prediction of steady state flow of real gases in randomly heterogeneous porous media abstract we consider steady state flow of real gases through bounded randomly heterogeneous porous media such flow is described by a nonlinear partial differential equation with the random coefficient medium s permeability and source terms subject to randomly prescribed boundary conditions prior to applying stochastic analysis the problem is linearized by means of the kirchhoff transformation which allows us to obtain the exact expressions for an effective upscaled gas permeability in particular for one two and three dimensional mean uniform flows in infinite statistically homogeneous and isotropic domains the resulting effective permeability is given by harmonic geometric and arithmetic averages respectively the influence of statistical anisotropy of the random permeability field and domain s boundaries on the effective gas permeability is also investigated

statistical theory of groundwater flow and transport pore to laboratory laboratory to formation and formation to regional scale the statistical approach has been applied increasingly to groundwater flow problems in the last decade as is illustrated in figure 3 by the cumulative number of articles published in this field in water resources reseach this development has been motivated by the recognition of the fact that porous formations are heterogeneous i e with properties which vary in an irregular manner in space flow domains are characterized by the length scale l of their spatial extent and three such scales of a fundamental nature are introduced the laboratory the local and the regional scale heterogeneity is characterized by the spatial correlation scale i of the property of interest the three scales corresponding to the above ones being the pore scale the log hydraulic conductivity and the log transmissivity integral scales the medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements they enjoy some type of stationarity and i l an additional scale d is the measurement or computational scale characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes in both cases the interest resides in the space average of the flow variable over a volume or area of length scale d the primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space averaged variables given the statistical structure of the spatially variable property the main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach along this line the specific aspects of each scale are examined separately and areas of interest for future research are indicated in the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory beyond the stage of an ad hoc modeling technique

multilayered leaky aquifer systems 2 boundary element solutions the boundary element algorithm for the multilayered leaky aquifer system of neuman witherspoon and herrera is presented by applying the laplace transform to the integrodifferential equations of herrera a set of coupled two dimensional elliptic equations results the boundary integral representation is derived from green s theorem green s functions governed by an adjoint system of the original governing equations are found from the pumping well solutions given in part 1 by a coefficient exchange the resultant numerical system requires a discretization only at the boundaries of the planar aquifers and is free from aquitard unknowns several numerical examples including one dimensional flow pumping well problems and two dimensional problems are provided for the purpose of validation and demonstration

learning prototype hyperplanes for face verification in the wild in this paper we propose a new scheme called prototype hyperplane learning phl for face verification in the wild using only weakly labeled training samples i e we only know whether each pair of samples are from the same class or different classes without knowing the class label of each sample by leveraging a large number of unlabeled samples in a generic data set our scheme represents each sample in the weakly labeled data set as a mid level feature with each entry as the corresponding decision value from the classification hyperplane referred to as the prototype hyperplane of one support vector machine svm model in which a sparse set of support vectors is selected from the unlabeled generic data set based on the learnt combination coefficients to learn the optimal prototype hyperplanes for the extraction of mid level features we propose a fisher s linear discriminant like fld like objective function by maximizing the discriminability on the weakly labeled data set with a constraint enforcing sparsity on the combination coefficients of each svm model which is solved by using an alternating optimization method then we use the recent work called side information based linear discriminant sild analysis for dimensionality reduction and a cosine similarity measure for final face verification comprehensive experiments on two data sets labeled faces in the wild lfw and youtube faces demonstrate the effectiveness of our scheme

theoretical and computational aspects of the stochastic boundary element method basic concepts of the stochastic boundary element computational methodology are presented stochastic boundary integral equations are discretized into a system of random algebraic equations using the midpoint and the boundary averaging methods the influence of the chosen boundary mesh geometry on covariances is tested applications of stochastic boundary elements to static and dynamic problems with random boundary conditions stochastic media stochastic sensitivity analysis and identification of internal defects are presented

deep transfer metric learning conventional metric learning methods usually assume that the training and test samples are captured in similar scenarios so that their distributions are assumed to be the same this assumption does not hold in many real visual recognition applications especially when samples are captured across different data sets in this paper we propose a new deep transfer metric learning dtml method to learn a set of hierarchical nonlinear transformations for cross domain visual recognition by transferring discriminative knowledge from the labeled source domain to the unlabeled target domain specifically our dtml learns a deep metric network by maximizing the inter class variations and minimizing the intra class variations and minimizing the distribution divergence between the source domain and the target domain at the top layer of the network to better exploit the discriminative information from the source domain we further develop a deeply supervised transfer metric learning dstml method by including an additional objective on dtml where the output of both the hidden layers and the top layer are optimized jointly to preserve the local manifold of input data points in the metric space we present two new methods dtml with autoencoder regularization and dstml with autoencoder regularization experimental results on face verification person re identification and handwritten digit recognition validate the effectiveness of the proposed methods

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

stokes flows under random boundary velocity excitations a viscous stokes flow over a disc under random fluctuations of the velocity on the boundary is studied we give exact karhunen loeve k l expansions for the velocity components pressure stress and vorticity and the series representations for the corresponding correlation tensors both the white noise fluctuations and general homogeneous random excitations of the velocities prescribed on the boundary are studied we analyze the decay of correlation functions in angular and radial directions both for exterior and interior stokes problems numerical experiments show the fast convergence of the k l expansions the results indicate that ignoring the stochastic fluctuations in boundary conditions dramatically underestimates the variance of the velocity and pressure in the interior exterior of the domain

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

boundary element solution for stochastic groundwater flow random boundary condition and recharge this paper presents an integral equation technique for solving stochastic boundary value problems in groundwater flow the aquifer considered has deterministic hydraulic conductivity but is subject to random boundary condition and domain recharge using the distribution of fictitious sources and dipoles stochastic integral equations for the mean and covariance of head and flux are derived an iterative boundary element technique is applied for numerical solution two one dimensional examples are examined and compared with exact solution a two dimensional problem is then presented

stochastic boundary elements for two dimensional potential flow in non homogeneous media abstract a stochastic boundary element formulation is presented for the treatment of two dimensional problems of steady state potential flow in non homogeneous media that involve a random operator in the governing differential equation the randomness is introduced through the material parameter of the domain which is described as a non homogeneous random field the random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem this leads to differential equations for the unknown potential and its first and second order derivatives respectively evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field an approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order these solutions are introduced into the differential equations and upon equating similar order terms a sequence of poisson s equations is obtained for each order a transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables the resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first and second order derivatives which are then used to compute the desired response statistics quadratic conforming boundary elements are used in the boundary integration and four node quadrilateral cells are used in the domain integration strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object the singular domain integrals are calculated analytically direct solution techniques are used to calculate the response variables and their derivatives respectively a number of example problems are presented and the results are compared with those obtained from monte carlo simulations a good agreement of the results is observed

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1 constant head boundary the spatial variability of log transmissivity y and head h in aquifers assumes a major role in determining the solutions of the direct flow problem as well as of the inverse problem investigation of the joint y h variability has been cast in the past in numerical and analytic frames analytic solutions for the y h statistical moments in two dimensions assumed the flow to occur in an unbounded domain in this study exact analytical expressions for the y h moments are derived which account for the presence of a given head boundary in a half plane this result is achieved by a first order approximation and for an exponential y covariance the result offers the opportunity to examine the effects of boundaries on the semivariogram of h and on the y h cross covariance the assumption of unbounded domain is examined and shown to be accurate at distances larger than two y integral scales from the boundary by using conditional probabilities a general and simple method to estimate the 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 analytic solution showing excellent results

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

stochastic analysis of three dimensional flow in a bounded domain a commonly accepted first order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of poisson s equation this form allows for the advantageous use of green s functions to solve for the random output hydraulic heads in terms of a convolution over the random input the logarithm of hydraulic conductivity a solution for steady state three dimensional flow in an aquifer bounded above and below is presented consideration of these boundaries is made possible by use of green s functions to solve poisson s equation within the bounded domain the medium hydraulic conductivity is assumed to be a second order stationary random process as represented by a simple three dimensional covariance function upper and lower boundaries are taken to be no flow boundaries the mean flow vector lies entirely in the horizontal dimensions the resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions comparisons are made with existing infinite domain solutions

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

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

fuzzy boundary element method for the analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches the uncertainties present may be associated with parameters that are vague imprecise or linguistic for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical system a fuzzy boundary element method for the analysis of imprecisely defined systems is developed starting from the basic concepts of fuzzy sets and fuzzy arithmetic the various steps of the boundary element method are redefined using fuzzy concepts the resulting fuzzy equations are solved using a fuzzified version of gaussian elimination procedure coupled with truncation the truncation method limits the growth of interval ranges of response parameters to obtain realistic and accurate solutions the procedure is illustrated by considering the analysis of a potential flow problem the methodology is applicable to uncertain systems that are described in linguistic terms as well as those that are described by incomplete information the approach represents a unique methodology that enables the analysis and design of many engineering systems more realistically

solution of stokes flow using an iterative drbem based on compactly supported positive definite radial basis function abstract this paper describes an iterative dual reciprocity boundary element method drbem based on the compactly supported positive definite radial basis function for the solution of stokes flow problems the method involves the solution of laplace equations for vorticity and poisson equations for velocity with the solenoidal components of vorticity as right hand sides these equations are solved using iterative bem and drbem for the drbem the compactly supported positive definite radial basis function is used to approximate the body force term to convert it to boundary integrals a relaxation method is devised such that the resulting algebraic equations are solved without the need of assembling a matrix hence large systems of equations such as the fine discretization needed for stokes flow problems can be handled easily the results of stokes flow in a square cavity and a circular cavity are presented and compared with other numerical model results the iterative drbem solution is found to be satisfactory

a fast learning algorithm for deep belief nets we show how to use complementary priors to eliminate the explaining away effects that make inference difficult in densely connected belief nets that have many hidden layers using complementary priors we derive a fast greedy algorithm that can learn deep directed belief networks one layer at a time provided the top two layers form an undirected associative memory the fast greedy algorithm is used to initialize a slower learning procedure that fine tunes the weights using a contrastive version of the wake sleep algorithm after fine tuning a network with three hidden layers forms a very good generative model of the joint distribution of handwritten digit images and their labels this generative model gives better digit classification than the best discriminative learning algorithms the low dimensional manifolds on which the digits lie are modeled by long ravines in the free energy landscape of the top level associative memory and it is easy to explore these ravines by using the directed connections to display what the associative memory has in mind

a distributed parameter approach for evaluating the accuracy of groundwater model predictions 2 application to groundwater flow the distributed parameter theory presented by mclaughlin and wood this issue is used to evaluate the accuracy of a groundwater flow model the special case investigated assumes that the model s prediction errors are due primarily to data limitations in this case approximate prediction error moments mean and covariance are obtained by solving two sets of coupled partial differential equations which have the same basic structure as the original flow equation solutions can be obtained with spectral green s function or numerical techniques depending on the assumptions made two examples are solved using a finite element approach the first two dimensional example confirms steady state infinite domain results obtained with spectral methods the second three dimensional example investigates the influence of spatial variability sampling strategy and suboptimal estimation on model prediction accuracy

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

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

complex variable boundary element method for potential flow with thin objects a complex variable boundary element method cvbem is developed for potential flow in the presence of thin objects singularities caused by such objects are expressed analytically through the conformal mapping technique and the resultant singular solution is coupled with the cvbem by the principle of superposition this can be done by virtue of the semi analytic nature of the cvbem the numerical results for example problems including flow barriers and high conductivity fractures confirm that the proposed model yields accurate solutions of the velocity potential and the stream function within the flow domain

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

flow and transport in highly heterogeneous formations 1 conceptual framework and validity of first order approximations 1 flow of uniform mean velocity u takes place in a formation of spatially variable random conductivity k x advective transport of a plume of an inert solute is investigated by the lagrangean approach the aim of the study is to determine the spatial moments of the plume i e of fluid particle trajectories for highly heterogeneous aquifers for which y 1 where y ln k a multi indicator model of the permeability structure which is different from the common multi gaussian one is proposed the formation is modeled as a collection of n blocks of different k j the structure is defined by the distribution of k j the blocks shape and the coordinates of their centroids the following simplifications are adopted the blocks are inclusions of a regular shape circles spheres for isotropic media investigated here defined by the radius a and the inclusions are not overlapping and their centroids are distributed uniformly and independently in space at the continuous limit the model is characterized by the joint pdf f y a the model is shown to be quite general and to comprise binary bimodal indicator variograms and unimodal distributions of y as particular cases the study is focused on the latter case with y normal n y y2 and stationary covariance of given integral scale iy these are the parameters commonly estimated for sedimentary formations this leaves still freedom in selecting the pdf f a the simple model selected for semianalytical and numerical analysis is that of inclusions of radius r and volume fraction n submerged in a matrix of effective conductivity kef the latter represents the effect of inclusions of much smaller radius which appear as a nugget in the log conductivity two point covariance an approximate analytical solution of the flow is obtained by using a self consistent approximation while a fully numerical one is derived in part 3 jankovic et al 2003a transport is solved by particle tracking and the time dependent spatial moments trajectories variance skewness kurtosis are presented in part 2 fiori et al 2003 in the self consistent approximation the asymptotic longitudinal macrodispersivity l which is a function of y shows strong nonlinear effects inclusions of large positive y lead to a finite l whereas l grows unbounded for those of negative y this effect is not captured by the common first order approximation in y which is symmetrical and overestimates l for y 0 and underestimates it for y 0 as a result the second spatial moment is predicted accurately by the first order approximation by cancellation of errors provided that f y is symmetrical however the transient regime and higher order moments are not captured by the first order approximation

boundary element solution for stochastic groundwater flow temporal weakly stationary problems this paper continues the development of boundary integral equation technique for solving stochastic groundwater flow problems the case of aquifers under deterministic hydraulic conductivity but subject to random boundary condition and domain recharge has been investigated in an earlier paper under the assumption of steady state governing equation in the current paper the governing equation is transient but the temporal variation of second order statistical moments is assumed to be weakly stationary several examples of fundamental importance to the understanding of stochastic process are examined both by the analytical and by the numerical boundary element techniques

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

supertori are algebraic curves super riemann surfaces of genus 1 with arbitrary spin structures are shown to be the sets of zeroes of certain polynomial equations in projective superspace we conjecture that the same is true for arbitrary genus properties of superelliptic functions and super theta functions are discussed the boundary of the genus 1 super moduli space is determined

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

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

discriminative deep metric learning for face and kinship verification this paper presents a new discriminative deep metric learning ddml method for face and kinship verification in wild conditions while metric learning has achieved reasonably good performance in face and kinship verification most existing metric learning methods aim to learn a single mahalanobis distance metric to maximize the inter class variations and minimize the intra class variations which cannot capture the nonlinear manifold where face images usually lie on to address this we propose a ddml method to train a deep neural network to learn a set of hierarchical nonlinear transformations to project face pairs into the same latent feature space under which the distance of each positive pair is reduced and that of each negative pair is enlarged to better use the commonality of multiple feature descriptors to make all the features more robust for face and kinship verification we develop a discriminative deep multi metric learning method to jointly learn multiple neural networks under which the correlation of different features of each sample is maximized and the distance of each positive pair is reduced and that of each negative pair is enlarged extensive experimental results show that our proposed methods achieve the acceptable results in both face and kinship verification

two dimensional stochastic analysis and optimal estimation in aquifers random recharge this paper investigates two dimensional aquifer flow under naturally variable recharge and its application to optimal estimation in groundwater the adopted framework is a stochastic one in which the governing stochastic equation is solved quasi analytically using the galerkin finite element method and matrix exponentials the continuous time stochastic solution relates head perturbations to random initial bead and a convolution of the pertinent stochastic recharge process based on the quasi analytical solution continuous time autocorrelation matrices for aquifer head are developed conforming to 1 white noise recharge fluctuations in time and 2 fully correlated recharge fluctuations in time from a stochastic point of view transient aquifer flow in its conventional meaning is only inferred when recharge fluctuations are fully correlated in time provided that the mean flow approaches steady state application to optimal state feedback estimation is demonstrated by adopting kalman filtering and forecasting to a numerical experiment consisting of two dimensional aquifer flow instigated by variable leakage filtering results demonstrate that under scarce measurements statistical conditioning on available measurements can result in an estimated aquifer hydraulic response that captures the actual variability that may exist under natural field conditions forecasted aquifer heads however are sufficient for a relatively short period they converge asymptotically to their respective average values

learning discriminant face descriptor local feature descriptor is an important module for face recognition and those like gabor and local binary patterns lbp have proven effective face descriptors traditionally the form of such local descriptors is predefined in a handcrafted way in this paper we propose a method to learn a discriminant face descriptor dfd in a data driven way the idea is to learn the most discriminant local features that minimize the difference of the features between images of the same person and maximize that between images from different people in particular we propose to enhance the discriminative ability of face representation in three aspects first the discriminant image filters are learned second the optimal neighborhood sampling strategy is soft determined third the dominant patterns are statistically constructed discriminative learning is incorporated to extract effective and robust features we further apply the proposed method to the heterogeneous cross modality face recognition problem and learn dfd in a coupled way coupled dfd or c dfd to reduce the gap between features of heterogeneous face images to improve the performance of this challenging problem extensive experiments on feret cas peal r1 lfw and hfb face databases validate the effectiveness of the proposed dfd learning on both homogeneous and heterogeneous face recognition problems the dfd improves poem and lqp by about 4 5 percent on lfw database and the c dfd enhances the heterogeneous face recognition performance of lbp by over 25 percent

discriminative multimanifold analysis for face recognition from a single training sample per person conventional appearance based face recognition methods usually assume that there are multiple samples per person mspp available for discriminative feature extraction during the training phase in many practical face recognition applications such as law enhancement e passport and id card identification this assumption however may not hold as there is only a single sample per person sspp enrolled or recorded in these systems many popular face recognition methods fail to work well in this scenario because there are not enough samples for discriminant learning to address this problem we propose in this paper a novel discriminative multimanifold analysis dmma method by learning discriminative features from image patches first we partition each enrolled face image into several nonoverlapping patches to form an image set for each sample per person then we formulate the sspp face recognition as a manifold manifold matching problem and learn multiple dmma feature spaces to maximize the manifold margins of different persons finally we present a reconstruction based manifold manifold distance to identify the unlabeled subjects experimental results on three widely used face databases are presented to demonstrate the efficacy of the proposed approach

an efficient deterministic probabilistic approach to modeling regional groundwater flow 1 theory an efficient probabilistic model is developed and cascaded with a deterministic model for predicting water table elevations in regional aquifers the objective is to quantify model uncertainty where precise estimates of water table elevations may be required the probabilistic model is based on the two point probability method which only requires prior knowledge of uncertain variables mean and coefficient of variation the two point estimate method is theoretically developed and compared with the monte carlo simulation method the results of comparisons using hypothetical determinisitic problems indicate that the two point estimate method is only generally valid for linear problems where the coefficients of variation of uncertain parameters for example storage coefficient and hydraulic conductivity is small the two point estimate method may be applied to slightly nonlinear problems with good results provided coefficients of variation are small in such cases the two point estimate method is much more efficient than the monte carlo method provided the number of uncertain variables is less than eight

analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1 steady flow the statistical structures of the hydraulic conductivity head gradient specific discharge and head fields are analyzed with the aid of a simplified model consisting of a collection of spherical in three dimensions and cylindrical in two dimensions blocks of different conductivity and diameter set at random and independently in space results of a previous work for the effective conductivity and unbounded formations dagan 1979 are recovered and given a rational derivation the influence of the aquifer boundary and nonuniform average flows are examined for the first time the growth of the head variance with distance from boundary in two dimensional flows is compared with that of the monte carlo simulations of smith and freeze 1979

wavelet based model for stochastic analysis of beam structures a wavelet based model for stochastic analysis of beam structures is presented in this model the random processes representing the stochastic material and geometric properties are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the process is represented as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients the correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets and the integral eigenvalue problem is converted to a finite dimensional eigenvalue problem the energy principle based finite element approach is used to obtain the equilibrium and boundary conditions neumann expansion of the stiffness matrix is used to write the nodal displacement vector in terms of random coefficients the expectation operator is applied to the nodal displacements and their squares to obtain the mean and standard deviation of the displacements studies show that the results obtained using this method compare well with monte carlo and semianalytical techniques

statistics of block conductivity through a simple bounded stochastic medium a monte carlo approach is employed to estimate the distribution of an equivalent conductivity measure the block conductivity which characterizes the total flow rate through a two dimensional bounded domain and which is itself a random variable chi square goodness of fit tests indicate that the lognormal distribution is an appropriate choice for the block conductivity distribution for square domains the mean and variance of the block conductivity are seen to be closely approximated using the statistics of local averages of log conductivity

first order analysis of uncertainty in numerical models of groundwater flow part 1 mathematical development in part 1 of this paper first and second order analysis of uncertainty is applied to numerical models of groundwater flow the models are cast in state space form with boundary conditions and inputs that are functionally dependent but statistically independent of time using a compact matrix calculus notation first and second order taylor series expansions of the model equations are derived and used to estimate the mean and variance covariance properties of piezometric head predictions given corresponding statistics for aquifer parameters material properties initial conditions boundary conditions and inputs the mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty and sensitivity of the predictions to the parameters furthermore the first order estimate of the piezometric head is identical to the dete ministic result part 2 of this paper to be presented later will illustrate these and other results through numerous applications of the methodology

role of boundary conditions in convergence and nonlocality of solutions to stochastic flow problems in bounded domains steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme the effect of boundary conditions on a two dimensional flow with arbitrary variation of the mean flow allowing large gradients is investigated using analytical expressions for the head and velocity covariance functions boundary conditions were decomposed into deterministic and stochastic components two flow cases with the same zero order and different first order boundary conditions were analyzed boundary conditions are deterministic for the first case and random for the second significant differences between the two cases indicate random or absence of randomness processes must be modeled at boundaries first order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived the resulting integral kernels involve greens functions and they are evaluated by numerical integration boundary conditions for higher order problems influence the absolute value and shape of the kernels and therefore of the head and velocity variance and covariance functions it is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition f2 1 where f is the variance of the log hydraulic conductivity assumed gaussian and weakly homogeneous in space the type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem milder head gradients also allow larger values of f2 nonlocality is also contingent on boundary type weakly homogeneous log fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous the inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity higher gradient regions induce higher head and velocity variances in the presence of space varying gradients nonlocal effects are most important away from the boundaries lpar center of the domain small local head gradients result in small head and velocity gradients and therefore where observations are made in stagnation regions data should be analyzed taking into account this effect the results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

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

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

generation of random fields characterized by discrete regions of constant value a new simulator is introduced which provides realizations of parameter fields characterized by randomly sized regions within which the parameter value is constant this generator produces second order stationary random fields with specified mean and correlation structure the structure of the simulated field is significantly different than that which can be obtained by previously existing techniques this new structure provides a valuable simulation tool for discretely varying fields such as required when modeling complex lithologies one application of this method utilizing ovals in two dimensions and ellipsoids in three dimensions is demonstrated for grids in which a finite number of simulation points are to be generated this technique is compared with a recently published turning bands routine and is faster in terms of cpu time for large values of i l where i is the integral scale of the correlation structure and l is the length of the longest side of the simulation grid for small values of i l the turning bands routine is faster conditional simulation based on kriging has been studied for this method the resulting random field varies continuously rather than discretely a second method of conditioning which produces discretely varying random fields is suggested but not examined computationally

a kernel approach for semisupervised metric learning while distance function learning for supervised learning tasks has a long history extending it to learning tasks with weaker supervisory information has only been studied recently in particular some methods have been proposed for semisupervised metric learning based on pairwise similarity or dissimilarity information in this paper we propose a kernel approach for semisupervised metric learning and present in detail two special cases of this kernel approach the metric learning problem is thus formulated as an optimization problem for kernel learning an attractive property of the optimization problem is that it is convex and hence has no local optima while a closed form solution exists for the first special case the second case is solved using an iterative majorization procedure to estimate the optimal solution asymptotically experimental results based on both synthetic and real world data show that this new kernel approach is promising for nonlinear metric learning

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

analysis of flow through heterogeneous random aquifers 2 unsteady flow in confined formations the direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space closed form expressions are derived for the effective conductivity or storativity and for the head variogram two types of unsteady flow are considered transients and periodic flows in the case of transients it is assumed that initially the head is constant no flow and that ultimately the flow becomes uniform it is found that the effective conductivity and transmissivity are time dependent and drop from the arithmetic mean initially to the steady state value during a relaxation time the latter is much larger for one dimensional flows than for two or three dimensional ones the head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag while in three dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale hence steady state analysis for transients might not be warranted for regional transient flows in the case of periodic flows of seasonal nature the head gradient is assumed to be made up from a steady component and a periodic one if the amplitude of the latter is sufficiently small compared to the steady one steady state value of effective transmissivity and storativity can be adopted the head variogram has also its steady state form for large lags and quasisteady for small lags hence the variogram lies between the steady one steady component of head gradient and the quasisteady one instantaneous head gradient

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

classical yang baxter equation and the a constraint abstract we show that elliptic solutions of classical yang baxter equation cybe can be obtained from triple massey products on an elliptic curve we introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions we also study some degenerations of these solutions