elements of a function analytic approach to probability we first provide a detailed motivation for using probability theory as a mathematical context in which to analyze engineering and scientific systems that possess uncertainties we then present introductory notes on the function analytic approach to probabilistic analysis emphasizing the connections to various classical deterministic mathematical analysis elements lastly we describe how to use the approach as a means to augment deterministic analysis methods in a particular hilbert space context and thus enable a rigorous framework for commingling deterministic and probabilistic analysis tools in an application setting

new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

stochastic uncertainty analysis for unconfined flow systems 1 a new stochastic approach proposed by zhang and lu 2004 called the karhunen loeve decomposition based moment equation klme has been extended to solving nonlinear unconfined flow problems in randomly heterogeneous aquifers this approach is on the basis of an innovative combination of karhunen loeve decomposition polynomial expansion and perturbation methods the random log transformed hydraulic conductivity field lnks is first expanded into a series in terms of orthogonal gaussian standard random variables with their coefficients obtained as the eigenvalues and eigenfunctions of the covariance function of lnks next head h is decomposed as a perturbation expansion series h m where h m represents the mth order head term with respect to the standard deviation of lnks then h m is further expanded into a polynomial series of m products of orthogonal gaussian standard random variables whose coefficients are deterministic and solved sequentially from low to high expansion orders using modflow 2000 finally the statistics of head and flux are computed using simple algebraic operations on a series of numerical test results in 2 d and 3 d unconfined flow systems indicated that the klme approach is effective in estimating the mean and co variance of both heads and fluxes and requires much less computational effort as compared to the traditional monte carlo simulation technique

stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

efficient geostatistical inverse methods for structured and unstructured grids 1 the geostatistical inverse approach of kitanidis 1995 requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix the explicit computation of the full covariance matrix is prohibitive for large scale problems discrete spectral methods which help to reduce computational costs dramatically are restricted to regular grids to make the geostatistical inverse approach applicable to unstructured grids which may be of great interest for practical applications we use the functional parameterization of the spatial random field by the karhunen loeve kl expansion in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function in the inverse approach the estimation of the spatial variable becomes estimating the weights of base functions the prior covariance matrix of the weights is the identity matrix the base functions are continuous in space and can be discretized in any fashion for the derivation of the base functions we embed the domain into a larger unit cell of a periodic domain then the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling for illustration we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids the results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the kl expansion we also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method

numerical solutions to statistical moment equations of groundwater flow in nonstationary bounded heterogeneous media in this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers we derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions due to their mathematical complexity we solve the moment equations by the numerical technique of finite differences the numerical approach has flexibility in handling moderately irregular geometry different boundary conditions various trends in the mean log hydraulic conductivity spatial variabilities in the magnitude and direction of mean flow and different covariance functions all of which are important factors to consider for real world applications the effect of boundaries on the first two statistical moments involving head is strong and persistent for example in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain as in many other stochastic models the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small accounting for a spatially varying large scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem it justifies the small variance assumption for many aquifers

variational iteration method for coupled nonlinear schrodinger equations in this paper we apply the variational iteration method proposed by ji huan he to simulate numerically a system of two coupled nonlinear one dimensional schrodinger equations subjected initially to a prescribed periodic wave solution test examples are given to demonstrate the accuracy and capability of the method with different wave wave interaction coefficients the accuracy of the method is verified by ensuring that the energy conservation remains almost constant the numerical results obtained with a minimum amount of computation show that the variational iteration method is much easier more convenient and efficient for solving nonlinear partial differential equations

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

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

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

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

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

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

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

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

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

probabilistic inference of multi gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction 2015 american geophysical union all rights reserved we present a bayesian inversion method for the joint inference of high dimensional multi gaussian hydraulic conductivity fields and associated geostatistical parameters from indirect hydrological data we combine gaussian process generation via circulant embedding to decouple the variogram from grid cell specific values with dimensionality reduction by interpolation to enable markov chain monte carlo mcmc simulation using the matern variogram model this formulation allows inferring the conductivity values simultaneously with the field smoothness also called matern shape parameter and other geostatistical parameters such as the mean sill integral scales and anisotropy direction s and ratio s the proposed dimensionality reduction method systematically honors the underlying variogram and is demonstrated to achieve better performance than the karhunen loeve expansion we illustrate our inversion approach using synthetic error corrupted data from a tracer experiment in a fairly heterogeneous 10 000 dimensional 2 d conductivity field a 40 times reduction of the size of the parameter space did not prevent the posterior simulations to appropriately fit the measurement data and the posterior parameter distributions to include the true geostatistical parameter values overall the posterior field realizations covered a wide range of geostatistical models questioning the common practice of assuming a fixed variogram prior to inference of the hydraulic conductivity values our method is shown to be more efficient than sequential gibbs sampling sgs for the considered case study particularly when implemented on a distributed computing cluster it is also found to outperform the method of anchored distributions mad for the same computational budget key points joint bayesian inference of gaussian conductivity fields and their variograms a dimensionality reduction that systematically honors the underlying variogram distributed multiprocessor implementation is straightforward

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

on an elliptic equation related to the blow up phenomenon in the nonlinear schrodinger equation this paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in n the equation is derived from the blow up problem in the non linear schrodinger equation

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

probabilistic characterization of transport in heterogeneous media abstract 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 random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

perturbation based moment equation approach for flow in heterogeneous porous media applicability range and analysis of high order terms we present detailed comparisons between high resolution monte carlo simulation mcs and low order numerical solutions of stochastic moment equations smes for the first and second statistical moments of pressure the objective is to quantify the difference between the predictions obtained from mcs and sme natural formations with high permeability variability and large spatial correlation scales are of special interest for underground resources e g oil and water consequently we focus on such formations we investigated fields with variance of log permeability y2 from 0 1 to 3 0 and correlation scales normalized by domain length of 0 05 to 0 5 in order to avoid issues related to statistical convergence and resolution level we used 9000 highly resolved realizations of permeability for mcs we derive exact discrete forms of the statistical moment equations formulations based on equations written explicitly in terms of permeability k based and log transformed permeability y based are considered the discrete forms are applicable to systems of arbitrary variance and correlation scales however equations governing a particular statistical moment depend on higher moments thus while the moment equations are exact they are not closed in particular the discrete form of the second moment of pressure includes two triplet terms that involve log permeability or permeability and pressure we combined mcs computations with full discrete sme equations to quantify the importance of the various terms that make up the moment equations we show that second moment solutions obtained using a low order y based sme formulation are significantly better than those from k based formulations especially when y2 1 as a result y based formulations are preferred the two triplet terms are complex functions of the variance level and correlation length the importance contribution of these triplet terms increases dramatically as y2 increases above one we also show that one of the triplet terms is much more important than the other when comparing k based mcs with y based sme model differences must be taken into consideration these differences model errors are due to embedded assumptions and differences in implementing the discrete forms of the equations

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

transient behavior of collapsing ring solutions in the critical nonlinear schrodinger equation abstract the critical nonlinear schrodinger equation nls possesses nearly self similar ring profile solutions we address the question of whether this profile is maintained all the way until the point of singularity a perturbative analysis of the rescaled pde and the resulting self similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak shaped collapse instead a numerical scheme capable of resolving self similar behavior to high resolutions confirms our analysis we also consider ring type blowup arising either from azimuthally polarized solutions of a coupled nls system or as vortex solutions of the usual nls in this case the ring profile is maintained up to the time of singularity

polynomial chaos expansion for nonlinear geophysical inverse problems abstractthere are lots of geophysical problems that include computationally expensive functions forward models polynomial chaos pc expansion aims to approximate such an expensive equation or system with a polynomial expansion on the basis of orthogonal polynomials evaluation of this expansion is extremely fast because it is a polynomial function this property of the pc expansion is of great importance for stochastic problems in which an expensive function needs to be evaluated thousands of times we have developed pc expansion as a novel technique to solve nonlinear geophysical problems to better evaluate the methodology we use pc expansion for automating the velocity analysis for this purpose we define the optimally picked velocity model as an optimizer of a variational integral in a semblance field however because computation of a variational integral with respect to a given velocity model is rather expensive it is impossible to use stochastic methods to search for the optimal velocity mo

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

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

stability of isotropic singularities for the nonlinear schro dinger equation abstract we describe a method by which the fully two and three dimensional cubic schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities in both cases anisotropic initial data collapse very rapidly towards isotropic singularities in three dimensions the solutions become self similar with a blowup rate t t 1 2 in two dimensions the self similarity is weakly broken and the blowup rate is t t ln ln 1 t t 1 2 the stability of the singular isotropic solutions is very firmly backed by the numerical results

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

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

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

a weight function for the associated jacobi laurent polynomials abstract under some additional conditions on the parameters a weight function on the unit circle in the complex plane is given for the associated jacobi laurent polynomials as introduced in hendriksen 1990

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

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

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

karhunen loeve decomposition of random fields based on a hierarchical matrix approach summary the simulation of the behavior of structures with uncertain properties is a challenging issue because it requires suitable probabilistic models and adequate numerical tools nowadays it is possible to perform probabilistic investigations of the structural performance which take into account a space variant uncertainty characterization of the structures given a structural solver and the probabilistic models the reliability analysis of the structural response depends on the continuous random fields approximation which is carried out by means of a finite set of random variables the paper analyzes the main aspects of discretization in the case of 2d problems the combination of the well known karhunen loeve series expansion the finite element method and the hierarchical matrices approach is proposed in the paper copyright 2013 john wiley sons ltd

a flexible uncertainty quantification method for linearly coupled multi physics systems abstract this paper presents a novel approach to building an integrated uncertainty quantification uq methodology suitable for modern day component based approach for multi physics simulation development our hybrid uq methodology supports independent development of the most suitable uq method intrusive or non intrusive for each physics module by providing an algorithmic framework to couple these stochastic modules for propagating global uncertainties we address algorithmic and computational issues associated with the construction of this hybrid framework we demonstrate the utility of such a framework on a practical application involving a linearly coupled multi species reactive transport model

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

conditional stochastic averaging of steady state unsaturated flow by means of kirchhoff transformation we consider the effect of measuring randomly varying soil hydraulic properties on one s ability to predict steady state unsaturated flow subject to random sources and or initial and boundary conditions our aim is to allow optimum unbiased prediction of system states pressure head water content and fluxes deterministically without upscaling and without linearizing the constitutive characteristics of the soil it has been shown by neuman et al 1999 that such prediction is possible by means of first ensemble moments of system states and fluxes conditioned on measured values of soil properties when the latter scale in a linearly separable fashion as proposed by vogel et al 1991 the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments the derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations which may lead to major inaccuracies when these relations are highly nonlinear as is often the case in nature when the scaling parameter of pressure head is a random variable independent of location the steady state unsaturated flow equation can be linearized by means of the kirchhoff transformation for gravity free flow linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head for the latter case we develop exact conditional first and second moment equations which are nonlocal and therefore non darcian we solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical monte carlo simulations our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble averaged unsaturated flow except when gravity is the sole driving force

a two stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media a new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability for pressure estimation it combines the dimension adaptive stochastic collocation method on smolyak sparse grids and the kronrod patterson hermite nested quadrature formulas for saturation estimation an additional stage is developed in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples to address the low regularity problem of the saturation numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two stage adaptive stochastic collocation method on nested sparse grids

asymptotics of multibump blow up self similar solutions of the nonlinear schrodinger equation this paper examines blow up self similar solutions of the cubic nonlinear schrodinger equation close to the critical dimension d 2 it gives a formal asymptotic theory for self similar solutions with multiple maxima in which the solution close to each maximum takes the form of a rescaled one dimensional soliton as d rightarrow 2 the maxima move to infinity and are centered close to the point log d 2 2 pi 3 sqrt 3 4 however the shape of the solution close to each maxima changes little in this limit leading to an interesting nonuniform bifurcation the formulae derived from the asymptotic theory are strongly supported by some numerical calculations

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

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

blowup and solitary wave solutions with ring profiles of two component nonlinear schrodinger systems abstract blowup ring profiles have been investigated by finding non vortex blowup solutions of nonlinear schrodinger equations nlses cf fibich et al 2005 7 and fibich et al 2007 8 however those solutions have infinite l 2 norm so one may not maintain the ring profile all the way up to the singularity to find h 1 non vortex blowup solutions with ring profiles we study the blowup solutions of two component systems of nlses with nonlinear coefficients and j j 1 2 when 0 and 1 2 0 the two component system can be transformed into a multi scale system with fast and slow variables which may produce h 1 blowup solutions with non vortex ring profiles we use the localized energy method with symmetry reduction to construct these solutions rigorously on the other hand these solutions may describe steady non vortex bright ring solitons various types of ring profiles including m ring and ring ring profiles are presented by numerical solutions

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

numerical study of blow up to the purely elliptic generalized daveystewartson system blow up solutions for the purely elliptic generalized daveystewartson system are studied by using a relaxation numerical method the numerical method is based on an implicit finite difference scheme with a second order accuracy in both time and space the stability of the numerical method is analyzed by investigating the linear stability of plane wave solutions to evaluate the ability of the relaxation method to detect blow up numerical simulations are conducted for several test problems a particular attention is paid to the gap interval neither a global existence nor a blow up result is established the monotonicity properties of blow up time on the coupling parameter are also investigated numerically

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

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

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

associated jacobi laurent polynomials abstract the jacobi laurent polynomials have been introduced by hendriksen and van rossum 1986 in the present paper explicit formulas for the orthogonal laurent polynomials satisfying the recurrency for the jacob laurent polynomials with n replaced by n b are given these new orthogonal laurent polynomials are called associated jacobi laurent polynomials using these associated laurent polynomials the denominator and the numerator of certain two point pade approximants to the pair of functions z f a b 1 c b 1 z f a b c b z at o and c b a b 1 f c 1 b 1 a b 2 z 1 f c 1 b 1 z l at are given also some confluent cases are considered

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

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

stochastic analysis of flow in a heterogeneous unsaturated saturated system 1 in this study we develop a stochastic model for transient unsaturated saturated flow in randomly heterogeneous media with the method of moment equations we first derive partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implement these equations under general conditions with the method of finite differences the stochastic model developed is applicable to the entire domain of a bounded multidimensional unsaturated saturated system in the presence of random or deterministic recharge and sink source and in the presence of multiscale nonstationary medium features the unsaturated and saturated zones are coupled through the water table whose position is random in randomly heterogeneous porous media the presence of the water table renders the flow moments strongly nonstationary even in the absence of medium nonstationary features this finding is confirmed with monte carlo simulations the resulting first two moments of flow may be used to approximate confidence intervals for the flow quantities in addition the integrated stochastic flow model provides the prerequisite quantities for realistically studying contaminant transport in unsaturated saturated systems with a stochastic approach

variational iteration method for solving cubic nonlinear schrodinger equation the variational iteration method is applied to solve the cubic nonlinear schrodinger cnls equation in one and two space variables in both cases we will reduce the cnls equation to a coupled system of nonlinear equations numerical experiments are made to verify the efficiency of the method comparison with the theoretical solution shows that the variational iteration method is of high accuracy

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

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

conditional moment analysis of steady state unsaturated flow in bounded randomly heterogeneous soils 1 we consider steady state unsaturated flow in bounded randomly heterogeneous soils under the influence of random boundary and source terms our aim is to predict pressure heads and fluxes without resorting to monte carlo simulation upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head we represent this relationship through gardner s exponential model while treating its exponent as a random constant and saturated hydraulic conductivity ks as a spatially correlated random field we linearize the steady state unsaturated flow equations by means of the kirchhoff transformation and integrate them in probability space to obtain exact integro differential equations for the conditional mean and variance covariance of transformed pressure head and flux after approximating these equations recursively to second order in the standard deviation y of y ln ks we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane with and without conditioning on measured y values comparison with monte carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 it yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 2 this accords well with a previous theoretical analysis which shows that the solution may remain asymptotic for values of y2 as large as 2

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