dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

hierarchical preconditioning for the stochastic galerkin method one of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic galerkin method where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables related systems of linear equations are thus usually huge studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning for a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened cauchy bunyakowsky schwarz cbs constants with respect to the scalar product defined by the operator of the weak form of the underlying equation small values of the cbs constant indicate that certain additive or multiplicative two by two block preconditioners reduce enough the condition number of the system moreover we show that a recursive multiplicative two by two block preconditioning can be used resulting in the algebraic multilevel iterative amli method we present the conditions under which the amli preconditioning is of an optimal order numerical experiments confirm the introduced theoretical estimates

scheduled control for robust attenuation of non stationary sinusoidal disturbances with measurable frequencies attenuation of sinusoidal disturbances with uncertain yet online measurable frequencies is considered the disturbances are modeled as the outputs of an undisturbed parameter dependent exogenous system with a skew symmetric system matrix obtained in response to nonzero initial conditions the problem is formulated for a parameter dependent plant as the synthesis of a parameter dependent controller in a way to ensure internal stability as well as a desired level of steady state disturbance attenuation in the face of all admissible parameter variations the solvability of this problem is first related to the existence of bounded solutions to a matrix differential regulator equation subject to an asymptotic norm constraint reformulating this as a parameter dependent state feedback like synthesis based on which suitable solutions to the differential regulator equation can be obtained online tractable solvability conditions are then provided in the form of parameter dependent matrix inequalities controllers that solve the generalized asymptotic regulation problem are also parameterized in terms of the suitable solutions of the differential regulator equation and some free parameter dependent matrices that are to be designed off line to ensure stability a procedure is then developed to design the free parameters in a way to achieve desirable transient behavior the use of the developed synthesis procedure is illustrated on a simplified version of the course control problem in ship steering

a multigrid solver for two dimensional stochastic diffusion equations steady and unsteady diffusion equations with stochastic diffusivity coefficient and forcing term are modeled in two dimensions by means of stochastic spectral representations problem data and solution variables are expanded using the polynomial chaos system the approach leads to a set of coupled problems for the stochastic modes spatial finite difference discretization of these coupled problems results in a large system of equations whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time to accelerate the convergence of the iterative technique a multigrid method based on spatial coarsening is implemented numerical experiments show good scaling properties of the method both with respect to the number of spatial grid points and the stochastic resolution level

on the divergence of gradient expansions for kinetic energy functionals in the potential functional theory we consider the density of a fermionic system as a functional of the potential in one dimensional case where it is approximated by the thomas fermi term plus semiclassical corrections through the gradient expansion we compare this asymptotic series with the exact answer for the case of the harmonic oscillator and the morse potential it is found that the leading thomas fermi term is in agreement with the exact density but the subdominant term does not agree in terms of the asymptotic behavior because of the presence of oscillations in the exact density but their absence in the gradient expansion however after regularization of the density by convolution with a gaussian the agreement can be established even in the subdominant term moreover it is found that the expansion is always divergent and its terms grow proportionally to the factorial function of the order similar to the well known divergence of perturbation series in field theory and the quantum anharmonic oscillator pade hermite approximants allow summation of the series and one of the branches of the approximants agrees with the density

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

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

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

taylor dispersion in curved channels the taylor dispersion problem is to determine the long time behavior of the concentration density of a passive scalar contaminant diffusing in an incompressible channel flow given an initial injection of contaminant which is slowly varying order epsilon in the longitudinal direction taylor showed that the cross channel mean of the concentration experiences an enhanced diffusion relative to the mean flow this is due to mechanical rather than molecular events mercer and roberts have treated this problem by a formal application of infinite dimensional center manifold theory and extended the analysis to the case of a curved channel which is slowly varying in the longitudinal direction it is shown that after the introduction of appropriate multiple scales their center manifold technique can be used to carry out the taylor dispersion calculations for a channel whose cross section has large order 1 variation in the longitudinal directions also more general underlying flows can be allowed than was heretofore possible a heuristic argument is given to show the existence of some channel shape and flow field for which effective diffusion is reduced rather than enhanced a similar advection diffusion problem but with no channel boundary has been rigorously analyzed by homogenization theory and discussed in a recent paper of majda andmore mclaughlin the two problems are briefly compared less

equivalence of bicontinuum and second order transport in heterogeneous soils and aquifers the mathematical equivalence of a closed second order transport equation for reactive solutes in saturated heterogeneous porous media e g soils and aquifers and a two region mobile mobile formulation is demonstrated in two ways 1 by averaging the bicontinuum equations and 2 by transforming the second order equation to canonical form the derivation of this equivalence is limited to media with heterogeneities in a plane orthogonal to the flow direction which represents a bidirectional generalization of the stratified case results indicate how the parameters of the bicontinuum formulation can be obtained from statistical descriptions of this heterogeneity and establish a stochastic interpretation for the bicontinuum equations one dimensional analytical solutions of the equivalent transport equations are compared to a three dimensional numerical simulation of transport in a spatially autocorrelated lognormal random conductivity field results indicate that the upscaled equations provide good approximations of the dynamics of the mean resident concentration the flux averaged concentration and their variances

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities

prevalence expansion in nimfa abstract the n intertwined mean field approximation nimfa is a reasonably accurate approximation of the exact sis epidemic process on a network the average fraction of infected nodes in the nimfa steady state also called the steady state prevalence in terms of the effective infection rate can be expanded into a power series around the nimfa epidemic threshold in this paper we investigate the convergence of the steady state prevalence taylor expansion we determine the radius of convergence in some special types of graphs we also show that the radius of convergence of the steady state prevalence expansion depends upon the network topology in particular the average degree of the network and the spectral gap of the adjacency matrix play a role

predictability of tracer dilution in large open channel flows analytical solution for the coefficient of variation of the depth averaged concentration a large time analytical solution is proposed for the spatial variance and coefficient of variation of the depth averaged concentration due to instantaneous cross sectionally uniform solute sources in pseudorectangular open channel flows the mathematical approach is based on the use of the green functions and on the fourier decomposition of the depth averaged velocities coupled with the method of the images the variance spatial trend is characterized by a minimum at the center of the mass and two mobile decaying symmetrical peaks which at very large times are located at the inflexion points of the average gaussian distribution the coefficient of variation which provides an estimate of the expected percentage deviation of the depth averaged point concentrations about the section average exhibits a minimum at the center which decays like t 1 and only depends on the river diffusive time scale the defect of cross sectional mixing quickly increases with the distance from the center and almost linearly at large times accurate numerical lagrangian simulations were performed to validate the analytical results in preasymptotic and asymptotic conditions referring to a particularly representative sample case for which cross sectional depth and velocity measurements were known from a field survey in addition in order to discuss the practical usefulness of computing large time concentration spatial moments in river flows and resorting to directly measured input data the order of magnitude of section averaged concentrations and corresponding coefficients of variation was estimated in field conditions and for hypothetical contamination scenarios considering a unit normalized mass impulsively injected across the transverse section of 81 u s rivers

plume scale dependent dispersion in heterogeneous aquifers 1 lagrangian analysis in a stratified aquifer an analytical approach is developed for describing the ensemble average of the second moment of a solute plume in heterogeneous porous media the growth of the ensemble average second moment is related to the growth of the mean square separation of a pair of particles exact lagrangian expressions are developed for the growth of the mean square separation in a perfectly stratified aquifer these exact expressions are made possible by an exact relation between the lagrangian and eulerian velocity covariances in the perfectly stratified aquifer the ensemble average second moment is shown to depend on the initial vertical dimensions of the concentration distribution as the vertical size increases the second moment growth rate is larger the second moment expressions are contrasted with the expressions for the mean square displacement of a single particle while all scales of heterogeneity contribute to the mean square displacement of a single particle only scales of heterogeneity smaller than the plume size contribute to the second moment asymptotic large displacement expressions for the case of a gaussian covariance function describing the hydraulic conductivity variations are derived these expressions indicate that at large displacement the second moment grows as the 3 2 power of displacement however for any finite source size no matter how large the second moment growth rate approaches a different asymptote from the rate of growth of the mean square displacement of a single particle this is in conflict with the traditional notion of ergodicity which leads to the expectation that for large source sizes the dispersivity approaches the rate of growth of the mean square displacement of a single particle

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

hyperbolic homogenized models for thermal and solutal dispersion we formulate a general theory based on a lyapunov schmidt expansion for averaging thermal and solutal dispersion phenomena in multiphase reactors with specific attention to the important taylor mechanism due to transverse intraphase and interphase capacitance weighted velocity gradients we show that the classical taylor dispersion phenomena are better described in terms of low dimensional models that are hyperbolic and contain an effective local time or length scale in place of the traditional taylor dispersion coefficient this description eliminates the use of an artificial exit boundary condition associated with parabolic homogenized equations as well as the classical upstream feedback and infinite propagation speed anomalies our approach is also applicable for describing steady dispersion in the presence of reaction and thermal generation or consumption for two phase systems maximum dispersion is found to exist at an optimum fraction epsilon f of the lower capacitance phase for the disp

the influence of pore scale dispersion on concentration statistical moments in transport through heterogeneous aquifers transport of an inert solute in a heterogeneous aquifer is governed by two mechanisms advection by the random velocity field v x and pore scale dispersion of coefficients ddij the velocity field is assumed to be stationary and of constant mean u and of correlation scale i much larger than the pore scale d it is assumed that ddij diju are constant the relative effect of the two mechanisms is quantified by the peclet numbers peij u ddij i dij which as a rule are much larger than unity the main aim of the study is to determine the impact of finite though high pe on c and c2 the concentration mean and variance respectively the solution derived in the past for pe is reconsidered first by assuming a normal x probability density function p d f closed form solutions are obtained for c and c2 recasting the problem in an eulerian framework leads to the same results if certain closure conditions are adopted the concentration moments for a finite pe are derived subsequently in a lagrangean framework the pore scale dispersion is viewed as a brownian motion type of displacement xd of solute subparticles of scale smaller than d added to the advective displacements x by adopting again a normal p d f for the latter explicit expressions for c and c2 are obtained in terms of quadratures over the joint p d f of advective two particles trajectories while the influence of high pe on c is generally small it has a significant impact on c2 simple results are obtained for a small v0 for which trajectories are fully correlated in particular the concentration coefficient of variation at the center tends to a constant value for large time comparison of the present solution obtained in terms of a quadrature with the monte carlo simulations of graham and mclaughlin 1989 shows a very good agreement

a new derivation of the taylor aris theory of solute dispersion in a capillary the transport of a dilute solute under viscous motion in a straight capillary is described at three distinct space time scales these are the kinetic fluid mechanical and taylorian scales the transition from one scale to the next higher scale is shown to be a consequence of the central limit theorem clt of probability theory the taylor aris dispersion equation is derived by an application of a recently proved clt for markov processes an alternative computation of the dispersion coefficient is given using a differential equation averaging approach a third method which also applies to solute transport in porous media is illustrated for an approximate computation of the dispersion coefficient

nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension two abstract the interaction of two different tearing modes in slab geometry is examined for certain physical parameter values the linearized problem has a four dimensional nullspace corresponding to two different marginal tearing modes with the center manifold theory the original partial differential equations could be reduced to a four dimensional system of ordinary ones these amplitude equations are equivalent under o 2 actions due to the symmetries of the physical problem because of these symmetries there exist many interesting spatial and spatio temporal solutions in the neighbourhood of the critical parameters such as standing travelling and modulated waves and a structural stable heteroclinic orbit see ambruster guckenheimer and holmes physica d 29 1988 257 282

schwarz preconditioners for stochastic elliptic pdes abstract increasingly the spectral stochastic finite element method ssfem has become a popular computational tool for uncertainty quantification in numerous practical engineering problems for large scale problems however the computational cost associated with solving the arising linear system in the ssfem still poses a significant challenge the development of efficient and robust preconditioned iterative solvers for the ssfem linear system is thus of paramount importance for uncertainty quantification of large scale industrially relevant problems in the context of high performance computing the preconditioner must scale to a large number of processors therefore in this paper a two level additive schwarz preconditioner is described for the iterative solution of the ssfem linear system the proposed preconditioner can be viewed as a generalization of the mean based block diagonal preconditioner commonly used in the literature for the numerical illustrations two dimensional steady state diffusion and elasticity problems with spatially varying random coefficients are considered the performance of the algorithm is investigated with respect to the geometric parameters strength of randomness dimension and order of the stochastic expansion

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

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

new stochastic mode reduction strategy for dissipative systems we present a new methodology for studying non hamiltonian nonlinear systems based on an information theoretical extension of a renormalization group technique using a modified maximum entropy principle we obtain a rigorous dimensionally reduced description for such systems the neglected degrees of freedom by this reduction are replaced by a systematically defined stochastic process under a constraint on the second moment this then forms the basis of a computationally efficient method numerical computations for the generalized kuramoto sivashinsky equation support our method and reveal that the long time underlying stochastic process of the fast unresolved modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle

pore scale dilution of conservative solutes an example we simulate flow and transport of a conservative nonsorbing tracer in an idealized periodic pore channel using finite element techniques the concentration is computed then the slowly varying concentration mean variance coefficient of variation and reactor ratio are calculated through averaging over every cell the coefficient of variation and reactor ratio are related and quantify the degree of dilution then a novel methodology is developed for the evaluation of macroscopic parameters homogenization including the variance decay coefficient which measures the rate with which small scale concentration fluctuations tend to diminish and the large time coefficient of proportionality between the concentration variance and the square of the mean concentration variance the methodology is based on the solution of a steady advection dispersion problem in a single cell which acts as a representative elementary volume the computed result is then integrated in order to compute the macroscopic parameters these parameters are compared with the parameters computed through direct simulation on a cell by cell basis and they are found to be in reasonably good agreement when the macroscopic parameters are used in the macroscopic equations they produce estimates of the concentration mean and variance that are in agreement with the results of direct simulation

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

adaptive stochastic galerkin fem abstract a framework for residual based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic pdes with random coefficients is presented a parametric deterministic elliptic boundary value problem on an infinite dimensional parameter space is discretized by means of a galerkin projection onto finite generalized polynomial chaos gpc expansions and by discretizing each gpc coefficient by a fem in the physical domain an anisotropic residual based a posteriori error estimator is developed it contains bounds for both contributions to the overall error the error due to gpc discretization and the error due to finite element discretization of the gpc coefficients in the expansion the reliability of the residual estimator is established based on the explicit form of the residual estimator an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic galerkin discretization and embedded in the gpc adaptation loop also the finite element mesh refinement of the gpc coefficients in the physical domain asynchronous mesh adaptation for different gpc coefficients is permitted subject to a minimal compatibility requirement on meshes used for different gpc coefficients details on the implementation with the open source software framework alea are presented it is generic and is based on available stiffness and mass matrices of a fem for the deterministic nonparametric nominal problem evaluated in the fenics environment preconditioning of the resulting matrix equation and iterative solution are discussed numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented they indicate substantial savings in total computational complexity due to fe mesh coarsening in high gpc coefficients

multiple time scale analysis of macrotransport processes abstract convective and diffusive transport of a brownian tracer corpuscle is analyzed within a multidimensional space decomposed into local internal and global external subspaces multiple time scale methods are employed to successively eliminate from the kinetic equation governing the multidimensional microtransport process its dependence upon each of the internal coordinates the resulting long time equation describing the residual global space macrotransport process derived by this systematic perturbation procedure is shown to accord with the well established results of generalized taylor dispersion theory heretofore obtained by ad hoc arguments by way of example this macrotransport description is used to analyze the taylor dispersion of a solute in a poiseuille type solvent flow occuring within a rectangular duct of small but non zero aspect ratio elementary dispersion results obtained by ignoring the side walls apply only for relatively short times for longer times the perturbing presence of the side walls acts to substantially increase the axial dispersivity even in the limit of zero aspect ratio where intuition would strongly suggest otherwise

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

on the exact analytical solution for the spatial moments of the cross sectional average concentration in open channel flows 1 this paper shows how an exact analytical solution for the transient state spatial moments of the cross sectional average tracer concentration in large open channel flows can be derived from the depth averaged advection diffusion equation resorting to the method of green s functions without any simplifying assumption about the regularity of the actual concentration field the smallness of the fluctuations or the large space time scale of variation of the average concentration gradient justifying the a priori localization of the problem which were the basis of the classic taylor dispersion theory the results reveal that in agreement with the findings by aris 1956 and later by others for flows within a conduit there are an initial centroid displacement and a variance deficit dependent on the specific position and dimension of the initial injection the second central moment asymptotically tends to the linearly increasing function predictable on the basis of taylor s classic theory and the skewness which is constantly zero for the cross sectionally uniform injection in the case of nonuniform initial distributions tends to slowly vanish after having reached a maximum thus the persistent asymmetry exhibited by the field concentration data as well as the retardations and the accelerations in the peak trajectory can be justified without making any a priori assumption about the physical mechanism underlying their appearance like transient storage phenomena just by rigorously solving the governing equation for the cross sectional average concentration in the presence of nonuniform asymmetrically located solute injections

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

internal model based design of repetitive and iterative learning controllers for linear multivariable systems repetitive and iterative learning control are two modern control strategies for tracking systems in which the signals are periodic in nature this paper discusses repetitive and iterative learning control from an internal model principle point of view this allows the formulation of existence conditions for multivariable implementations of repetitive and learning control it is shown that repetitive control can be realized by an implementation of a robust servomechanism controller that uses the appropriate internal model for periodic distrubances the design of such controllers is discussed next it is shown that iterative learning control can be implemented in the format of a disturbance observer compensator it is shown that the resulting control structure is dual to the repetitive controller and that both constitute an implementation of the internal model principle consequently the analysis and design of repetitive and iterative learning control can be generalized to the powerful analysis and design

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

stochastic analysis of the concentration variability in a three dimensional heterogeneous aquifer a spectrally based perturbation approach is used to evaluate the concentration variability for a steady concentration field in a three dimensional statistically homogeneous and anisotropic aquifer the analysis assumes small locally stationary concentration perturbations and consequently is valid only after mean displacements that are large compared to the scale of aquifer heterogeneity it is shown that the concentration variance is proportional to the square of the local mean concentration gradient and the variance and the correlation scales of log hydraulic conductivity ln k and is inversely proportional to the local dispersivity the concentration covariance function is highly anisotropic with the largest correlation length aligned to the mean flow direction another important finding is the sensitivity of the longitudinal persistence of the concentration field to the high wave number behavior of the input ln k spectrum the commonly employed exponential type covariance function which corresponds to a nondifferentiable random field results in extremely large concentration correlation lengths along the flow direction input spectra corresponding to differentiable fields with more rapid high wave number cutoffs produce a significant reduction of the longitudinal correlation length

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

semi coarsening amli preconditioning of anisotropic trilinear fem systems optimal order algebraic multilevel iteration amli preconditioners based on recursive application of two level finite element fe methods and polynomial stabilization have been introduced and analyzed in axelsson and vassilevski 1989 1990 the construction follows the natural hierarchical splitting using the fact that the finite element spaces corresponding to two successive mesh refinements are nested uniform estimates for the constant in the strengthened cauchy bunyakowski schwarz cbs inequality are very important for the derivation of optimal order methods the value of the upper bound for 0 1 is a part of the construction of the multilevel extension of the related two level method in this paper algebraic two level and multilevel preconditioning algorithms for second order anisotropic elliptic boundary value problems are constructed here we allow big jumps in the coefficients and varying the direction of dominating anisotropy from one element to another in the coarse triangulation the discretization is done by trilinear conforming finite elements where the semi coarsening mesh refinement strategy is applied a new uniform estimates for the related cbs constants are derived the additive preconditioning strategy for the system with the pivot block in the hierarchical two level splitting is proposed where the related sub problems have two dimensional structure

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

brief paper robust high order repetitive control optimal performance trade offs high order repetitive control has previously been introduced to either improve the robustness for period time uncertainty or reduce the sensitivity for non periodic inputs of standard repetitive control schemes this paper presents a systematic semidefinite programming based approach to compute high order repetitive controllers that yield an optimal trade off between these two performance criteria the methodology is numerically illustrated through trade off curves for various controller orders and levels of period time uncertainty moreover existing high order repetitive control approaches are shown to correspond to specific points on these curves

concentration variance and spatial covariance in second order stationary heterogeneous conductivity fields the concentration variance and spatial covariance resulting from convective dispersive transport driven by a uniform mean flow in second order stationary conductivity fields was derived in a lagrangian framework based on a first order approximation of solute particle trajectories the approximation of the concentration co variances for large injection volumes is considerably simplified by defining the concentration at a certain location x and time t in terms of the backward solute trajectory probability distribution this is the probability that the trajectory of a fictitious microscopic and indivisible solute particle which is in a volume x centered around x at time t was at the time of solute injection t0 in the injection volume v0 the approximated concentration co variances were validated against concentration covariances derived from transport simulations in generated second order stationary conductivity fields the approximate solutions reproduced fairly well the effects of local scale dispersion and of the spatial variability of the hydraulic conductivity on the concentration co variance the effect of the spatial structure of the hydraulic conductivity field on the spatial covariance of the concentrations was investigated in order to identify parameters that can be unequivocally determined from the structure of the concentration field for a given spreading of the solute plume in the mean flow direction x11 t a given spatial correlation length of the loge transformed hydraulic conductivity in the transverse to flow direction i 2 and a given local scale dispersion dd the concentration covariance was nearly invariant and hardly influenced by the anisotropy of the covariance function e and the variance f2 of the loge transformed conductivity as a result e and f2 cannot be unequivocally determined from the spatial structure of the concentration field the concentration variance and spatial covariance in the transverse to mean flow direction are predominantly determined by the lateral component of the local scale dispersion dd22 and by the spatial correlation length of the loge transformed conductivity in the transverse to flow direction i 2 these two parameters might be unequivocally derived from the concentration co variance

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

large time behavior of concentration variance and dilution in heterogeneous formations consider the advection and dispersion of a conservative nonsorbing solute in a spatially variable but statistically homogeneous velocity field a lagrangian approach leads to expressions for the large time mean and variance of concentration the expressions require only the macroscopic velocity vector um the macrodispersion tensor dm and an additional tensor the macroscopic velocities and macrodispersivities are well known from numerous previous studies but is introduced here for the first time the tensor is needed to describe the kinetics of dilution of a plume two cases with the same um anddm have different dilution characteristics depending on the characteristic times of dilution are given by tensor d 1m it is demonstrated for the first time through a lagrangian approach that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1 t as was previously shown in the eulerian theory ofkapoor and gelhar 1994a b expressions for the geometric mean of the dilution index and the reactor ratio are derived numerical simulations support the validity of the approach

energy norm a posteriori error estimation for parametric operator equations stochastic galerkin approximation is an increasingly popular approach for the solution of elliptic pde problems with correlated random data a typical strategy is to combine conventional h finite element approximation on the spatial domain with spectral p approximation on a finite dimensional manifold in the stochastic parameter domain the issues involved in a posteriori error analysis of computed solutions are outlined in this paper using an abstract setting of parametric operator equations a novel energy error estimator that uses a parameter free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space we prove that our error estimator is reliable and efficient we also discuss different strategies for enriching the approximation space and prove two sided estimates of the error reduction for the corresponding enhanced approximations these give computable estimates of the error reduction that depend only on

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

backward error and condition number of a generalized sylvester equation with application to the stochastic galerkin method the governing equation of the stochastic galerkin method can be formulated as a generalized sylvester equation therefore developing solvers for it is attracting a lot of attention from the uncertainty quantification community in this regard krylov subspace based iterative solvers that are used for standard linear systems are being used for the generalized sylvester equations as well this is achieved by converting the generalized sylvester equation to a standard linear system using the kronecker product accordingly the residual norm is used as a stopping criterion for the iterations and the condition number of linear systems is used for the generalized sylvester equations as well for a linear system a small residual norm implies a small backward error and hence using residual norm as a stopping criterion is justified in this work we prove that this need not be the case for the generalized sylvester equation we introduce two definitions for the backward error and then derive an upper bound on each of them we also verify the predictions of the analysis using numerical experiments for the special case of the stochastic galerkin method we show that the upper bound on the backward error can be computed with minimal computational overhead and hence it can be used as a stopping criterion in the iterative solvers for the matrices stemming from the stochastic galerkin method we numerically demonstrate that the actual backward error can be up to 3 orders of magnitude higher than the relative residual finally by taking into account the structure of the equation we derive an expression for the condition number and discuss an algorithm for its computation in the special case of the stochastic galerkin method

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

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

harmonic generation in adaptive feedforward cancellation schemes the paper investigates the generation of harmonics in adaptive feedforward cancellation schemes specifically it is shown that an algorithm designed to reject a certain number of frequency components may in fact be capable of reducing higher order harmonics as well this effect originates in the time variation of the adaptive parameters a consequence is that the adaptive system may perform better than the algorithm with the fixed ideal parameters surprisingly the response to higher order harmonics can be calculated precisely using a laplace transform analysis the origin of the numerical procedure is in the equivalence between the adaptive feedforward cancellation scheme and a control scheme based on the internal model principle the implication of the results on design and implementation issues is discussed

brief paper design of noise and period time robust high order repetitive control with application to optical storage repetitive control is useful if periodic disturbances or setpoints act on a control system perfect asymptotic disturbance rejection is achieved if the period time is exactly known the improved disturbance rejection at the periodic frequency and its harmonics is achieved at the expense of a degraded system sensitivity at intermediate frequencies a convex optimization problem is defined for the design of high order repetitive controllers where a trade off can be made between robustness for changes in the period time and for reduction of the error spectrum in between the harmonic frequencies the high order repetitive control algorithms are successfully applied in experiments with the tracking control of a cd player system

almost periodic passive tracer dispersion the authors investigate the impact of external sources on the pattern formation of concentration profiles of passive tracers in a two dimensional shear flow by using the pullback attractor technique for the associated nonautonomous dynamical system it is shown that a unique temporally almost periodic concentration profile exists for the temporally almost periodic external source

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

first order based cumulative distribution function for solute concentration in heterogeneous aquifers theoretical analysis and implications for human health risk assessment quantifying the uncertainty of solute concentration in heterogeneous aquifers is an important step in both human health and ecological risk analysis the need for a probabilistic representation of transport is justified by the incomplete characterization of the subsurface we derive the one point concentration cumulative distribution function cdf while taking into account the spatial statistical structure of the hydraulic conductivity space dimensionality the injection source size the peclet number and the sampling volume at the monitoring location the cdf is application oriented and derived at first order in the log conductivity variance we illustrate how several key parameters control the shape of the concentration cdf the cdf shape is important since it reflects both uncertainty and the dilution state of the plume the transition from a bimodal to a unimodal cdf is examined and results are further supported by analyzing the concentration coefficient of variation results indicate the significance of the statistical anisotropy ratio i e the ratio between the hydraulic conductivity correlation scales in determining the cdf shape the importance of the sampling volume in the tails of the concentration cdf and a comparison between the proposed model with the cdf approach i e beta distribution are also shown finally we illustrate how the framework could be used in applications by evaluating the human health risk cdf our results are formally valid for low to moderate heterogeneous aquifers and source sizes small as compared to the hydraulic conductivity correlation length the proposed approach can serve as a benchmark tool for other methods

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

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

volume averaging for determining the effective dispersion tensor closure using periodic unit cells and comparison with ensemble averaging in this work we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions although this rectification is something that is generally believed to be true there has been very little research that explores this issue explicitly we show that under certain limiting but mild assumptions the closure problem provides a fourier series solution for the effective dispersion tensor when second order spatial stationarity is imposed on the velocity field the method yields a simple fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity this limiting result is identical to the quasi fickian forms that have been developed previously via ensemble averaging by deng et al 1993 and recently by fiori and dagan 2000 except in the definition of the averaging operation as a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics this second objective is complimentary in many ways to recent research reported by rubin et al 1999 via ensemble averaging and by wang and kitanidis 1999 via volume averaging on the block averaged effective dispersion tensor the variability of the effective dispersion tensor from realization to realization is assessed by computing the volume averaged effective dispersion tensor for an ensemble of finite fields with the same ensemble statistics ensembles were generated using three different sizes of unit cells all three unit cell sizes yield similar results for the value of the mean effective dispersion tensor however the coefficient of variation depends strongly upon the size of the unit cell and our results are consistent with those developed by fiori 1998 from the ensemble averaging perspective this implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics and this variation should be considered when applying macrodispersion to real world systems

an analytical solution of the three component transport equation with application to third order transport 1 an explicit analytical solution is derived and illustrated for multidimensional advective dispersive reactive transport in systems with three distinct velocities coupled by first order interactions or reactions the solution is an exponentially scaled space time convolution of an advective dispersive reactive kernel with a coupled advective transport kernel explicit forms are provided for both kernels the solution process for the coupled advective kernel uses one sided two dimensional laplace transforms and introduces two constraints into the transport problem which are to be relaxed in future work the behavior of the solution is illustrated with a third order approximation of single species transport in a heterogeneous aquifer results indicate that the early time behavior of the third order system is that of noninteracting advective dispersive reactive transport the large time mean behavior is that of a single advective dispersive reactive transport equation with constant effective parameters results further demonstrate that the third order approximation provides accurate predictions of the mean and uncertainty of concentration distributions calculated from detailed two dimensional transport simulations at all times

numerical evaluation of solute dispersion and dilution in unsaturated heterogeneous media 1 we investigate flow and solute transport in a heterogeneous unsaturated soil under gravity driven flow conditions using a direct simulation monte carlo method we account for the cross correlation between the heterogeneous fields of unsaturated flow parameters representing a silty loam we determine through particle tracking the statistical moments of the travel time distribution for nonpoint injection of a conservative tracer the one particle mean and variance characterize the mean arrival time and spreading of solute integrated over an entire horizontal cross section of the domain whereas the two particle semivariogram characterizes the spreading observed at individual points from these quantities we derive macrodispersivities and effective dispersivities the latter reflect solute dilution owing to the nonlinearity of unsaturated flow the dispersivity values at a given depth vary with the mean infiltration rate the two particle semivariogram hardly differs from the one particle variance of travel time at low infiltration rates whereas dilution and macrodispersion rates differ significantly at high infiltration rates we compare the dispersivities determined through particle tracking with approximations from linear stochastic theories based on the covariance of the eulerian velocity field dagan 1988 fiori and dagan 2000 these linear stochastic theories slightly underestimate both dispersivity types we present parametric models for the two types of dispersivities as function of the infiltration rate and the depth

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations