first order variance of travel time in nonstationary formations 1 evaluating uncertainty in solute transport under nonstationary flow conditions is a computationally demanding task this is particularly true for cases with a two point covariance function of log conductivity depending on the actual positions of the points rather than their distance vector these may occur when the geological formation exhibits a trend nonstationarity can also be the result of uncertainty in the trend parameters of the mean log conductivity value or it may originate from conditioning of the log conductivity field to measurements of for example head or conductivity we present an efficient numerical method for evaluating the variance of travel time in such formations we cover cases in which the nonstationary covariance functions are constructed from stationary counterparts either by scaling functions or by summation with nonstationary functions resulting from marginalization or conditioning we apply a matrix based first order second moment fosm method for uncertainty propagation using the continuous adjoint state method for coupled systems to evaluate the sensitivity matrix the resulting matrix matrix multiplications are accelerated by fast fourier transformation fft techniques after periodic embedding of the covariance matrices referring to the stationary counterparts the combination of these methods makes it possible to compute the travel time uncertainty in domains discretized by several hundred thousand log conductivity values on standard personal computers within a reasonable time frame for demonstration we apply the method to a binary medium and a medium exhibiting a continuous trend in the covariance function in the latter application we also demonstrate the effects of marginalization and conditioning

infinite dimensional features of matrices and pseudospectra abstract given a hilbert space operator t the level sets of function t z t z 1 1 determine the so called pseudospectra of t we set t to be zero on the spectrum of t after giving some elementary properties of t which as it seems were not noticed before we apply them to the study of the approximation we prove that for any operator t there is a sequence t n of finite matrices such that t n z tends to t z uniformly on c in this proof quasitriangular operators play a special role this is merely an existence result we do not give a concrete construction of this sequence of matrices one of our main points is to show how to use infinite dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size in particular we get a result which means in a sense that the pseudospectrum of a nilpotent matrix can be anything one can imagine we also study the norms of the multipliers in the context of cowen douglas class operators we use these results to show that to the opposite to the function s the function s z for certain finite matrices s may oscillate arbitrarily fast even far away from the spectrum

truncations of a class of pseudo hermitian tridiagonal matrices abstract we consider the class of non hermitian operators represented by infinite tridiagonal matrices selfadjoint in an indefinite inner product space with one negative square we approximate them with their finite truncations both infinite and truncated matrices have eigenvalues of nonpositive type either a single one on the real axis or a couple of complex conjugate ones as a tool to evaluate the reliability of the use of truncations in numerical simulations we give bounds for the rate of convergence of their eigenvalues of nonpositive type numerical examples illustrate our results

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

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

minimum relative entropy forward probabilistic modeling the pioneering work of jaynes in bayesian maximum entropy methods has been successfully explored in many disciplines the principle of maximum entropy pme is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution minimum relative entropy mre is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included in this paper we use mre to determine the prior probability density function pdf of a set of model parameters based on limited information the resulting pdf is used in monte carlo simulations to provide expected values in field variables such as concentration and confidence limits we compare the probabilistic results from a traditional advection dispersion ade model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process the results suggest that although naff s 1990 model satisfies the observed data to a better degree than ade model the upper and lower confidence bands about the mean value are larger than the ade results this result we attribute to the fact that naff s 1990 model simply contains more parameters each of which is unknown and has to be estimated there is no statistical difference between the expected values of second spatial moments for the two models the examples presented in this paper illustrate problems associated with assigning gaussian pdfs as priors in a probabilistic model first such an assumption for the input parameters actually injects more information into the problem than may actually exist whether consciously or unconsciously this fact is borne out by comparison with minimum relative entropy theory second the output parameters as suggested from the monte carlo analysis cannot be assumed to be gaussian distributed even when the prior pdf is gaussian in form in a practical setting the significance of this result and the approximation of gaussian form would depend on the toxicity and environmental standards that apply to the problem

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

spectral value sets of closed linear operators we study how the spectrum of a closed linear operator on a complex banach space changes under affine perturbations of the form a a a d e here a d and e are given linear operators whereas is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation d e the union of the spectra of the perturbed operators a with the norm of smaller than a given 0 is called the spectral value set of a at level in this paper we extend a known characterization of these sets for the matrix case to infinite dimensions and in so doing present a framework that allows for unbounded perturbations of closed linear operators on banach spaces the results will be illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition

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

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

quasi diagonality and the finite section method quasi diagonal operators on a hilbert space are a large and important class containing all self adjoint operators for instance they are also perfectly suited for study via the finite section method a particular galerkin method indeed the very definition of quasi diagonality yields finite sections with good convergence properties moreover simple operator theory techniques yield estimates on certain rates of convergence in the case of quasi diagonal band operators both the finite sections and rates of convergence are explicitly given

maximum of the resolvent over matrices with given spectrum abstract in numerical analysis it is often necessary to estimate the condition number c n t t t 1 and the norm of the resolvent t 1 of a given n n matrix t we derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds we recover the fact that the supremum of c n t over all matrices with t 1 and minimal absolute eigenvalue r min t 0 is the kronecker bound 1 r n this result is subsequently generalized by computing for given in the closed unit disc the supremum of t 1 where t 1 and the spectrum t of t is constrained to remain at a pseudo hyperbolic distance of at least r 0 1 around we find that the supremum is attained by a triangular toeplitz matrix this provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically the occurring toeplitz matrices are so called model matrices i e matrix representations of the compressed backward shift operator on the hardy space h 2 to a finite dimensional invariant subspace

bayesian geostatistical design task driven optimal site investigation when the geostatistical model is uncertain 1 geostatistical optimal design optimizes subsurface exploration for maximum information toward task specific prediction goals until recently most geostatistical design studies have assumed that the geostatistical description i e the mean trends covariance models and their parameters is given a priori this contradicts as emphasized by rubin and dagan 1987a the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort we believe that geostatistical design should 1 avoid unjustified a priori assumptions on the geostatistical description 2 instead reduce geostatistical model uncertainty as secondary design objective 3 rate this secondary objective optimal for the overall prediction goal and 4 be robust even under inaccurate geostatistical assumptions bayesian geostatistical design follows these guidelines by considering uncertain covariance model parameters we transfer this concept from kriging like applications to geostatistical inverse problems we also deem it inappropriate to consider parametric uncertainty only within a single covariance model the matern family of covariance functions has an additional shape parameter controlling model shape by a parameter converts covariance model selection to parameter identification and resembles bayesian model averaging over a continuous spectrum of covariance models this is appealing since it generalizes bayesian model averaging from a finite number to an infinite number of models we illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases the underlying scenarios are to minimize the prediction variance of 1 contaminant concentration or 2 arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function

towards a coherent philosophy for modelling the environment the predominant philosophy underlying most environmental modelling is a form of pragmatic realism the limitations of this approach in practical applications are discussed in particular in relati

a full bayesian approach to parameter inference from tracer travel time moments and investigation of scale effects at the cape cod experimental site a method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer the approach combines a lagrangian rubin and dagan 1992 solute transport framework with full bayesian hydrogeological parameter inference in the full bayesian approach the noise values in the observed data are treated as hyperparameters and their effects are removed by marginalization the prior probability density functions pdfs for the model parameters horizontal integral scale velocity and log k variance and noise values are represented by prior pdfs developed from minimum relative entropy considerations analysis of the cape cod massachusetts field experiment is presented inverse results for the hydraulic parameters indicate an expected value for the velocity variance of log hydraulic conductivity and horizontal integral scale of 0 42 m d 0 26 and 3 0 m respectively while these results are consistent with various direct field determinations the importance of the findings is in the reduction of confidence range about the various expected values on selected control planes we compare observed travel time frequency histograms with the theoretical pdf conditioned on the observed travel time moments we observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows we also test the hypothesis that there is no scale dependence of the integral scale with the scale of the experiment at cape cod we adopt two strategies the first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source the second approach is from the viewpoint of entropy concentration no increase in integral scale with distance is inferred from either approach over the range of the cape cod tracer experiment

on n ϵ pseudospectra of operators on banach spaces abstract in this paper we extend the concept of the n pseudospectra of hansen to the case of bounded linear operators on banach spaces and prove several relations to the usual spectrum we particularly discuss the approximation by rectangular finite sections and the impact of the fundamental result of shargorodsky on jumping pseudospectra

a note on head and velocity covariances in three dimensional flow through heterogeneous anisotropic porous media a procedure for the characterization of the head and velocity fields for saturated flow in heterogeneous statistically anisotropic formations is developed the fields are characterized through a series of spatial covariances as well as the velocity head velocity log conductivity and head log conductivity cross covariances all these functions are given in terms of the parameters which characterize the input log conductivity field and the average head garadient a general procedure is outlined followed by explicit derivation of head and velocity covariances for heterogeneous statistically anisotropic formations closed form analytic results are obtained for the statistically isotropic formations while the results for the anisotropic ones required taking quadrature over analytic expressions the results may be used for transport predictions and for conditioning on measurements and may be particularly useful for testing of numerical codes

continuous groundwater velocity fields and path lines in linear bilinear and trilinear finite elements darcy velocity fields obtained from finite element solutions of heads in groundwater flow exhibit discontinuities at element boundaries thus giving rise to inaccuracies in path line construction several methods have been developed to resolve those inaccuracies including global postprocessing of velocity fields mixed hybrid finite element formulations and stream function formulations all of these techniques either lead to a considerable increase in the computational effort or are not general enough for all purposes the point of view taken here is that standard finite element flow models yield exact water balances which in the usual approach to path line computation are not fully used by starting out from patches with flux conserving boundaries a new postprocessing procedure is derived that allows construction of a continuous flux distribution over the whole model domain in this procedure only the head gradients of adjacent elements are used thus avoiding the solution of another global system of equations furthermore application of a stream function interpolation inside each patch allows the fast computation of path lines without any time stepping the path lines computed with the proposed procedure are compared to path lines computed by usual particle tracking and examples are presented showing that the new method is vastly superior to traditional particle tracking

probabilistic collocation method for flow in porous media comparisons with other stochastic methods 1 an efficient method for uncertainty analysis of flow in random porous media is explored in this study on the basis of combination of karhunen loeve expansion and probabilistic collocation method pcm the random log transformed hydraulic conductivity field is represented by the karhunen loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points the procedure is straightforward and analogous to the monte carlo method but the number of simulations required in pcm is significantly reduced steady state flows in saturated random porous media are simulated with the probabilistic collocation method and comparisons are made with other stochastic methods monte carlo method the traditional polynomial chaos expansion pce approach based on galerkin scheme and the moment equation approach based on karhunen loeve expansion klme this study reveals that pcm and klme are more efficient than the galerkin pce approach while the computational efforts are greatly reduced compared to the direct sampling monte carlo method the pcm and klme approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head

homogenization of richards equation in permeability fields with different connectivities 1 large scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone upscaled flow models need to capture the impact of small scale heterogeneities which are not resolved by the model on large scale flow we perform upscaling of the richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques we restrict the analysis to flow regimes in which the capillary equilibrium assumption holds on the small scale in order to account for effects of capillary entry pressure we apply the brooks corey model for the soil retention and relative permeability curves and consider leverett scaling for the coupling of intrinsic permeability and entry pressure for this model we derive and analyze the ensemble averaged parameter functions for the macroscopic flow equations the effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability we compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties these results illustrate that soils with well connected coarse materials differ in the relative permeability from those with well connected fine materials or those without particular connectedness

conditional estimation of solute travel time in heterogeneous formations impact of transmissivity measurements solute transport through heterogeneous formations is modeled by the travel time approach i e the time it takes a solute particle to travel from the source to a control plane due to uncertainty is a random variable characterized by its cumulative probability density function g this function is used by regulatory agencies in pollution site assessment the main aim of the present study is to examine the impact of data namely measured transmissivities upon g and reduction of uncertainty this is achieved for two dimensional steady flow of average uniform head gradient by using a lagrangian approach developed in the past dagan 1982 1984 1989 rubin 1990 1991 the impact of data is seen in conditioning on measured values and in estimation of parameters characterizing flow and formation properties the approach is illustrated by a simulation based on synthetic data the results show how uncertainty is reduced by increasing the density of measurements for a control plane sufficiently far from the source

stochastic finite element method for probabilistic analysis of flow and transport in a three dimensional heterogeneous porous formation 1 a probabilistic study is attempted to analyze the flow and transport in a three dimensional 3 d porous formation where the governing parameters are varying randomly in space it is assumed that the soil parameters namely hydraulic conductivity dispersivity molecular diffusion porosity sorption coefficient and decay rate are random fields a stochastic finite element method sfem which is based on perturbation technique is developed the method developed here uses an alternate approach for obtaining improved computational efficiency the derivatives of the concentration with respect to random parameters are obtained by using the derivatives of local matrices instead of global matrices this approach increases the computational efficiency of the present method by several orders with respect to standard sfem both accuracy and computational efficiency of this method are compared with that of commonly used monte carlo simulation method mcsm it is observed that for moderate values of coefficient of variations of the random parameters the mean and standard deviation match reasonably well with mcsm results using this method the excessive computational effort required by mcsm can be avoided in the present study both 1 d as well as 3 d problems are solved to show the advantages of sfem over mcsm the correlation scale of the random field is found to be an important parameter for the range of this parameter studied here it is found that as correlation scale increases the standard deviation increases the results obtained for two particular problems in this study show that the coefficient of variation of concentration is higher for the 1 d problem than the 3 d problem

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

best unbiased ensemble linearization and the quasi linear kalman ensemble generator 1 linearized representations of the stochastic groundwater flow and transport equations have been heavily used in hydrogeology e g for geostatistical inversion or generating conditional realizations the respective linearizations are commonly defined via jacobians numerical sensitivity matrices this study will show that jacobian based linearizations are biased with nonminimal error variance in the ensemble sense an alternative linearization approach will be derived from the principles of unbiasedness and minimum error variance the resulting paradigm prefers empirical cross covariances from monte carlo analyses over those from linearized error propagation and points toward methods like ensemble kalman filters enkfs unlike conditional simulation in geostatistical applications enkfs condition transient state variables rather than geostatistical parameter fields recently modifications toward geostatistical applications have been tested and used this study completes the transformation of enkfs to geostatistical conditioning tools on the basis of best unbiased ensemble linearization to distinguish it from the original enkf the new method is called the kalman ensemble generator keg the new context of best unbiased ensemble linearization provides an additional theoretical foundation to enkf like methods such as the keg like enkfs and derivates the keg is optimal for gaussian variables toward increased robustness and accuracy in non gaussian and nonlinear cases sequential updating acceptance rejection sampling successive linearization and a levenberg marquardt formalism are added state variables are updated through simulation with updated parameters always guaranteeing the physicalness of all state variables the keg combines the computational efficiency of linearized methods with the robustness of enkfs and accuracy of expensive realization based methods while drawing on the advantages of conditional simulation over conditional estimation such as adequate representation of solute dispersion as proof of concept a large scale numerical test case with 200 synthetic sets of flow and tracer data is conducted and analyzed

explicit upper bounds for the spectral distance of two trace class operators abstract given two trace class operators a and b on a separable hilbert space we provide an upper bound for the hausdorff distance of their spectra involving only the distance of a and b in operator norm and the singular values of a and b by specifying particular asymptotics of the singular values our bound reproduces or improves existing bounds for the spectral distance the proof is based on lower and upper bounds for determinants of trace class operators of independent interest

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

a hypothesis driven approach to optimize field campaigns 1 most field campaigns aim at helping in specified scientific or practical tasks such as modeling prediction optimization or management often these tasks involve binary decisions or seek answers to yes no questions under uncertainty e g is a model adequate will contamination exceed a critical level in this context the information needs of hydro geo logical modeling should be satisfied with efficient and rational field campaigns e g because budgets are limited we propose a new framework to optimize field campaigns that defines the quest for defensible decisions as the ultimate goal the key steps are to formulate yes no questions under uncertainty as bayesian hypothesis tests and then use the expected failure probability of hypothesis testing as objective function our formalism is unique in that it optimizes field campaigns for maximum confidence in decisions on model choice binary engineering or management decisions or questions concerning compliance with environmental performance metrics it is goal oriented recognizing that different models questions or metrics deserve different treatment we use a formal bayesian scheme called predia which is free of linearization and can handle arbitrary data types scientific tasks and sources of uncertainty e g conceptual physical geo statistical measurement errors this reduces the bias due to possibly subjective assumptions prior to data collection and improves the chances of successful field campaigns even under conditions of model uncertainty we illustrate our approach on two instructive examples from stochastic hydrogeology with increasing complexity

probability density functions of hydraulic head and velocity in three dimensional heterogeneous porous media 1 in this study we assess probability density functions of hydraulic heads and specific discharges in three dimensional bounded heterogeneous porous media by monte carlo mc simulation we discuss their empirical shapes and demonstrate that the intuitive use of obvious information on boundedness leads to parametric distribution functions which fit surprisingly well on the basis of statistical moments of hydraulic heads and velocities up to fourth order we discuss the spatial dependence of the empirical distributions and their dependence on the variance of log conductivity comparison of the first and second central moment to the results from classical numerical first order second moment fosm analysis reveals that fosm predicts these moments surprisingly close for hydraulic heads on the basis of this fact we demonstrate that fitting the chosen parametric distributions for hydraulic heads to fosm moments is promising for the sake of estimating exceedance probabilities our mc scenarios vary in variance of log conductivity 0 125 to 5 0 in the type of multivariate dependence in correlation scale and types of boundary conditions our study illustrates that in contrast to the common assumption fosm is a reasonable choice for evaluating multivariate and univariate moments for heads if used in conjunction with additional information on distribution shapes in the absence of utilizable additional information we demonstrate that second moment methods are mostly inadequate for assessing distributions accurately significant deviations from gaussian distributions occurred for discharge components even at a variance of log conductivity as low as 0 125 and we found that the distributions of transverse discharge components are extremely fat tailed the observed non gaussianity questions the results of approximate approaches in solute flux and dispersion studies where velocity fields are assumed to be multi gaussian and then directly represented by or generated from their covariances the main implication is to apply more accurate schemes such as exact non local methods extensive mc or higher order stochastic galerkin approaches and to include higher order moments at least if no additional assumptions on the shape of distributions are available or justifiable

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

c algebras and numerical linear algebra given a self adjoint operator a on a hilbert space suppose that one wishes to compute the spectrum of a numerically in practice these problems often arise in such a way that the matrix of a relative to a natural basis is sparse for example discretized second order differential operators can be represented by doubly infinite tridiagonal matrices in these cases it is easy and natural to compute the eigenvalues of large n n submatrices of the infinite operator matrix and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of a numerical analysts call this the galerkin method while this hope is often realized in practice it often fails as well and it can fail in spectacular ways the sequence of eigenvalue distributions may not converge as n or they may converge to something that has little to do with the original operator a at another level even the meaning of convergence has not been made precise in general in this paper we determine the proper general setting in which one can expect convergence and we describe the asymptotic behavior of the n n eigenvalue distributions in all but the most pathological cases under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of a is recovered from the sequence of eigenvalues of the n n compressions in broader terms our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms ofc algebras indeed it is only when the single operator a is viewed as an element of an appropriate c algebra a that one can see the precise nature of the limit of the n n eigenvalue distributions the limit is associated with a tracial state on a normally a is highly noncommutative and in our main applications it is a simple c algebra having a unique tracial state we obtain precise asymptotic results for operators which represent discretized hamiltonians of one dimensional quantum systems with arbitrary continuous potentials

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d dimensional knapsack problems this study analyzes multiobjective d dimensional knapsack problems mod kp within a comparative analysis of three multiobjective evolutionary algorithms moeas the e nondominated sorted genetic algorithm ii e nsgaii the strength pareto evolutionary algorithm 2 spea2 and the e nondominated hierarchical bayesian optimization algorithm e hboa this study contributes new insights into the challenges posed by correlated instances of the mod kp that better capture the decision interdependencies often present in real world applications a statistical performance analysis of the algorithms uses the unary e indicator the hypervolume indicator and success rate plots to demonstrate their relative effectiveness efficiency and reliability for the mod kp instances analyzed our results indicate that the e hboa achieves superior performance relative to e nsgaii and spea2 with increasing number of objectives number of decisions and correlative linkages between the two performance of the e hboa suggests that probabilistic model building evolutionary algorithms have significant promise for expanding the size and scope of challenging multiobjective problems that can be explored

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

on the solvability complexity index the n pseudospectrum and approximations of spectra of operators we show that it is possible to compute spectra and pseudospectra of linear operators on separable hilbert spaces given their matrix elements the core in the theory is pseudospectral analysis and in particular the n pseudospectrum and the residual pseudospectrum we also introduce a new classification tool for spectral problems namely the solvability complexity index this index is an indicator of the difficultness of different computational spectral problems

strategic placement of localization devices such as pilot points and anchors in inverse modeling schemes 1 localization schemes using devices such as pilot points or anchors localization devices are used in inversion schemes as a tool for gleaning information from measurements that can be used to constrain inversion schemes with the added benefit in the form of a potential for reduction in the number of parameters employed in the inversion scheme this paper proposes and demonstrates a method for strategic placement of the localization devices the method combines stochastic singular value decomposition with parameter rejection methods to produce a map of intensity scores which allows the inverse modeler to place the localization devices at locations defined by their probability for producing informative probabilistic constraints on the statistical distributions of the target variables we also show that the method can be used for selecting measurement locations when designing data acquisition campaigns the method is demonstrated for the case of steady state flow in a heterogeneous conductivity field using mad method of anchored distributions in summary the proposed localization scheme intends to increase the information gleaned from measurements while reducing the associated computational costs associated with stochastic inversion

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

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

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

an optimal bound for the spectral variation of two matrices abstract we establish a bound for the spectral variation of two complex n n matrices a b in terms of a b and a b here denotes the spectral norm it is always better than a bound previously given by bhatia and friedland and it is optimal we describe the set of pairs a b for which the bound is attained

symmetries and conservation laws of two dimensional hydrodynamics abstract we shall briefly review the hamiltonian and lie algebraic structure of 2 d hydrodynamics and its known conservation laws energy enstrophy the moments of vorticity then we utilize the eigenmode expansion of the euler equation to represented it as an infinite ode system and the detailed analysis of the structure coefficients to show that there are no other nontrivial conserved integrals among all linear and quadratic functional of vorticity this result provides a partial justification of the kraichnan energy enstrophy theory of 2 d turbulence

bayesian assessment of the expected data impact on prediction confidence in optimal sampling design 1 incorporating hydro geo logical data such as head and tracer data into stochastic models of subsurface flow and transport helps to reduce prediction uncertainty because of financial limitations for investigation campaigns information needs toward modeling or prediction goals should be satisfied efficiently and rationally optimal design techniques find the best one among a set of investigation strategies they optimize the expected impact of data on prediction confidence or related objectives prior to data collection we introduce a new optimal design method called predia gnosis preposterior data impact assessor predia derives the relevant probability distributions and measures of data utility within a fully bayesian generalized flexible and accurate framework it extends the bootstrap filter bf and related frameworks to optimal design by marginalizing utility measures over the yet unknown data values predia is a strictly formal information processing scheme free of linearizations it works with arbitrary simulation tools provides full flexibility concerning measurement types linear nonlinear direct indirect allows for any desired task driven formulations and can account for various sources of uncertainty e g heterogeneity geostatistical assumptions boundary conditions measurement values model structure uncertainty a large class of model errors via bayesian geostatistics and model averaging existing methods fail to simultaneously provide these crucial advantages which our method buys at relatively higher computational costs we demonstrate the applicability and advantages of predia over conventional linearized methods in a synthetic example of subsurface transport in the example we show that informative data is often invisible for linearized methods that confuse zero correlation with statistical independence hence predia will often lead to substantially better sampling designs finally we extend our example to specifically highlight the consideration of conceptual model uncertainty

a solute flux approach to transport in heterogeneous formations 1 the general framework it is common to represent solute tranport in heterogeneous formations in terms of the resident concentration c x t regarded as a random space function the present study investigates the alternative representation by q the solute mass flux at a point of a control plane normal to the mean flow this representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface a general framework to compute the statistical moments of q and of the associated total solute discharge q and mass m is established with x the direction of the mean flow a solute particle is crossing the control plane at y z and at the travel arrival time the associated expected solute flux value is proportional to the joint probability density function pdf g1 of and whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles in turn the statistical moments of and depend on those of the velocity components through a system of stochastic ordinary differential equations for a steady velocity field and neglecting the effect of pore scale dispersion a major simplification of the problem results in the independence of the random variables and as a consequence the pdf of and can be derived independently of a few approximate approaches to derive the statistical moments of and are outlined these methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of and

estimating structured correlation matrices in smooth gaussian random field models this article considers the estimation of structured correlation matrices in infinitely differentiable gaussian random field models the problem is essentially motivated by the stochastic modeling of smooth deterministic responses in computer experiments in particular the log likelihood function is determined explicitly in closed form and the sieve maximum likelihood estimators are shown to be strongly consistent under mild conditions

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

scales of fluctuation and the propagation of uncertainty in random porous media the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

prediction of velocity statistics in three dimensional multi gaussian hydraulic conductivity fields 1 to study statistics of velocity fields in three dimensional heterogeneous multi gaussian saturated hydraulic conductivity fields and the accuracy of their prediction we performed high resolution monte carlo mc analyses the mc analyses included variances of the log hydraulic conductivity in the range of 0 5 y2 3 0 and anisotropy ratios in the range of 0 017 e 1 the statistics of the velocity fields from the mc analyses are compared with analytical solutions of the first and second order approximations of the stochastic flow equation this paper shows that the second order approximations fit significantly better to the univariate statistics of the darcy velocity from the mc analyses for isotropic cases the second order approximations correspond fairly well to the univariate statistics of the velocity for anisotropic cases the accordance is given only for the mean velocity and the variance of the transverse vertical component of the velocity the mc analyses show that the spatial correlation of the velocity decreases more rapidly with increasing y2 this was more pronounced for the anisotropic than for the isotropic case the negative correlations in absolute terms of the transverse velocity components simultaneously decrease with increasing y2 this is in contrast to the first order approximation of the spatial correlations of the velocity it is assumed that the discrepancies between approximate solutions of the stochastic flow equation and the results of the mc analyses are strongly dependent on the nonnormality of the probability density distributions of the velocity

upscaling for unsaturated flow for non gaussian heterogeneous porous media 1 large scale models of transient flow processes in the unsaturated zone require in general upscaling of the flow problem in order to capture the impact of heterogeneities on a small scale which cannot be resolved by the model effective parameters for the upscaled models are often derived from second order stochastic properties of the parameter fields such properties are good quantifications for parameter fields which are multi gaussian however the structure of soil does rarely resemble these kinds of fields the non multi gaussian field properties can lead to strong discrepancies between predictions of upscaled models and the averaged real flow process in particular the connected paths of parameter ranges of the medium are important features which are usually not taken into account in stochastic approaches they are determined here by the euler number of one cut indicator fields methods to predict effective parameters are needed that incorporate this type of information we discuss different simple and fast approaches for estimating the effective parameter for upscaled models of slow transient flow processes in the unsaturated zone where connected paths of the material may be taken into account upscaled models are derived with the assumption of capillary equilibrium the effective parameters are calculated using effective media approaches we also discuss the limits of the applicability of these methods

high resolution monte carlo simulation of flow and conservative transport in heterogeneous porous media 1 methodology and flow results in this the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via monte carlo methods various aspects of the modelling effort are examined in particular the need to save on core memory causes one to use only specific realizations that have certain initial characteristics in effect these transport simulations are conditioned by these characteristics also the need to independently estimate length scales for the generated fields is discussed the statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x y and z directions finally specific features of the velocity field itself are illuminated in this first paper in particular these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform comparisons are made with first and second order perturbation analyses the mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model finally the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion

on the approximation of spectra of linear operators on hilbert spaces abstract we present several new techniques for approximating spectra of linear operators not necessarily bounded on an infinite dimensional separable hilbert space our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized to an infinite dimensional setting to provide approximations of spectra of elements in a large class of operators we conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n pseudospectrum and argue how that can be used as an approximation to the spectrum

tests for complete k spectral sets abstract let be a family of functions analytic in some neighborhood of a complex domain and let t be a hilbert space operator whose spectrum is contained in our typical result shows that under some extra conditions if the closed unit disc is complete k spectral for t for every then is complete k spectral for t for some constant k in particular we prove that under a geometric transversality condition the intersection of finitely many k spectral sets for t is again k spectral for some k k these theorems generalize and complement results by mascioni stessin stampfli badea beckermann crouzeix and others we also extend to non convex domains a result by putinar and sandberg on the existence of a skew dilation of t to a normal operator with spectrum in as a key tool we use the results from our previous paper 11 on traces of analytic uniform algebras