a methodology to determine optimal transmissivity measurement locations in groundwater quality management models with scarce field information a new criterion to determine transmissivity measurement location for groundwater quality management models is proposed the transmissivity is represented as a random field with known mean and covariance function conditioned to existing measurements values the criterion is the minimization of the coefficient of variation of the optimal remediation cost distribution this distribution is computed numerically by means of monte carlo simulations in which each optimal remediation cost is evaluated by solving a deterministic management problem where the transmissivity is a single realization of the random field the coefficient of variation is equivalent to the relative error in the prediction of the optimal cost and is shown to be invariant with respect to the actual unknown measurement value according to some limiting conditions in the more general case the coefficient of variation of the optimal cost is less sensitive than the expected optimal cost to the actual measurement value the proposed criterion becomes a useful tool to determine the best transmissivity measurement location when only few measurements have been performed

sensitivity analysis for steady state groundwater flow using adjoint operators adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system for groundwater flow systems performance measures of interest include piezometric heads in the vicinity of a waste site velocities or travel time in aquifers and mass discharge to biosphere points the parameters include recharge discharge rates prescribed boundary heads or fluxes formation thicknesses and hydraulic conductivities the derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity to calculate sensitivities adjoint sensitivity equations are formulated from the equations describing the primary problem the solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients in this study adjoint sensitivity theory is developed for equations of two dimensional steady state flow in a confined aquifer both the primary flow equation and the adjoint sensitivity equation are solved using the galerkin finite element method the developed computer code is used to investigate the regional flow parameters of the leadville formation of the paradox basin in utah the results illustrate the sensitivity of calculated local heads to the boundary conditions alternatively local velocity related performance measures are more sensitive to hydraulic conductivities

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

an extended identifiability in aquifer parameter identification and optimal pumping test design an extended identifiability called identifiability is developed for groundwater modeling and management some fundamental concepts are developed for establishing a criterion in connection with the problem of optimal experimental design such as the design of an optimum pumping test to assist aquifer parameter identification in practice the quantity and quality of data prohibits a unique solution of the inverse problem of parameter identification the proposed identifiability is based upon the concept of weak uniqueness the determination of the admissibility of a given design is formulated as a nonlinear programming problem the original constrained problem is transformed into solving a sequence of unconstrained problems by a penalty function method numerical experiments are conducted to illustrate the proposed concept and algorithm

boolean rings and reciprocal eigenvalue properties abstract let a be the adjacency matrix of the zero divisor graph r of a finite commutative ring r containing nonzero zero divisors in this paper it is shown that r is the zero divisor graph of a boolean ring if and only if det a 1 also a is similar to plus or minus its inverse whenever r is a boolean ring as a consequence it is proved that r is the zero divisor graph of a boolean ring if and only if the set of eigenvalues including multiplicities of r can be partitioned into 2 element subsets of the form 1 furthermore any finite boolean ring r is characterized by the degree and coefficients of the characteristic polynomial of a

behavior of sensitivities in the one dimensional advection dispersion equation implications for parameter estimation and sampling design the spatial and temporal variability of sensitivities has a significant impact on parameter estimation and sampling design for studies of solute transport in porous media physical insight into the behavior of sensitivities is offered through an analysis of analytically derived sensitivities for the one dimensional form of the advection dispersion equation when parameters are estimated in regression models of one dimensional transport the spatial and temporal variability in sensitivities influences variance and covariance of parameter estimates several principles account for the observed influence of sensitivities on parameter uncertainty 1 information about a physical parameter may be most accurately gained at points in space and time with a high sensitivity to the parameter 2 as the distance of observation points from the upstream boundary increases maximum sensitivity to velocity during passage of the solute front increases and the consequent estimate of velocity tends to have lower variance 3 the frequency of sampling must be in phase with the s shape of the dispersion sensitivity curve to yield the most information on dispersion 4 the sensitivity to the dispersion coefficient is usually at least an order of magnitude less than the sensitivity to velocity 5 the assumed probability distribution of random error in observations of solute concentration determines the form of the sensitivities 6 if variance in random error in observations is large trends in sensitivities of observation points may be obscured by noise and thus have limited value in predicting variance in parameter estimates among designs 7 designs that minimize the variance of one parameter may not necessarily minimize the variance of other parameters 8 the time and space interval over which an observation point is sensitive to a given parameter depends on the actual values of the parameters in the underlying physical system

estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion tension disc infiltrometers are becoming increasingly popular devices for in situ measurement of the unsaturated hydraulic properties of soil tension infiltration data are generally used to evaluate the parametersks and ain gardner s exponential model of the unsaturated hydraulic conductivity either two measurements using different disc diameters or measurements with a single disc but using multiple pressure heads are then used in this paper we describe a parameter estimation procedure which combines the levenberg marquardt nonlinear parameter optimization method involving weighted least squares with a quasi three dimensional numerical model which solves the variably saturatedflow equation by numerical inversion of richards equation the unknown parameters in van genuchten s model of the unsaturated soil hydraulic properties are estimated from observed cumulative infiltration data during transient waterflow additional measurements of the pressure head or water content as well as a penalty function for constraining the unknown parameters to remain in some feasible region bayesian estimation can be optionally included into the parameter estimation procedure the problem of optimal sampling design that is selecting the best points in space and time for making measurements is addressed by studying the sensitivity of the objective function to changes in the optimized hydraulic parameters we calculate objective functions based on available cumulative infiltration pressure head and water content measurements and also on several combinations of these data the behavior of the objective function in three dimensional parameter space is evaluated by means of a series of two dimensional response surfaces the utility of the parameter estimation procedure is demonstrated using numerically generated data the sensitivity of the procedure to different initial estimates of the model parameters is also discussed

sensitivity methods for time continuous spatially discrete groundwater contaminant transport models existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time stepping numerical algorithms and cannot be readily used with time continuous approaches to transport simulation such as the laplace transform galerkin technique we develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the laplace domain and inverted numerically to the time domain the methods are computationally efficient when used in conjunction with time continuous transport equations the relative efficiency of the two methods depends on the number of model parameters number of performance measures and number of spatial discretization nodes the adjoint method is favored when the number of performance measures is much smaller than the number of model parameters the adjoint method is limited in that performance measures are restricted to being linear functions of state variables a two dimensional transport example is developed in detail and sensitivities with respect to nodal hydraulic conductivities are computed in the problems analyzed the direct and adjoint methods are 9 to 156 times faster than the perturbation method with the computational savings increasing as the size of the problem is increased

experimental design and model parameter estimation for locating a dissolving dense nonaqueous phase liquid pool in groundwater 1 this paper investigates the experimental design problem in the context of estimating the location and dimensions of a dissolving dense nonaqueous phase liquid dnapl pool in an aquifer the design problem deals with the selection of the number and location of observation wells that minimizes the parameter uncertainty while minimizing installation and sampling costs the solution of this multiobjective optimization problem is achieved by solving a series of combinatorial optimization problems in which for each imposed budget constraint the optimal combination of wells is found the combinatorial optimization problems are solved by a genetic algorithm for two different design strategies one where the prior estimates are not updated and one where the parameter updating is performed at the end of each design stage sequential design in the latter case parameter updating is obtained by executing an inverse model based on concentration values collected at the locations specified by the previous optimal design the algorithms are tested for a controlled bench scale pool dissolution experiment the results demonstrate that the sequential design performs better in terms of final parameter estimates three optimality criteria associated with experimental design for model parameter estimation are compared the a optimality d optimality and e optimality among the three criteria the a and e optimalities produce better estimates of the pool location and size which is justified by appreciably different variances associated with the parameters and their high degree of correlation

a statistical methodology for estimating transport parameters theory and applications to one dimensional advectivec dispersive systems a simulation nonlinear multiple regression methodology for estimating parameters that characterize the transport of contaminants is developed and demonstrated finite difference contaminant transport simulation is combined with a nonlinear weighted least squares multiple regression procedure the technique provides optimal parameter estimates and gives statistics for assessing the reliability of these estimates under certain general assumptions about the distributions of the random measurement errors monte carlo analysis is used to estimate parameter reliability for a hypothetical homogeneous soil column for which concentration data contain large random measurement errors the value of data collected spatially versus data collected temporally was investigated for estimation of velocity dispersion coefficient effective porosity first order decay rate and zero order production the use of spatial data gave estimates that were 2 3 times more reliable than estimates based on temporal data for all parameters except velocity comparison of estimated linear and nonlinear confidence intervals based upon monte carlo analysis showed that the linear approximation is poor for dispersion coefficient and zero order production coefficient when data are collected over time in addition examples demonstrate transport parameter estimation for two real one dimensional systems first the longitudinal dispersivity and effective porosity of an unsaturated soil are estimated using laboratory column data we compare the reliability of estimates based upon data from individual laboratory experiments versus estimates based upon pooled data from several experiments second the simulation nonlinear regression procedure is extended to include an additional governing equation that describes delayed storage during contaminant transport the model is applied to analyze the trends variability and interrelationship of parameters in a mourtain stream in northern california

sequential ensemble based optimal design for parameter estimation the ensemble kalman filter enkf has been widely used in parameter estimation for hydrological models the focus of most previous studies was to develop more efficient analysis estimation algorithms on the other hand it is intuitively understandable that a well designed sampling data collection strategy should provide more informative measurements and subsequently improve the parameter estimation in this work a sequential ensemble based optimal design seod method coupled with enkf information theory and sequential optimal design is proposed to improve the performance of parameter estimation based on the first and second order statistics different information metrics including the shannon entropy difference sd degrees of freedom for signal dfs and relative entropy re are used to design the optimal sampling strategy respectively the effectiveness of the proposed method is illustrated by synthetic one and two dimensional unsaturated flow case studies it is shown that the designed sampling strategies can provide more accurate parameter estimation and state prediction compared with conventional sampling strategies optimal sampling designs based on various information metrics perform similarly in our cases the effect of ensemble size on the optimal design is also investigated overall larger ensemble size improves the parameter estimation and convergence of optimal sampling strategy although the proposed method is applied to unsaturated flow problems in this study it can be equally applied in any other hydrological problems this article is protected by copyright all rights reserved

a geostatistical approach to the conditional estimation of spatially distributed solute concentration and notes on the use of tracer data in the inverse problem this paper presents a theoretical framework for deriving the moments of the concentration based on the lagrangian approach and using a stochastic framework conditional to measurements of conductivity and hydraulic head the method consists of deriving the spatial correlations between concentration and travel time and hydrologic variables such as the conductivity and the hydraulic head these correlations allow the conditioning of the moments of the concentration on measurements by conditioning the concentration the uncertainty associated with its estimation can be reduced substantially consequently difficulties associated with estimation of the extent of contamination can be alleviated the theoretical framework and derivations may also be used to condition the moments of the conductivity on tracer data such as concentrations travel times and displacements an application of such an approach would require a configuration of sources and samplers we show that measured concentration is inferior to travel time and displacements in terms of efficient conditioning

discrimination among one dimensional models of solute transport in porous media implications for sampling design a methodology is developed for discrimination among models of transient solute transport in porous media the method utilizes nonlinear regression on observations of solute concentration discrimination requires comparisons of model predictions to observations systematic error in residuals stability in parameter estimates from regression on different observation sets and other measures of model fit among hypothesized models of transport the set of observations of solute concentration to which models are fitted strongly influences the assessment of these discrimination criteria the most desirable observation set for discrimination amplifies the weaknesses of those models that appear to describe existing conditions but are in fact unsuitable for prediction the inadequacies of various observation sets are illustrated in four examples of discrimination between one dimensional models of solute transport our purpose in these examples is to understand the physical deterministic basis of sampling design for model discrimination in addition to physical attributes such as transport processes boundary conditions and flow geometry the assumed distribution of random error in the regression model is also treated as a model attribute to be tested by the designed experiment a common problem in field studies occurs when the set of available observations does not include sufficient information with which to discriminate among hypothesized models hence supporting the need to design a second round of sampling specifically for discrimination a proposed objective function in the sampling design problem favors design points at locations and times when two hypothesized transport models display the greatest differences in predicted concentration two hypothetical examples demonstrate the effectiveness of the objective function and the application of the discrimination criteria

estimation of distributional parameters for censored trace level water quality data 1 estimation techniques a recurring difficulty encountered in investigations of many metals and organic contaminants in ambient waters is that a substantial portion of water sample concentrations are below limits of detection established by analytical laboratories several methods were evaluated for estimating distributional parameters for such censored data sets using only uncensored observations their reliabilities were evaluated by a monte carlo experiment in which small samples were generated from a wide range of parent distributions and censored at varying levels eight methods were used to estimate the mean standard deviation median and interquartile range criteria were developed based on the distribution of uncensored observations for determining the best performing parameter estimation method for any particular data det the most robust method for minimizing error in censored sample estimates of the four distributional parameters over all simulation conditions was the log probability regression method with this method censored observations are assumed to follow the zero to censoring level portion of a lognormal distribution obtained by a least squares regression between logarithms of uncensored concentration observations and their z scores when method performance was separately evaluated for each distributional parameter over all simulation conditions the log probability regression method still had the smallest errors for the mean and standard deviation more but the lognormal maximum likelihood method had the smallest errors for the median and interquartile range when data sets were classified prior to parameter estimation into groups reflecting their probable parent distributions the ranking of estimation methods was similar but the accuracy of error estimates was markedly improved over those without classification less

exact solutions of population balance equations for breakage and growth processes using group analysis abstract population balance equations have been applied to model a wide range of processes including breakage crystal nucleation growth agglomeration comminution of ores and other solids but the lack of exact solutions necessitates the use of numerical methods a class of population balance equations with time independent but size dependent growth rate and breakage rate is investigated analytically all invariant solutions explicit exact solutions reduced equations and optimal system of subalgebras for the corresponding system of equations are presented in this paper

optimal experimental design for parameter estimation in column outflow experiments 1 inverse modeling is increasingly used for water flow and reactive solute transport in the unsaturated zone to determine unknown parameters e g soil hydraulic parameters degradation rates or dispersion coefficients however experimental data are frequently not suitable for parameter estimation leading instead to practically singular estimation problems optimal experimental design aims at identifying experimental conditions and sampling schemes that deliver measurement data which are most sensitive to unknown parameters a new numerical approach to optimize experimental designs for parameter estimation in coupled partial differential equations pdes and ordinary differential equation odes arising from transport and degradation processes is presented the optimal experimental design problem is embedded in the framework of optimal control theory this results in an intricate nonlinear constrained optimization problem that is solved by a direct approach based on a structured sqp method these methods are used to investigate several scenarios for a column outflow experiment in a simulation study simultaneously experimental conditions such as irrigation schemes substance concentrations and sampling schemes are optimized result is an enormous reduction of parameter uncertainty additionally it is demonstrated that it is possible to estimate both soil hydraulic and solute transport parameters together in one experiment on the basis of leachate data only the usefulness of the numerical approach for practical applications is discussed

linearization criteria for systems of two second order stochastic ordinary differential equations we provide the necessary and sufficient conditions for the linearization of systems of two second order stochastic ordinary differential equations the linearization criteria are given in terms of coefficients of the system followed by some illustrations this paper gives a new treatment for the linearization of two second order stochastic ordinary differential equations and with some examples

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

determining soil hydraulic functions from evaporation experiments by a parameter estimation approach experimental verifications and numerical studies a parameter estimation method is developed for the determination of unsaturated soil hydraulic properties from evaporation experiments under laboratory conditions variables used in the inversion procedure are soil water pressure heads at various positions and total soil weights both measured as a function of time during the experiment unknown parameters of different analytical expressions used to describe the soil hydraulic properties are estimated by coupling a finite difference solution of the richards equation with a nonlinear optimization problem this problem is formulated by minimizing the deviations between the numerical solution of the transient flow process and the real system response measured during the experiment minimization of the objective function is performed using a version of the levenberg marquardt s method the procedure also provides information about the uncertainty in parameter estimates the performance of the selected parametric relationships is evaluated and the applicability and accuracy of the proposed inverse method are shown by comparing estimated water retention and hydraulic conductivity functions with experimental results obtained via the instantaneous profile method parameter sensitivity analyses and stability of the inverse solutions are discussed with reference to the originally designed evaporation experiment and to an evaporation method that is developed with a view to reducing experimental efforts further insights into the properties of existence and uniqueness of inverse solutions are gained by examining contour plots of the objective functions under varying experimental conditions the results confirm the reliability and flexibility of the proposed method and suggest that the evaporation flux imposed at the upper soil surface may determine the well posedness of the optimization problem

on reciprocal eigenvalue property of weighted trees an undirected unweighted graph g is called nonsingular if its adjacency matrix a g is nonsingular a graph is said to have property r if with every eigenvalue of a g 1 is also an eigenvalue if further the multiplicity of and 1 as eigenvalues of a g are the same then g is said to have property sr in a previous paper the classes of unweighted trees with property r and sr where shown to be the same and that this common class coincides with the class of corona trees that is trees which have been obtained from smaller trees by adding a pendent edge to each vertex in this paper we continue the study of nonsingular trees in two ways first we characterize the set of all graphs whose inverses are nonsingular trees second we investigate the extent of property r for weighted trees and characterize the property for all trees with 8 vertices or more under some conditions on the weights

on some graphs which satisfy reciprocal eigenvalue properties abstract we consider only simple graphs consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite on the class h g of such graphs g the equivalence of the following statements are known i the reciprocal of the spectral radius of the adjacency matrix a g is the least positive eigenvalue of the adjacency matrix ii the graph is isomorphic to its inverse where the inverse of a graph g is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g via a diagonal matrix of 1s iii the graph has the reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g iv the graph has the strong reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g and they both have the same multiplicities v the graph is a corona graph that is it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex it is known that the statements iii and iv are not equivalent if we allow nonbipartite graphs however the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings where we relax the condition of bipartiteness of the contracted graph in this article we establish the equivalence of the first four statements for a class larger than h g we then supply a constructive structural description of those graphs

a optimality for regression designs abstract use is made of a result of horn amer j math 76 1954 620 630 on the existence of a symmetric matrix with prescribed diagonal elements and eigenvalues a necessary and sufficient condition is then given for the existence of an a optimal design for a regression experiment in the dorogovcev selected transl math statist probab 10 1971 35 41 setting

a comparison of several methods of solving nonlinear regression groundwater flow problems computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear regression steady state groundwater flow problems the fastest methods were the marquardt and quasi linearization methods which required almost identical computer times and numbers of iterations the next fastest was the quasi newton method and last was the fletcher reeves method which did not converge in 100 iterations for two of the problems the fastest method per iteration was the fletcher reeves method and this was followed closely by the quasi newton method the marquardt and quasi linearization methods were slower for all four methods the speed per iteration was directly related to the number of parameters in the model however this effect was much more pronounced for the marquardt and quasi linearization methods than for the other two hence the quasi newton and perhaps fletcher reeves method might be more efficient than either the marquardt or quasi linearization methods if the number of parameters in a particular model were large although this remains to be proven the marquardt method required somewhat less central memory than the quasi linearization metilod for three of the four problems for all four problems the quasi newton method required roughly two thirds to three quarters of the memory required by the marquardt method and the fletcher reeves method required slightly less memory than the quasi newton method memory requirements were not excessive for any of the four methods

optimal data acquisition strategy for the development of a transport model for groundwater remediation the reliability of groundwater quality management algorithms is limited in large part by the uncertainty present in the model parameters because the field parameter measurement costs and the remediation costs must be supplied by the same financial source the classical optimization procedure does not minimize the real total remediation investment this research presents an algorithm able to find the total minimum for the sum of both the measurement and the pumping costs a chance constrained technique is used to cast the optimization problem in stochastic form relating the concentration covariance matrix to the log transmissivity covariance matrix by means of the transport equations and a first order approximation for the uncertainty the simulation model solves the steady state flow equations on a finite element triangular mesh and the transport equations using the backward method of characteristics the resulting nonlinearly constrained optimization problem is solved using the quasi linearity algorithm this algorithm is designed to find a good initial point for the local minimum search when the feasible domain is not convex

further comments on sensitivities parameter estimation and sampling design in one dimensional analysis of solute transport in porous media sensitivities of solute concentration to parameters associated with first order chemical decay boundary conditions initial conditions and multilayer transport are examined in one dimensional analytical models of transient solute transport in porous media a sensitivity is a change in solute concentration resulting from a change in a model parameter sensitivity analysis is important because minimum information required in regression on chemical data for the estimation of model parameters by regression is expressed in terms of sensitivities nonlinear regression models of solute transport were tested on sets of noiseless observations from known models that exceeded the minimum sensitivity information requirements results demonstrate that the regression models consistently converged to the correct parameters when the initial sets of parameter values substantially deviated from the correct parameters on the basis of the sensitivity analysis several statements may be made about design of sampling for parameter estimation for the models examined 1 estimation of parameters associated with solute transport in the individual layers of a multilayer system is possible even when solute concentrations in the individual layers are mixed in an observation well 2 when estimating parameters in a decaying upstream boundary condition observations are best made late in the passage of the front near a time chosen by adding the inverse of an hypothesized value of the source decay parameter to the estimated mean travel time at a given downstream location 3 estimation of a first order chemical decay parameter requires observations to be made late in the passage of the front preferably near a location corresponding to a travel time of times the half life of the solute and 4 estimation of a parameter relating to spatial variability in an initial condition requires observations to be made early in time relative to passage of the solute front

strongly self dual graphs we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs if the h graphs g and h with adjacency matrices m g and m h satisfy m g 1 sm h s where s is a signature matrix we refer to h as the dual of g the dual is a type of graph inverse if the h graph g is isomorphic to its dual via a particular isomorphism we refer to g as strongly self dual we investigate the structural and spectral properties of strongly self dual graphs with a particular emphasis on identifying when such a graph has 1 as an eigenvalue

a reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from 1 i abstract a weighted directed graph is a directed graph g whose underlying undirected graph is simple and whose edges have nonzero directional complex weights that is the presence of an edge u v of weight w is as good as the presence of the edge v u with weight w the complex conjugate of w let g be a weighted directed graph on vertices 1 2 n denote by w u v the weight of an edge u v e g the adjacency matrix a g of g is an n n matrix with entries a i j w i j or w j i or 0 depending on whether i j e g or j i e g or otherwise respectively we supply a characterization of those unicyclic weighted directed graphs g whose edges have weights from the set 1 i and whose adjacency matrix a g satisfies the following property is an eigenvalue of a g with multiplicity k if and only if 1 is an eigenvalue of a g with the same multiplicity

strong reciprocal eigenvalue property of a class of weighted graphs let h be the class of connected bipartite graphs g with a unique perfect matching m for g h let wg be the set of weight functions w on the edge set e g such that w e 1 for each matching edge and w e 0 for each nonmatching edge let gw denote the weighted graph with g h and w wg the graph gw is said to satisfy the reciprocal eigenvalue property property r if 1 is an eigenvalue of the adjacency matrix a gw whenever is an eigenvalue of a gw moreover if the multiplicities of the reciprocal eigenvalues are the same we say gw has the strong reciprocal eigenvalue property property sr let hg g h g m is bipartite where g m is the graph obtained from g by contracting each edge in m to a vertex recently in 12 it was shown that if g hg then gw has property sr for some w wg if and only if gw has property sr for each w wg if and only if g is a corona graph obtained from another graph h by adding a new pendant vertex to each vertex of h now we have the following questions is there a graph g h hg such that gw has property sr for each w wg are there graphs g h hg such that gw never has property sr not even for one w wg are there graphs g h such that gw has property sr for some w wg but not for all w wg in this article we supply answers to these three questions we also supply a graph class larger than hg where for any graph g if gw has property sr for one w wg then g is a corona graph

parameter space methods in joint parameter estimation for groundwater flow models the joint estimation of model parameters using hydraulic head and tracer concentration data is examined using methods based on parameter space analysis response surfaces and confidence regions can indicate how multiple data sets interact to reduce parameter uncertainty each data set produces a unique response surface and confidence region if these surfaces are oriented differently for the two data sets parameter uncertainties are significantly reduced when the second data set is included in the inversion if these surfaces are similar for the two data sets the second data set will not reduce the parameter uncertainty significantly the axes of the linearized confidence ellipsoid are analyzed to determine the difference in orientation of the confidence regions for the two data sets the use of confidence regions can be extended to predict the value of a second data set in reducing the uncertainty of parameter estimates before the data are collected parameter space approaches are also introduced for selecting the relative weights for the individual data sets in joint parameter estimation the conventional method based on the analysis of data residuals is compared to three other methods with the weights selected to maximize parameter stability to minimize the volume of the confidence region or to minimize the longest axis of the confidence region each criterion can lead to substantially different weights applied to each of the data sets the application of these methods is demonstrated using hydraulic head measurements and 14c concentrations to calibrate a model of groundwater flow in the san juan basin new mexico

multiobjective sampling design for parameter estimation and model discrimination in groundwater solute transport sampling design for site characterization studies of solute transport in porous media is formulated as a multiobjective problem optimal design of a sampling network is a sequential process in which the next phase of sampling is designed on the basis of all available physical knowledge of the system three objectives are considered model discrimination parameter estimation and cost minimization for the first two objectives physically based measures of the value of information obtained from a set of observations are specified in model discrimination value of information of an observation point is measured in terms of the difference in solute concentration predicted by hypothesized models of transport points of greatest difference in predictions can contribute the most information to the discriminatory power of a sampling design sensitivity of solute concentration to a change in a parameter contributes information on the relative variance of a parameter estimate inclusion of points in a sampling design with high sensitivities to parameters tends to reduce variance in parameter estimates cost minimization accounts for both the capital cost of well installation and the operating costs of collection and analysis of field samples sensitivities discrimination information and well installation and sampling costs are used to form coefficients in the multiobjective problem in which the decision variables are binary zero one each corresponding to the selection of an observation point in time and space the solution to the multiobjective problem is a noninferior set of designs to gain insight into effective design strategies a one dimensional solute transport problem is hypothesized then an approximation of the noninferior set is found by enumerating 120 designs and evaluating objective functions for each of the designs trade offs between pairs of objectives are demonstrated among the models the value of an objective function for a given design is shown to correspond to the ability of a design to actually meet an objective

sensitivity analysis of distributed parameter systems a method is developed for the calculation of sensitivity coefficients of general distributed parameter systems to the variation of spatially and temporally varying parameters appearing in the system equations and initial and boundary conditions

optimum experimental design for parameter identification in groundwater hydrology the experimental design for parameter identification of a confined groundwater system is formulated as a nonlinear mixed integer programming problem the decision variables considered are the number and locations of the pumping and observation wells and the pumping rates the constraints include the allowable drawdown the restricted locations of pumping wells and observation wells the permissible pumping rates and the required information the amount of the required information is determined by the prescribed reliability of the response for the future operation of the groundwater system the galerkin finite element crank nicolson approach is used to develop the simulation models for solving the groundwater governing equation for the system responses the sensitivity coefficients are computed by the influence coefficient method the principle of superposition is used to combine the simulation models with the optimization model the characteristics of the optimization problem have been investigated and are clarified and its difficulties are pointed out an efficient heuristic solution approach is proposed for solving this problem a test problem is constructed to demonstrate the usefulness of the proposed methodology observations generated from the optimal design are used to identify the parameters a covariance analysis is then carried out to verify the validity of the assumptions made in the formulation of the optimization problem the results obtained indicate that the proposed methodology is feasible and can be used to solve complicated experimental design problems associated with a confined groundwater system

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

optimal pumping test design for the parameter identification of groundwater systems the overall goal of this research is to identify a least cost pumping test design whose data will yield parameter estimates of a required reliability the reliability is characterized by the statistical concept of the d optimality which in turn is used as the acceptability criterion for experimental design classically d optimality seeks an experiment which minimizes the determinant of the parameter estimates covariance matrix thereby maximizing their reliability an upper bound on the determinant of the covariance matrix is derived by using the assumed asymptotic normal distribution of the parameter estimates as formulated the least cost problem is a nonlinear mixed integer programming problem where the decision variables are the location of the pumping observation well s the number of wells the pumping rate and the frequency of observations a heuristic design algorithm is developed the algorithm seeks to minimize the number of pumping and observation wells since a heuristic algorithm is used and because the inverse problem is nonlinear the algorithm will identify a local d optimal pumping test design this algorithm is then applied to a hypothetical aquifer system a sequential design algorithm is developed for circumventing the nonlinearity of the inverse problem of parameter estimation two case studies are performed the first case study verifies the sequential design algorithm under noiseless conditions the second case study yields realistic results under noisy conditions

symmetries of population balance equations for aggregation breakage and growth processes constructing invariant solutions of integro differential equation group analysis of population balance equation preliminary group classification of an integro differential equation the integro differential population balance equation describing aggregation processes was proposed almost 100 years ago aggregation is an important size enlargement process in many industries the modeling and design of the process can be done using the population balance framework however it is typically impossible to obtain analytical solutions in almost every case a numerical solution of the equations must be obtained in this paper we present the developed group analysis method for the one dimensional population balance equation for aggregation in a well mixed batch system including a crystal growth term the determining equations are solved the optimal system invariant solutions and all the reduced equations are obtained furthermore finding the determining equation by use of the preliminary group classification is also considered

coupled inverse problems in groundwater modeling 2 identifiability and experimental design parameter identifiability and experimental design in the context of solving the coupled inverse problem in groundwater modeling are considered in part 2 of the two paper series the following three new definitions of extended identifiability for a distributed parameter system are presented interval identifiability ini prediction equivalence identifiability pei and management equivalence identifiability mei the uniqueness requirement of the inverse solution is relaxed by these definitions an identified parameter is said to be extendedly identifiable if it is able to satisfy an accuracy requirement in a specific application of the model when a generalized least squares norm is used the proposed identifiabilities are applicable to any coupled problems in which the state variables and parameters may have different order of magnitudes a salient feature of the new definitions is that the extended identifiabilities are closely related to experimental design consequently the sufficiency of an experimental design for assuring a given level of an extended identifiability can be estimated prior to field experiments the methodology developed can be used to determine the reliability of an experimental design even for nonlinear models a numerical example on the management of a wastewater disposal is given to explain how the concepts and methods presented in this paper can be used for solving practical problems

on unimodular graphs abstract we study graphs whose adjacency matrices have determinant equal to 1 or 1 and characterize certain subclasses of these graphs graphs whose adjacency matrices are totally unimodular are also characterized for bipartite graphs having a unique perfect matching we provide a formula for the inverse of the corresponding adjacency matrix and address the problem of when that inverse is diagonally similar to a nonnegative matrix special attention is paid to the case that such a graph is unicyclic