the turning bands method for simulation of random fields using line generation by a spectral method the turning bands method tbm for the simulation of multidimensional random fields is presented these fields commonly occur in the monte carlo simulation of hydrologic processes particularly groundwater flow and mass transport the general tbm equations for two and three dimensional fields are derived with particular emphasis on the more complicated two dimensional case for stationary two dimensional fields the unidimensional line process is generated by a simple spectral method a technique which can be generally applied to any two dimensional covariance function and which is easily extended to anisotropic and areal averaged processes theoretically and by example the tbm is shown to be ergodic even for a finite number of lines and it is demonstrated that it rapidly converges to the true statistics of the field guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments the tbm is compared to other methods in terms of cost and accuracy demonstrating that the tbm is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion such as the nearest neighbor approach the unidimensional spectral technique presented here permits for the first time the inexpensive and accurate tbm simulation of any proper two dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes

on the equations ax q and sx xt q abstract in this paper we first study solutions of the functional equation ax q where a is the generator of a certain semigroup using mean ergodic theory then we apply the result to study the solvability of the operator equation sx xt q where s and t are generators of some semigroups of operators we give a characterization of the solvability using weak operator limit of an integral involving the tensor product semigroup of the two semigroups generated respectively by s and t

stochastic modeling of large scale transient unsaturated flow systems a new framework for modeling large scale transient unsaturated flow systems in spatially variable soils is proposed in order to overcome the problem of limited information about the local details of spatial soil variability a stochastic approach which assumes that local soil properties are realizations of three dimensional random fields is followed for derivation of a large scale model representation structure the three dimensionality of the local flow and the nonlinear dependence of the local flow output on the local soil properties are considered the large scale model structure is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields the resulting mean model representation is in the form of a partial differential equation in which averaged or effective model parameters occur these effective model parameters i e effective hydraulic conductivity and effective specific moisture capacity are evaluated using a quasi linearized fluctuation equation and a spectral representation of stationary processes the large scale model representation considers the large scale effects of soil property variability and has relatively few parameters the general conclusions of this study are that soil property variability produces large scale hysteresis and anisotropy of the effective parameters the potential theoretical and practical ramifications of these results in the area of unsaturated flow modeling need to be investigated the general stochastic modeling framework developed here is applicable not only to unsaturated flow but also to other distributed parameter systems e g saturated flow and transport geothermal and oil reservoir modeling

stochastic analysis of one dimensional steady state unsaturated flow a comparison of monte carlo and perturbation methods the stochastic nature of moisture content in a soil profile under steady state unsaturated infiltration is examined where the saturated hydraulic conductivity is taken as a stationary stochastic process two different techniques are employed in determining the stochastic output these are monte carlo simulations and an analytic derivation from a first order perturbation solution in comparing the two techniques the liability of the perturbation expressions is investigated if the monte carlo results are assumed to exactly represent the stochastic nature of the moisture content in four test examples a relatively good comparison is obtained between the results of the two techniques the analytic nature of the perturbation solution then readily provides information concerning the general stochastic properties of the moisture content and in this way additional monte carlo runs are avoided from this analysis it is shown that the moisture content is a stationary stochastic process at distances far from boundary regions the distance from the boundary to regions of stationarity is dependent upon soil properties and the boundary conditions imposed

on a polynomial scalar perturbation of a schrodinger system in lp spaces abstract in the paper 10 the l p realization l p of the matrix schrodinger operator l u d i v q u v u was studied the generation of a semigroup in l p r d c m and characterization of the domain d l p has been established in this paper we perturb the operator l p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on l p r d c m with domain embedded in w 2 p r d c m we also study the analyticity compactness positivity and ultracontractivity of the semigroup and prove gaussian kernel estimates further kernel estimates and asymptotic behaviour of eigenvalues of the matrix schrodinger operator are investigated

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

a stochastic analysis of steady state two phase flow in heterogeneous media 1 we present a novel approach to modeling stochastic multiphase flow problems for example nonaqueous phase liquid flow in a heterogeneous subsurface medium with random soil properties in particular with randomly heterogeneous intrinsic permeability and soil pore size distribution a stochastic numerical model for steady state water oil flow in a random soil property field is developed using the karhunen loeve moment equation klme approach and is numerically implemented an exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure the log transformed intrinsic permeability y x and soil pore size distribution b x are assumed to be gaussian random functions with a separable exponential covariance function the perturbation part of these two log transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables xn the phase pressure capillary pressure and phase mobility are decomposed by polynomial expansions and the perturbation method combining these expansions of y x b x and dependent pressures the steady state water oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order these differential equations are solved numerically and the solutions are directly used to construct moments of phase pressure and capillary pressure we demonstrate the validity of the proposed klme model by favorably comparing firstand second order approximations to monte carlo simulations the significant computational efficiency of the klme approach over monte carlo simulation is also illustrated

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

kernel and eigenfunction estimates for some second order elliptic operators abstract for a potential v such that v x x with 2 we prove that the heat kernel k t x y associated to the uniformly elliptic operator a j k 1 n k a j k j v satisfies the estimate k t x y c e 0 t e c t b e 2 2 x 1 2 x 4 n 1 2 e 2 2 y 1 2 y 4 n 1 2 for large x y r n and all t 0 here 0 1 is an appropriate constant b 2 2 and 0 is the first eigenvalue of a we also obtain an estimate for large x of the eigenfunctions of a

on w continuous cosine operator functions on donne les proprietes fondamentales des theoremes de comparaison infinitesimaux et des theoremes ergodiques

internal model theory for distributed parameter systems in this paper we consider robust output regulation of distributed parameter systems and the internal model principle the main purpose is to generalize the internal model principle by francis and wonham for infinite dimensional systems and clarify the relationships between different generalizations of the internal model we also construct a signal generator capable of generating infinite dimensional polynomially increasing signals

stochastic analysis of three phase flow in heterogeneous porous media 1 spectral perturbation approach stochastic analysis of steady state three phase water oil and air flow in heterogeneous porous media was performed using the perturbation theory and spectral representation technique the governing equations describing flow of water oil and air are coupled and nonlinear the key stochastic input variables are intrinsic permeability k and the soil grain size distribution index three different stochastic combinations of these two input parameters were considered in the analysis the spectral perturbation approach was used to develop closed form expressions to describe stochastic variability of key output processes such as capillary and individual phase pressures the second paper of this two part series abdin and kaluarachchi this issue will discuss the applicability and the validity of the spectral perturbation technique to study the stochastic behavior of three phase flow using numerical simulations

a stochastic analysis of transient two phase flow in heterogeneous porous media 1 the karhunen loeve moment equation klme approach is implemented to model stochastic transient water napl two phase flow in heterogeneous subsurface media with random soil properties to describe the constitutive relationships between water saturation capillary pressure and phase relative permeability the widely used van genuchten model and parker and lenhard models are adopted the log transformed intrinsic permeability soil pore size distribution and van genuchten fitting parameter n are treated as normally distributed stochastic variables with a separable exponential covariance model the perturbation part of these three log transformed variables is decomposed via karhunen loeve expansion the dependent variables phase pressure phase mobility and capillary pressure are expanded by polynomial expansions and the perturbation method incorporating these expansions of random soil properties variables and dependent variables into the governing equations yields a series of differential equations in different orders we construct the moments of the dependent variables from the solutions of these differential equations we demonstrate the stochastic model with two dimensional examples of transient two phase flow we also conduct monte carlo simulations using the finite element heat and mass fehm transfer code whose results are considered true solutions the match between the results from fehm and klme indicates the validity of the proposed klme application in transient two phase flow the computational efficiency of the klme approach over monte carlo methods is at least an order of magnitude for transient two phase flow problems

on the range of the generator of a markovian semigroup on considere des semigroupes bornes fortement continus t t t 0 d operateurs lineaires bornes dans un espace de banach

stochastic analysis of unsaturated flow one dimensional monte carlo simulations and comparisons with spectral perturbation analysis and field observations a numerical experiment was designed to study the stochastic behavior of one dimensional transient unsaturated flow in a monte carlo setting soil hydraulic properties log saturated hydraulic conductivity ln ks pore size distribution parameter and the specific water capacity c are assumed to be statistically homogeneous random fields described by exponential correlation functions with identical correlation lengths fifty realizations of each soil hydraulic property with statistical properties obtained from a field experiment are generated by using a nearest neighbor model numerical solutions of the one dimensional flow equation are used repeatedly to obtain realizations of soil water pressure head and flux q corresponding to the realizations of in ks and c the dependence of and q on the statistical properties of ln ks and c and the magnitude of the hydraulic head gradient g prevailing at the lower boundary are investigated in addition results of monte carlo analysis and spectral perturbation analyses are compared with field observations the greatest variability of and q are observed when soil hydraulic properties are uncorrelated and have large variances and integral scales and when g at the lower boundary is unity the monte carlo and spectral perturbation analyses tend to agree reasonably well for the flow domains in which ergodicity of local soil hydraulic properties is assured results of the monte carlo and spectral perturbation analyses are also supported by field observations

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

weakly integrable semigroups on locally convex spaces abstract a class of weakly integrable semigroups on locally convex spaces is introduced and studied the results are illustrated by examples of semigroups of unbounded operators on a banach space which include fractional powers of a closed operator and spectral local semigroups

mean ergodic theorems and linear functional equations let a be a closed linear operator with domain d a and range r a in a banach space x let ax and b be two nets of bounded operators on x such that a m r a r b d a and b a ab a i w lim aa x 0 for all x x and s lim a ax 0 for all x d a we first formulate an abstract mean ergodic theorem which contains as special cases many known or new mean ergodic theorems the theorem is then used to study the approximate solutions of the functional equation ax y one of the results says that if y a d a r a then b y converges to the unique solution x in r a the general results provide a unified approach to many specific methods for solving ax y the cases treated here include 1 a i t with t b x 2 a is the generator of a y semigroup 3 a is the generator of a strongly continuous cosine operator function 4 the operator equation a1e ea2 b

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

ergodic properties of operator semigroups in general weak topologies abstract the aim of this paper is to study ergodic properties i e properties about the limit of cesaro averages of a semigroup of bounded linear operators on a banach space x which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of x then the results are applied to particular examples such as locally strongly integrable semigroups their dual semigroups and the tensor product semigroup of two c 0 semigroups

stochastic analysis of three phase flow in heterogeneous porous media 2 numerical simulations the first paper of this two part series abdin and kaluarachchi this issue developed a spectral perturbation analysis to predict the variability of individual and capillary pressures of three phase water oil and air flow in heterogeneous porous media the work in this manuscript is an extension of that work to validate the applicability of the perturbation analysis and to study the stochastic behavior of three phase flow the input stochastic variables were soil grain size distribution index and log k where k is the permeability in this work a one dimensional numerical solution was developed to solve the three coupled nonlinear flow equations the results using three different combinations of input stochastic variables showed good agreement between perturbation and numerical analyses over a wide range of flow conditions in addition the results showed a large influence of subsurface heterogeneity on the predicted capillary pressure variability in general this variability was highest with the uncorrelated and log k case followed by deterministic stochastic log k and correlated cases the results suggest that subsurface heterogeneity is important in predicting three phase flow in heterogeneous porous media finally the results of this work can be used as a preliminary step for stochastic analysis of three fluid phase flow