stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

multitarget error estimation and adaptivity in aerodynamic flow simulations important quantities in aerodynamic flow simulations are the aerodynamic force coefficients including the pressure induced and the viscous stress induced drag lift and moment coefficients in addition to the exact approximation of these quantities it is of increasing importance in particular in the field of uncertainty quantification to estimate the error in the computed quantities in recent years a posteriori error estimation and goal oriented refinement approaches have been developed for the accurate and efficient computation of single target quantities the current approaches are based on computing an adjoint solution related to each of the specific target quantities under consideration in this paper we extend this approach to the accurate and efficient computation of multiple target quantities instead of computing multiple adjoint solutions one for each target functional the new approach is based on the solution to one discrete adjoint problem and one discrete error problem this way only two auxiliary problems are required irrespective of the number of target functionals the practical performance of this approach is demonstrated for a laminar compressible flow in particular the proposed approach is compared to the standard approach of error estimation and goal oriented refinement as well as to residual based refinement the performance of the algorithms is measured in terms of computing resources required for meeting industrial as well as academic accuracy requirements on the computed force coefficients

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

instability for standing waves of nonlinear klein gordon equations via mountain pass arguments we introduce mountain pass type arguments in the context of orbital instability for klein gordon equations our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability instability for a power type nonlin earity we prove that the ground states of the associated stationary equation are minimizers of the functional action on a wide variety of constraints for a general nonlinearity we extend to the dimension n 2 the classical in stability result for stationary solutions of nonlinear klein gordon equations proved in 1985 by shatah in dimension n 3

flow to partially penetrating wells in unconfined heterogeneous aquifers mean head and interpretation of pumping tests 1 a partially penetrating well of length lw and radius rw starts to pump at constant discharge qw at t 0 from an unconfined aquifer of thickness d the aquifer is of random and stationary conductivity characterized by kg geometric mean y2 log conductivity variance and i and iv the horizontal and vertical integral scales the flow problem is solved under a few simplifying assumptions commonly adopted in the literature for homogeneous media rw lw 1 linearization of the free surface condition and constant drainable porosity n additionally it is assumed that rw i 1 and lw iv 1 to simplify the well boundary conditions and that a first order approximation in y2 extended to finite y2 on a conjectural basis is adopted the solution is obtained for the mean head field h r z t and the associated water table equation the main result of the analysis is that the flow domain can be divided into three zones for h 1 the neighborhood of the well r i where h qw lwka h0 r z tkefuv nd with h0 being the zero order solution pertaining to a homogeneous and isotropic aquifer ka being the conductivity arithmetic mean and kefuv being the effective vertical conductivity in mean uniform flow 2 an exterior zone r i in which h qw lwkefuh h0 r z tkefuv nd with kefuh being the horizontal effective conductivity and 3 an intermediate zone in which the solution requires a few numerical quadratures not carried out here the application to pumping tests reveals that identification of the aquifer parameters for homogeneous and anisotropic aquifers by commonly used methods can be applied for the drawdown measured in an observation well of length lowiv to ensure exchange of space and ensemble head averages in the second zone in order to identify kefuh kefuv and n in contrast the use of the drawdown in the well first zone leads to an overestimation of kefuh by the factor ka kefuh

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

solute transport in highly heterogeneous aquifers the paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity y ln k available solutions obtained from perturbation methods are limited to first and second order solutions valid only for small values of the log conductivity variance 2y published numerical investigations give comparative results with finite values of 2y but some discrepancies among the results generate doubts about the capability of numerical methods to capture high order effects for media with large variance values when these large 2y values are encountered in natural formations the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process the nonlinearity consequences are here investigated in two dimensional isotropic porous media by the monte carlo technique coupled with a finite element analysis the analysis includes 2y values from 0 05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution to dissipate the doubts related to the numerical approach the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors the numerical results confirm the validity of the first order and second order analyses when 2y 0 the second order solution captures the nonlinear effects in a small range of log conductivity variance close to zero the nonlinear terms neglected in the first order formulation for higher 2y values give 1 late travel time longitudinal dispersion values greater than the linear solution 2 notably non gaussian distribution of the lagrangian velocity and particle displacements and 3 travel times to approach the asymptotic fickian regime longer than those obtained using the linear solution

adaptive discontinuous galerkin finite element methods for nonlinear hyperbolic conservation laws we consider the a posteriori error analysis and adaptive mesh design for discontinuous galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws in particular we discuss the question of error estimation for general target functionals of the solution typical examples include the outflow flux local average and pointwise value as well as the lift and drag coefficients of a body immersed in an inviscid fluid by employing a duality argument we derive so called weighted or type i a posteriori error bounds these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated based on the resulting approximate type i bound we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest the performance of the proposed adaptive strategy and the quality of the approximate type i a posteriori error bound is illustrated by a series of numerical experiments in particular we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ type ii error indicators

estimating the variance and integral scale of the transmissivity field using head residual increments a modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals termed head residual increments hris are related to the variance and integral scale of the transmissivity field a first order solution is developed for the case of a transmissivity field which is isotropic and whose second order behavior can be characterized by an exponential covariance structure the estimates of the variance y2 and the integral scale of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the hri to its sample semivariogram this approach is applied to head data sampled from a series of two dimensional simulated aquifers with isotropic exponential covariance structures and varying degrees of heterogeneity y2 0 25 0 5 1 0 2 0 and 5 0 the results show that this method provided reliable estimates for both and y2 in aquifers with the value of y2 up to 2 0 but the errors in those estimates were higher for y2 equal to 5 0 it is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments

on asymptotic stability of stationary solutions to nonlinear wave and klein gordon equations we consider nonlinear wave and klein gordon equations with general nonlinear terms localized in space conditions are found which provide asymptotic stability of stationary solutions in local energy norms these conditions are formulated in terms of spectral properties of the schrodinger operator corresponding to the linearized problem they are natural extensions to partial differential equations of the known lyapunov condition for the nonlinear wave equation in three dimensional space we find asymptotic expansions as t of the solutions which are close enough to a stationary asymptotically stable solution

goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations

inverse modeling of multimodal conductivity distributions 1 we present a method for the calibration of multimodal hydraulic conductivity distributions and apply this method to the particular case of confining layers with a complex geological architecture the basis of our technique is the transformation of the original multimodal conductivity distribution to the standard normal distribution thus fulfilling the condition of normality which is required by the used representer based inverse algorithm valstar et al 2004 using this transformation a calibration that starts from a homogeneous prior field is shown to radically improve the estimation of the protective properties of the confining layer compared to a unimodal approach to the calibration the method is also used for the calibration of multimodal heterogeneous prior fields the inevitable distortion of the original parameter covariances in the posterior fields that results from the transformation process is absorbed by an iterative postprocessing procedure in which lithologic information obtained from the distorted calibrated fields is used to condition the generation of a new multimodal field that complies again with the original geostatistics after transformation this new field can be calibrated again and this process is repeated until the newly generated field agrees with the measurement information sufficiently well then the lithologic distribution of this new field is fixed and the intrafacies conductivity distributions are calibrated this approach is shown to preserve the original geostatistics both of the lithology field and of the intralithology hydraulic conductivity distributions

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

stochastic capture zone delineation within the generalized likelihood uncertainty estimation methodology conditioning on head observations a stochastic methodology to evaluate the predictive uncertainty in well capture zones in heterogeneous aquifers with uncertain parameters is presented the approach is based on the generalized likelihood uncertainty estimation methodology the hydraulic conductivity is modeled as a random space function allowing for the uncertainty that stems from the imperfect knowledge of the parameters of the correlation structure parameters are sampled from prior distributions and are used for the generation of a large number of hydraulic conductivity fields which are subsequently used to solve the groundwater flow equation a likelihood is calculated for every simulation based on some goodness of fit measure between simulated heads and available observations using inverse particle tracking a capture zone is determined which is assigned the likelihood calculated for that particular simulation statistical analysis of the ensemble of all simulations enables the predictive uncertainty of the well capture zones to be defined results are presented for a hypothetical test case and different likelihood definitions used in the conditioning process the results show that the delineated capture zones are most sensitive to the mean hydraulic conductivity and the variance whereas the integral scale of the variogram is the parameter with the smallest influence for all likelihood measures the prior uncertainty is reduced considerably by introducing the observation heads but the reduction is most effective for the very selective likelihood definition the method presented can be used in real applications to quantify the uncertainty in the location and extent of well capture zones when little or no information is available about the hydraulic properties through the conditioning on head observations

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

statistical estimation of polynomial generalized covariance functions and hydrologic applications minimum variance unbiased linear estimation theory has found many applications in the analysis of spatial rainfall and hydrogeologic data estimation of point values or areal averages calculation of variances of estimation error and network design however the important problem of inferring the spatial structure or interdependence of the function of interest has received relatively less attention three methods for the estimation of the parameters of matheron s polynomial generalized covariance function a useful model for nonstationary functions are presented and evaluated maximum likelihood ml minimum variance unbiased quadratic mvuq and minimum norm least squares the estimation error covariance matrix of the parameters is calculated with each of these methods ml and mvuq estimation have very good properties and for reasonably large samples give very similar results minimum norm estimation is appropriate for preliminary data analysis the application of these methodologies is illustrated with a set of rainfall data and problems of implementation are discussed it is shown that the estimation of the parameters of the covariance function involves sizeable errors it is suggested that in practice it may be rather difficult to identify the parameters of all but simple models it is important that future research evaluate the effects of parameter uncertainty in the circulation of estimation error variances of point or areal averages and in the design of hydrologic data collection networks

core scale solute transport model selection using monte carlo analysis 1 model applicability to core scale solute transport is evaluated using breakthrough data from column experiments conducted with conservative tracers tritium 3h and sodium 22 22na and the retarding solute uranium 232 232u the three models considered are single porosity double porosity with single rate mobile immobile mass exchange and the multirate model which is a deterministic model that admits the statistics of a random mobile immobile mass exchange rate coefficient the experiments were conducted on intact culebra dolomite core samples previously data were analyzed using single porosity and double porosity models although the culebra dolomite is known to possess multiple types and scales of porosity and to exhibit multirate mobile immobile domain mass transfer characteristics at field scale the data are reanalyzed here and null space monte carlo analysis is used to facilitate objective model selection prediction or residual bias is adopted as a measure of the model structural error the analysis clearly shows single porosity and double porosity models are structurally deficient yielding late time residual bias that grows with time on the other hand the multirate model yields unbiased predictions consistent with the late time 5 2 slope diagnostic of multirate mass transfer the analysis indicates the multirate model is better suited to describing core scale solute breakthrough in the culebra dolomite than the other two models

a bayesian criterion for sample size an i i d sample xm x1 xm is to be drawn from a distribution f of unknown form where the sample size m must be determined before sampling we wish to estimate f the mean of f the prior measure for f is p f the purposes of this note are firstly to suggest an intuitive criterion for the sample size and secondly to derive an upper bound for this quantity involving only certain simple aspects of p the motivation for the criterion is as follows suppose we are able to take observations sequentially having observed a sample xn xn we might stop sampling if we believed it to be unlikely that any further observations would cause us to substantially revise our current estimate thus a possible stopping criterion might be to stop sampling having observed x provided that for specified positiye constants a b

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

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

stochastic modeling of vertically averaged concentration uncertainty in a perfectly stratified aquifer solute concentration which results from transport in an aquifer with unknown variability in hydraulic conductivity can be subject to large predictive uncertainty this uncertainty can be modeled using a stochastic approach in which conductivity is represented by a random field with known probabilistic structure in this work we use monte carlo methods of stochastic modeling to study the uncertainty in vertically flux averaged concentration for transport in the simplified case of a perfectly stratified aquifer we assume steady flow parallel to bedding and negligible transverse dispersion we measure uncertainty with the ensemble coefficient of variation of concentration and find it to be large in most conditions examined we consider the validity of an ergodic assumption practical equivalence of ensemble mean and single realization concentrations by investigating the dependence of uncertainty upon the vertical averaging length even for averaging over a depth of 100 times the vertical correlation scale of conductivity uncertainty is large enough around 20 to be considered significant in many predictive situations we conclude that prediction based solely on the ensemble mean concentration would be inappropriate for this simple system and emphasize the importance of the ensemble variance for quantification of predictive uncertainty in stochastic modeling of solute transport

comment on application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity by r mose p siegel p ackerer and g chavent

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities the accuracy of the darcy velocity flux and stream function computed from lowest order triangle based control volume and mixed finite element approximations to the two dimensional pressure equation is considered the control volume finite element method similar to integrated finite difference methods and analogous to the interpolation of galerkin finite element results over control volumes is shown to yield a conservative velocity field and smooth streamlines the streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method which approximates the pressure and velocity variables separately it is shown that for systems with only moderate degrees of heterogeneity the control volume finite element method is the more computationally efficient alternative i e it provides more accurate flow results for a given number of unknowns for more variable or discontinuous permeability fields by contrast such as sand shale systems the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

continuous time stochastic analysis of groundwater flow in heterogeneous aquifers the problem of depth averaged groundwater flow in heterogeneous aquifers is looked at from a stochastic point of view the galerkin finite element version of the linearized stochastic equation governing the flow is solved analytically in time using an eigenvalue eigenvector technique the stochastic solution which is valid for small variance of aquifer log transmissivity relates spatial and temporal variabilities of aquifer head to aquifer heterogeneity stochastic recharge and random initial head the computational effort requires the evaluation of integrals of matrices whose elements are linear functions of the nodal mean heads the solutions are obtained for the exact i e continuous and quasi steady approximation of the mean head aquifer head temporal covariances are evaluated using standard matrix operations once the storage matrix is inverted and the associated generalized eigenvalue problem is solved implication of the level of spatial discretization on the performance of the solution is examined and the influence of aquifer heterogeneity on spatial and temporal variances of hydraulic heads is investigated in the presence of a semipervious boundary

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

transmissivity upscaling in numerical aquifer models of steady well flow unconditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t typically the numerical element scale r and the transmissivity integral scale i are both of the order of hundreds of meters then the upscaled block transmissivity is random and its statistical moments depend on those of t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we derive unconditional statistics of the upscaled ln accurate to the first order in y2 for strongly nonuniform flow in a circular block of radius r centered on a well of radius rw in an unbounded domain after a proper definition it was found that the ensemble mean is approximately but not exactly equal to ln tg the geometric mean and that the variance ratio 2 y2 which depends on rw i and the shape of the correlation drops slowly from unity for r i 0 to approximately 0 6 for r i 10 hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow additionally defining the equivalent teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge we find that in radial flow teq th the harmonic mean and grows slowly with increasing r i whereas for mean uniform flow teq tg application of the procedure is illustrated for an example of aquifer with selected values of parameters

isoinertial family of operators and convergence of kdv cnoidal waves to solitons abstract in this paper we show the convergence of korteweg de vries cnoidal waves to the limit soliton it is proved that the convergence is uniform and in h 2 norm as the period of the solutions tends to infinity families of hill operators are also studied we obtain a condition under which families of operators are isoinertial this condition is satisfied for classes of hill operators that are obtained by linearization our application is to the family of linearized operators at the kdv cnoidal waves it is proved that this family is isoinertial and also the value of the inertial index is calculated

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

quasi monte carlo methods for elliptic pdes with random coefficients and applications we devise and implement quasi monte carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients our motivation comes from fluid flow in random porous media where relevant functionals include the fluid pressure velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field our emphasis is on situations where a very large number of random variables is needed to model the coefficient field as an alternative to classical monte carlo we here employ quasi monte carlo methods which use deterministically chosen sample points in an appropriate usually high dimensional parameter space each realization of the pde solution requires a finite element fe approximation in space and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid not necessarily the same as the fe grid in the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using fft in this way we avoid the use of a truncated karhunen loeve expansion but introduce high nominal dimension in parameter space numerical experiments with 2 dimensional rough random fields high variance and small length scale are reported showing that the quasi monte carlo method consistently outperforms the monte carlo method with a smaller error and a noticeably better than o n 1 2 convergence rate where n is the number of samples moreover the rate of convergence of the quasi monte carlo method does not appear to degrade as the nominal dimension increases examples with dimension as high as 10 6 are reported

an indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity solute transport takes place in heterogeneous porous formations with the log conductivity y ln k modeled as a stationary random space function of given univariate normal probability density fu

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

a local anisotropic adaptive algorithm for the solution of low mach transient combustion problems a novel numerical algorithm for the simulation of transient combustion problems at low mach and moderately high reynolds numbers is presented these problems are often characterized by the existence of a large disparity of length and time scales resulting in the development of directional flow features such as slender jets boundary layers mixing layers or flame fronts this makes local anisotropic adaptive techniques quite advantageous computationally in this work we propose a local anisotropic refinement algorithm using for the spatial discretization unstructured triangular elements in a finite element framework for the time integration the problem is formulated in the context of semi lagrangian schemes introducing the semi lagrange galerkin slg technique as a better alternative to the classical semi lagrangian sl interpolation the good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem the flame vortex interaction first a premixed methane air flame vortex interaction with simplified transport and chemistry description test i is considered results are found to be in excellent agreement with those in the literature proving the superior performance of the slg scheme when compared with the classical sl technique and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement as a more realistic example we then conduct simulations of non premixed hydrogen air flame vortex interactions test ii using a more complex combustion model which involves state of the art transport and chemical kinetics in addition to the analysis of the numerical features this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature

bayesian and mixed bayesian likelihood criteria for sample size determination sample size estimation is a major component of the design of virtually every experiment in medicine prudent use of the available prior information is a crucial element of experimental planning most sample size formulae in current use employ this information only in the form of point estimates even though it is usually more accurately expressed as a distribution over a range of values in this paper we review several bayesian and mixed bayesian likelihood approaches to sample size calculations based on lengths and coverages of posterior credible intervals we apply these approaches to the design of an experiment to estimate the difference between two binomial proportions and we compare results to those derived from standard formulae consideration of several criteria can contribute to selection of a final sample size

closed form expressions for bayesian sample size sample size criteria are often expressed in terms of the concentration of the posterior density as controlled by some sort of error bound since this is done pre experimentally one can regard the posterior density as a function of the data thus when a sample size criterion is formalized in terms of a functional of the posterior its value is a random variable generally such functionals have means under the true distribution we give asymptotic expressions for the expected value under a fixed parameter for certain types of functionals of the posterior density in a bayesian analysis the generality of our treatment permits us to choose functionals that encapsulate a variety of inference criteria and large ranges of error bounds consequently we get simple inequalities which can be solved to give minimal sample sizes needed for various estimation goals in several parametric examples we verify that our asymptotic bounds give good approximations to the expected values of the functionals they approximate also our numerical computations suggest our treatment gives reasonable results

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

goal oriented a posteriori error estimation for the travel time functional in porous media flows in this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows the key application of this work is in the safety assessment of radioactive waste facilities in this setting the travel time functional measures the time taken for a nonsorbing radioactive solute transported by groundwater to travel from a potential site deep underground to the biosphere to ensure the computability of the travel time functional we employ a mixed formulation of darcy s law and conservation of mass together with raviart thomas h mathrm div omega conforming finite elements the proposed a posteriori error bound is derived based on a variant of the standard dual weighted residual approximation which takes into account the lack of smoothness of the underlying functional of interest the proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the

reply to comment on application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity by r mose p siegel p ackerer and g chavent

using the nonstationary spectral method to analyze flow through heterogeneous trending media this paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity this theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories the nonstationary theory is illustrated with a two dimensional analysis of a linear trend aligned with the mean flow direction in this case closed form solutions can be obtained for the effective hydraulic conductivity head covariance and log conductivity head cross covariance the effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity head cross covariance such changes have important implications for measurement conditioning or cokriging methods which rely on the head covariance and log conductivity head covariance the nonstationary spectral analysis is also compared with classical spectral analysis this comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media the stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity head cross covariance although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope the stationary and nonstationary results are the same when there is no trend in log conductivity the trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

reply to comments on comparison of gaussian conditional mean and kriging estimation in the geostatistical solution of the inverse problem by r j hoeksema and p k kitanidis

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory

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

evaluation of mixed crank nicolson scheme and tau method for the solution of klein gordon equation abstract numerical method based on the crank nicolson scheme and the tau method is proposed for solving nonlinear klein gordon equation nonlinear klein gordon equation is reduced by crank nicolson scheme to the system of ordinary differential equations then tau method is used to solve this system by using interpolating scaling functions and operational matrix of derivative the order of convergence is proposed and some numerical examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity selected groundwater flow scenarios are used in a two way comparison between the mixed hybrid finite element method and the standard finite element method also called the conforming finite element method the simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists the basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element by contrast the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential the conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity the mixed hybrid finite element method provides velocities everywhere in the field as well as potentials at the center of each element and each edge moreover the normal component of the velocity field is continuous between adjacent elements the results of the simulations are presented in the form of streamlines to avoid the problem of velocity discontinuity the method of cordes and kinzelbach 1992 is used it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method the comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy it is also superior to the conforming method in terms of computational effort the potential fields obtained by the mixed hybrid and the conforming finite element methods are the same

adaptive discontinuous galerkin finite element methods for the compressible euler equations in this paper a recently developed approach for the design of adaptive discontinuous galerkin finite element methods is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the euler equations of gas dynamics in particular we employ weighted type i a posteriori bounds to drive adaptive finite element algorithms for the estimation of the error measured in terms of general target functionals of the solution these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated this general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest as well as providing reliable and efficient error estimation the superiority of the proposed approach over mesh refinement algorithms which employ standard unweighted type ii error indicators which do not require the solution of an auxiliary problem are illustrated by a series of numerical experiments here we consider transonic flow through a nozzle as well as subsonic and supersonic flows around different airfoil geometries

adjoint based h p adaptive discontinuous galerkin methods for the 2d compressible euler equations in this paper we investigate and present an adaptive discontinuous galerkin algorithm driven by an adjoint based error estimation technique for the inviscid compressible euler equations this approach requires the numerical approximations for the flow i e primal problem and the adjoint i e dual problem which corresponds to a particular simulation objective output of interest the convergence of these two problems is accelerated by an hp multigrid solver which makes use of an element gauss seidel smoother on each level of the multigrid sequence the error estimation of the output functional results in a spatial error distribution which is used to drive an adaptive refinement strategy which may include local mesh subdivision h refinement local modification of discretization orders p enrichment and the combination of both approaches known as hp refinement the selection between h and p refinement in the hp adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values numerical results for the inviscid compressible flow over an idealized four element airfoil geometry demonstrate that both pure h refinement and pure p enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach but requiring fewer degrees of freedom the proposed hp adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom and results in significant savings in computational cost a high speed flow test case is used to demonstrate the ability of the hp refinement approach for capturing strong shocks or discontinuities while improving functional accuracy

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples