and equal to the given tw a summary of the two articles dagan and lessoff 2007 and the present one concludes the paper

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

on the polynomial equations for the continuous time lq optimization problem this paper discusses the role and the relationships among three coupled diophantine polynomial equations involved in the solution of the optimal continuous time lq regulation problem the emphasis is on establishing conditions under which the lq regulator can be obtained via the solution of a single uncoupled diophantine equation this analysis is motivated by the fact that although recent papers have clarified the matter for the discrete time domain some subtle differences exist in the continuous time domain as a consequence the extension of the discrete time results to the continuous time context requires a deeper analysis which is the main aim of this paper

polynomial solution of the continuous time gmv control problem abstract this paper deals with the solution of the generalized minimum variance gmv output tracking control problem for continuous time siso plants the problem is approached by means of the polynomial method and the solution is achieved in the general case by solving three diophantine equations furthermore since the existence of the solution cannot be ensured from the outset and the problem solvability does not ensure the properness of the solution sufficient conditions are given to guarantee the problem solvability and the properness of the optimal controller finally two simple examples clarify the theory and show the effectiveness of the gmv strategy in the case of very highly coloured measurement noise acting on the plant with control and noise bandwidths overlapped

saddlepoint approximation for mixture models two component mixture distributions with one component a point mass and the other a continuous density may be used as priors for bayesian inference when sparse representation of an underlying signal is required we show how saddlepoint approximation in such models can yield highly accurate quantiles for posterior distributions and illustrate this numerically using wavelet regression with point mass laplace and point mass normal prior distributions