a new mixed finite element for the stokes and elasticity problems the stokes problem is approximated by a mixed finite element method using a new finite el ement which has properties analogous to the finite volume methods namely the local conservation of the momentum and the mass estimates of optimal order are derived for the errors in the velocity the pressure and the gradient of the velocity this new finite element also works for the elasticity problem and all estimates are valid uniformly with respect to the compressibility finally some numerical results for the incompressible navier stokes equations are presented

the regularization method for an obstacle problem we give a relatively complete analysis for the regularization method which is usually used in solving non differentiable minimization problems the model problem considered in the paper is an obstacle problem in addition to the usual convergence result and a priori error estimates we provide a posteriori error estimates which are highly desired for practical implementation of the regularization method

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

analysis of accuracy of approximate simultaneous nonlinear confidence intervals on hydraulic heads in analytical and numerical test cases inaccuracies in parameter values parameterization stresses and boundary conditions of analytical solutions and numerical models of groundwater flow produce errors in simulated hydraulic heads these errors can be quantified in terms of approximate simultaneous nonlinear confidence intervals presented in the literature approximate confidence intervals can be applied in both error and sensitivity analysis and can be used prior to calibration or when calibration was accomplished by trial and error in this paper the method is expanded for use in numerical problems and the accuracy of the approximate intervals is evaluated using monte carlo runs four test cases are used to compare results in two numerical test cases the approximate simultaneous confidence intervals are generally accurate and are calculated efficiently enough to indicate that the method is useful for real world problems in one of the two analytical test cases considered the approximate intervals are too wide to be of practical interest improving the method used to approximate the uncertainty of the model inputs could make the confidence intervals useful in more problems

assessment of parametric uncertainty for groundwater reactive transport modeling the validity of using gaussian assumptions for model residuals in uncertainty quantification of a groundwater reactive transport model was evaluated in this study least squares regression methods explicitly assume gaussian residuals and the assumption leads to gaussian likelihood functions model parameters and model predictions while the bayesian methods do not explicitly require the gaussian assumption gaussian residuals are widely used this paper shows that the residuals of the reactive transport model are non gaussian heteroscedastic and correlated in time characterizing them requires using a generalized likelihood function such as the formal generalized likelihood function developed by schoups and vrugt 2010 for the surface complexation model considered in this study for simulating uranium reactive transport in groundwater parametric uncertainty is quantified using the least squares regression methods and bayesian methods with both gaussian and formal generalized likelihood functions while the least squares methods and bayesian methods with gaussian likelihood function produce similar gaussian parameter distributions the parameter distributions of bayesian uncertainty quantification using the formal generalized likelihood function are non gaussian in addition predictive performance of formal generalized likelihood function is superior to that of least squares regression and bayesian methods with gaussian likelihood function the bayesian uncertainty quantification is conducted using the differential evolution adaptive metropolis dream zs algorithm as a markov chain monte carlo mcmc method it is a robust tool for quantifying uncertainty in groundwater reactive transport models for the surface complexation model the regression based local sensitivity analysis and morris and dream zs based global sensitivity analysis yield almost identical ranking of parameter importance the uncertainty analysis may help select appropriate likelihood functions improve model calibration and reduce predictive uncertainty in other groundwater reactive transport and environmental modeling

role of the calibration process in reducing model predictive error 1 an equation is derived through which the variance of predictive error of a calibrated model can be calculated this equation has two terms the first term represents the contribution to predictive error variance that results from an inability of the calibration process to capture all of the parameterization detail necessary for the making of an accurate prediction if a model is uncalibrated with parameter values being supplied solely through outside information this is the only term required the second term represents the contribution to predictive error variance arising from measurement noise in an overdetermined system such as that which may be obtained through parameter lumping e g through the introduction of a spatial zonation scheme this is the only term required it is shown however that parameter lumping is a form of implicit regularization and that ignoring the implied first term of the predictive error variance equation can potentially lead to underestimation of predictive error variance a model s role as a predictor of environmental behavior can be enhanced if it is calibrated in such a way as to reduce the variance of those predictions which it is required to make it is shown that in some circumstances this can be accomplished through overfitting against historical field data it can also be accomplished by giving greater weight to those measurements which carry the greatest information content with respect to a required prediction this suggests that a departure may be necessary from the custom of using a single calibrated model for the making of many different predictions instead model calibration may need to be repeated many times so that in each case the calibration process is optimized for the making of a specific model prediction

analysis of non singular solutions of a mixed navier stokes formulation abstract a mixed finite element method is used for the numerical solution of the stationary navier stokes equations in this formulation the pressure the velocity and its gradient are the most relevant unknowns an existence and uniqueness results for the finite element solution together with the convergence results are proved near a non singular solution optimal error estimates and numerical results are presented

review and complements on mixed hybrid finite element methods for fluid flows mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems linear elasticity stokes problem navier stokes equations boussinesq equations etc have known a great development in the last few years these methods allow simultaneous computation of the original variable and its gradient both of them being equally accurate moreover they have local conservation properties conservation of the mass and the momentum as in the finite volume methods the purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of stokes and navier stokes equations and the linear elasticity problem further developments for a quasi newtonian flow obeying the power law are presented

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

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

inverse groundwater modeling with emphasis on model parameterization 1 this study develops an inverse method aiming to circumvent the subjective decision regarding model parameterization and complexity in inverse groundwater modeling the number of parameters is included as a decision variable along with parameter values a parameterization based on b spline surfaces bss is selected to approximate transmissivity and genetic algorithms were selected to perform error minimization a transform based on linear least squares lls is developed so that different parameterizations may be combined by standard genetic algorithm operators first three applications with isotropic anisotropic and zoned aquifer parameters are examined in a single objective optimization problem and the estimated transmissivity is found to be near the true one interestingly in the anisotropic case the algorithm converged to a solution with an anisotropic distribution of control points next a single objective optimization with regularization penalizing complex models is considered and last the problem is expressed in a multiobjective optimization framework moo where the goals are simultaneous minimization of calibration error and model complexity the result of moo is a pareto set of potential solutions where the user can examine the tradeoffs between calibration error and model complexity and select the most suitable model by comparing calibration with prediction errors it appears that the most promising models are the ones near a region where the rate of decrease of calibration error as model complexity increases drops bend of error curve this is a useful result of practical interest in real inverse modeling applications

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

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

numerical verification of solutions for obstacle problems this article is an extension of the previous paper numer math 81 1998 305 by the same authors we propose a method to prove the existence of solutions for more general obstacle problems by numerical computations the main task in this paper consists of the numerical determination of some constants which appear in a priori error estimations for the finite element approximation of a simple variational inequality we present a numerical example for verification

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

evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall runoff models influence of calibration data variability and length on model credibility the success of an automatic calibration procedure is highly dependent on the choice of the objective function and the nature quantity and quality of the data used the objective function should be selected on the basis of the stochastic properties of the errors present in the data and in the model also the data should be chosen so as to contain as much valuable information about the process as possible in this paper we compare the performance of two maximum likelihood estimators the amle which assumes the presence of first lag autocorrelated homogeneous variance errors and the hmle which assumes the presence of uncorrelated inhomogeneous variance errors to the commonly used simple least squares criterion sls the model calibrated was the soil moisture accounting model of the u s national weather service s river forecast system sma nwsrfs the results indicate that a properly chosen objective function can enhance the possibility of obtaining unique and conceptually realistic parameter estimates furthermore the sensitivity of the estimation results to various characteristics of the calibration data such as hydrologic variability and length are substantially reduced

analysis of regression confidence intervals and bayesian credible intervals for uncertainty quantification 1 confidence intervals based on classical regression theories augmented to include prior information and credible intervals based on bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties because both confidence and credible intervals are used in environmental modeling we seek to understand their differences and similarities this is of interest in part because calculating confidence intervals typically requires tens to thousands of model runs while bayesian credible intervals typically require tens of thousands to millions of model runs given multi gaussian distributed observation errors our theoretical analysis shows that for linear or linearized nonlinear models confidence and credible intervals are always numerically identical when consistent prior information is used for nonlinear models nonlinear confidence and credible intervals can be numerically identical if parameter confidence regions defined using the approximate likelihood method and parameter credible regions estimated using markov chain monte carlo realizations are numerically identical and predictions are a smooth monotonic function of the parameters both occur if intrinsic model nonlinearity is small while the conditions of gaussian errors and small intrinsic model nonlinearity are violated by many environmental models heuristic tests using analytical and numerical models suggest that linear and nonlinear confidence intervals can be useful approximations of uncertainty even under significantly nonideal conditions in the context of epistemic model error for a complex synthetic nonlinear groundwater problem the linear and nonlinear confidence and credible intervals for individual models performed similarly enough to indicate that the computationally frugal confidence intervals can be useful in many circumstances experiences with these groundwater models are expected to be broadly applicable to many environmental models we suggest that for environmental problems with lengthy execution times that make credible intervals inconvenient or prohibitive confidence intervals can provide important insight during model development when frequent calculation of uncertainty intervals is important to understanding the consequences of various model construction alternatives and data collection strategies strategic use of both confidence and credible intervals can be critical

exact scheffe type confidence intervals for output from groundwater flow models 2 combined use of hydrogeologic information and calibration data calibration data observed values corresponding to model computed values of dependent variables are incorporated into a general method of computing exact scheffe type confidence intervals analogous to the confidence intervals developed in part 1 cooley this issue for a function of parameters derived from a groundwater flow model parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data this distribution was obtained as a posterior distribution by applying bayes theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model computed dependent variables tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely the confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage

coupled inverse problems in groundwater modeling 1 sensitivity analysis and parameter identification some basic concepts and methods for solving the coupled inverse problem in groundwater modeling are presented in a series of two papers because there are crossover effects between state variables and parmeters coupled inverse problems are different from single inverse problems in several aspects especially in the selection of objective functions identifiability and experimental design this paper discusses the general definition of coupled inverse problems posed in the framework of multiobjective optimization in order to obtain gradient vectors of objective functions as well as all sensitivity coefficients a general procedure for deriving the adjoint state equations and their associated conditions is described using the rules of adjoint operation presented in this paper the derivation of adjoint problems becomes very easy even for nonlinear and transient coupled problems as examples the adjoint problem of flow mass transport or flow heat transport the adjoint problem of saltwater intrusion and the adjoint problem of two phase flow are derived numerical experiments on a two dimensional flow mass transport problem are given to explain the procedure of sensitivity analysis and parameter identification for coupled problems the hydraulic conductivity can be identified by using observations of head and or solute concentration

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

practical scheffe type credibility intervals for variables of a groundwater model simultaneous scheffe type credibility intervals the bayesian version of confidence intervals for variables of a groundwater flow model calibrated using a bayesian maximum a posteriori procedure were derived by cooley 1993b it was assumed that variances reflecting the expected differences between observed and model computed quantities used to calibrate the model are known whereas they would often be unknown for an actual model in this study the variances are regarded as unknown and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance the credibility intervals are calculated from the posterior distribution which was developed by considering each group variance to be a random variable about which nothing is known a priori then eliminating it by integration numerical experiments using two test problems illustrate some characteristics of the credibility intervals nonlinearity of the statistical model greatly affected some of the credibility intervals indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems the parameter characterizing the probability level for the credibility intervals was however accurately computed using a linear model approximation as compared with values calculated using second order and fully nonlinear formulations this allows the credibility intervals to be computed very efficiently

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

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

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

an analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields an analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity the analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model computed water pressures are small when this assumption is not met the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy the conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method so that when applied to a nonlinear model the method is subject to bootstrap errors these sources of error must be considered when using the method

on characterizations and regularity of the solution of bilateral obstacle problems in this paper we introduce a new reformulation of bilateral obstacle problems which allows us to characterize the domain of noncontact with the obstacles and give a new proof for h2 regularity of the solution

a mixed finite element method for a ladyzhenskaya model we study a mixed finite element approximation of a model proposed by ladyzhenskaya for stationary incompressible viscous flow we give existence and uniqueness results for the continuous problem and its approximation and we prove error bounds which improve the existing ones finally some numerical results are presented

improved methodology for parameter inference in nonlinear hydrologic regression models a new method is developed for the construction of reliable marginal confidence intervals and joint confidence regions for the parameters of nonlinear hydrologic regression models a parameter power transformation is combined with measures of the asymptotic bias and asymptotic skewness of maximum likelihood estimators to determine the transformation constants which cause the bias or skewness to vanish these optimized constants are used to construct confidence intervals and regions for the transformed model parameters using linear regression theory the resulting confidence intervals and regions can be easily mapped into the original parameter space to give close approximations to likelihood method confidence intervals and regions for the model parameters unlike many other approaches to parameter transformation the procedure does not use a grid search to find the optimal transformation constants an example involving the fitting of the michaelis menten model to velocity discharge data from an australian gauging station is used to illustrate the usefulness of the methodology

a global multilevel coordinate search procedure for estimating the unsaturated soil hydraulic properties we present a new inverse modeling procedure to characterize the hydraulic properties of partially saturated soils from soil moisture measurements during a natural transient flow experiment the inversion of the governing one dimensional richards equation is carried out using the global multilevel coordinate search optimization algorithm in sequential combination with the local nelder mead simplex algorithm gmcs nms we introduce this optimization method in the area of unsaturated zone hydrology since it is adapted for solving accurately and efficiently complex nonlinear problems several numerical experiments have been conducted to evaluate the proposed inversion method using synthetic error free and error contaminated data for different textured soils inversion of the simulated error free data and examination of the related response surfaces demonstrated the uniqueness of the inverse solution and the suitability of the gmcs nms strategy when identifying four key parameters of the hydraulic functions described by the mualem van genuchten model inversion of the error contaminated data proved further the good stability of the inverse solution that is consistent with the needs required by real experiments

predictive uncertainty analysis of a saltwater intrusion model using null space monte carlo 1 because of the extensive computational burden and perhaps a lack of awareness of existing methods rigorous uncertainty analyses are rarely conducted for variable density flow and transport models for this reason a recently developed null space monte carlo nsmc method for quantifying prediction uncertainty was tested for a synthetic saltwater intrusion model patterned after the henry problem saltwater intrusion caused by a reduction in fresh groundwater discharge was simulated for 1000 randomly generated hydraulic conductivity distributions representing a mildly heterogeneous aquifer from these 1000 simulations the hydraulic conductivity distribution giving rise to the most extreme case of saltwater intrusion was selected and was assumed to represent the true system head and salinity values from this true model were then extracted and used as observations for subsequent model calibration random noise was added to the observations to approximate realistic field conditions the nsmc method was used to calculate 1000 calibration constrained parameter fields if the dimensionality of the solution space was set appropriately the estimated uncertainty range from the nsmc analysis encompassed the truth several variants of the method were implemented to investigate their effect on the efficiency of the nsmc method reducing the dimensionality of the null space for the processing of the random parameter sets did not result in any significant gains in efficiency and compromised the ability of the nsmc method to encompass the true prediction value the addition of intrapilot point heterogeneity to the nsmc process was also tested according to a variogram comparison this provided the same scale of heterogeneity that was used to generate the truth however incorporation of intrapilot point variability did not make a noticeable difference to the uncertainty of the prediction with this higher level of heterogeneity however the computational burden of generating calibration constrained parameter fields approximately doubled predictive uncertainty variance computed through the nsmc method was compared with that computed through linear analysis the results were in good agreement with the nsmc method estimate showing a slightly smaller range of prediction uncertainty than was calculated by the linear method

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

analysis of a mixed formulation of a bilateral obstacle problem abstract in this paper we present a reformulation of a bilateral obstacle problem as a mixed formulation problem based on subdifferential of a continuous function of which the subdifferential can characterize the non contact domain then we present the analysis of the discrete problem we prove the convergence of the approximate solution to the exact one and we provide an error estimate this formulation was established in an abstract way then the theoretical results was applied to a bilateral obstacle problem

coupled estimation of flow and solute transport parameters a methodology for solving the coupled linear flow and transport inverse problem is presented it allows the estimation of aquifer parameters transmissivity storativity recharge boundary heads and flows leakage dispersivity molecular diffusion porosity retardation linear decay boundary concentrations using their prior estimates as well as head and concentration measurements as basic information transient flow and transport equations are solved using the finite element method in space and a weighted finite difference scheme in time flow and transport domains can be one two or quasi three dimensional applying linear one dimensional elements linear two dimensional triangles or bilinear two dimensional rectangles mixed at will time regimes can be transient or steady state in any equation flow or transport parameters are estimated on the basis of maximum likelihood theory which allows us to obtain information about parameters uncertainty minimization of the negative log likelihood is performed using marquardt s method the applicability of the method is shown by means of two real examples they illustrate that the proposed approach can successfully address model selection parameter uncertainty and nontrivial identifiability problems estimation of time dependent sources and the role of prior estimates are also addressed in the examples while the formulation of the problem leaves room for both stochastic and deterministic treatment of hydraulic parameters the latter has been adopted in both examples limitations of the approach are also apparent

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

evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions we tested the accuracy of 95 individual prediction intervals for hydraulic heads streamflow gains and effective transmissivities computed by groundwater models of two danish aquifers to compute the intervals we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error testing was accomplished by using a cross validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models the tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability 95 and that each tail probability is not significantly different from the assumed probability 2 5 in all cases tested these hypotheses were accepted at the 5 level of significance we therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate

strategic placement of localization devices such as pilot points and anchors in inverse modeling schemes 1 localization schemes using devices such as pilot points or anchors localization devices are used in inversion schemes as a tool for gleaning information from measurements that can be used to constrain inversion schemes with the added benefit in the form of a potential for reduction in the number of parameters employed in the inversion scheme this paper proposes and demonstrates a method for strategic placement of the localization devices the method combines stochastic singular value decomposition with parameter rejection methods to produce a map of intensity scores which allows the inverse modeler to place the localization devices at locations defined by their probability for producing informative probabilistic constraints on the statistical distributions of the target variables we also show that the method can be used for selecting measurement locations when designing data acquisition campaigns the method is demonstrated for the case of steady state flow in a heterogeneous conductivity field using mad method of anchored distributions in summary the proposed localization scheme intends to increase the information gleaned from measurements while reducing the associated computational costs associated with stochastic inversion

calibration and uncertainty analysis for computer simulations with multivariate output model calibration analysis is concerned with the estimation of unobservable modeling parameters using observations of system response when the model being calibrated is an expensive computer simulation special techniques such as surrogate modeling and bayesian inference are often fruitful in this paper we show how the flexibility of the bayesian calibration approach can be exploited to account for a wide variety of uncertainty sources in the calibration process we propose a straightforward approach for simultaneously handling gaussian and non gaussian errors as well as a framework for studying the effects of prescribed uncertainty distributions for model inputs that are not treated as calibration parameters further we discuss how gaussian process surrogate models can be used effectively when simulator response may be a function of time and or space multivariate output the proposed methods are illustrated through the calibration of a simulation of thermally decomposing foam

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

hydrologic characterization of heterogeneous geologic media with an inverse method based on iterated function systems one way to estimate the hydrologic properties of heterogeneous geologic media is to invert well test data using multiple observation wells pressure transients observed during a well test are compared to the corresponding values obtained by numerically simulating the test using a mathematical model the parameters of the mathematical model are varied and the simulation repeated until a satisfactory match to the observed pressure transients is obtained at which point the model parameters are accepted as providing a possible representation of the hydrologic property distribution restricting the search to parameters that represent self similar fractal hydrologic property distributions can improve the inversion process far fewer parameters are needed to describe a hierarchical medium improving the efficiency and robustness of the inversion additionally each parameter set produces a hydrologic property distribution with a hierarchical structure which mimics the multiple scales of inversion create fractal sets known as attractors using an iterated function system ifs an attractor is mapped to a distribution of transmissivity and storativity in the mathematical model thus the ifs inverse method searches for the parameters of the ifs typically tens of parameters rather than the values of the hydrologic property distribution directly typically hundreds to thousands of parameters applicationmore of the ifs inverse method to synthetic data shows that the method works well for simple heterogeneities application to field data from a sand clay sedimentary sequence and a fractured granite produces reasonable results 26 refs 32 figs 5 tabs less

simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model methods are presented for computing three types of simultaneous confidence and prediction intervals exact likelihood ratio and linearized on output from nonlinear regression models with normally distributed residuals the confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables and the prediction intervals can be placed on any number of future observations the confidence intervals are analogous to simultaneous scheffe intervals for linear models and the prediction intervals are analogous to the prediction intervals of hahn 1972 all three types of intervals can be computed efficiently by using the same straightforward lagrangian optimization scheme the prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the lagrangian scheme the methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer three different data sets are used to demonstrate the effect of sampling strategies on the intervals for all three data sets the linearized confidence intervals are inferior to the exact and likelihood ratio intervals with the latter two being very similar however all three types of prediction intervals yielded similar results the third data set time drawdown data at only a single observation well points out many of the problems that can arise from extreme nonlinear behavior of the regression model

calibration constrained monte carlo analysis of highly parameterized models using subspace techniques we describe a subspace monte carlo ssmc technique that reduces the burden of calibration constrained monte carlo when undertaken with highly parameterized models when monte carlo methods are used to evaluate the uncertainty in model outputs ensuring that parameter realizations reproduce the calibration data requires many model runs to condition each realization in the new ssmc approach the model is first calibrated using a subspace regularization method ideally the hybrid tikhonov tsvd superparameter approach described by tonkin and doherty 2005 sensitivities calculated with the calibrated model are used to define the calibration null space which is spanned by parameter combinations that have no effect on simulated equivalents to available observations next a stochastic parameter generator is used to produce parameter realizations and for each a difference is formed between the stochastic parameters and the calibrated parameters this difference is projected onto the calibration null space and added to the calibrated parameters if the model is no longer calibrated parameter combinations that span the calibration solution space are reestimated while retaining the null space projected parameter differences as additive values the recalibration can often be undertaken using existing sensitivities so that conditioning requires only a small number of model runs using synthetic and real world model applications we demonstrate that the ssmc approach is general it is not limited to any particular model or any particular parameterization scheme and that it can rapidly produce a large number of conditioned parameter sets

a frequentist approach to computer model calibration summary the paper considers the computer model calibration problem and provides a general frequentist solution under the framework proposed the data model is semiparametric with a non parametric discrepancy function which accounts for any discrepancy between physical reality and the computer model in an attempt to solve a fundamentally important but often ignored identifiability issue between the computer model parameters and the discrepancy function the paper proposes a new and identifiable parameterization of the calibration problem it also develops a two step procedure for estimating all the relevant quantities under the new parameterization this estimation procedure is shown to enjoy excellent rates of convergence and can be straightforwardly implemented with existing software for uncertainty quantification bootstrapping is adopted to construct confidence regions for the quantities of interest the practical performance of the methodology is illustrated through simulation examples and an application to a computational fluid dynamics model

a mixed formulation of boussinesq equations analysis of nonsingular solutions this paper is concerned with the mixed formulation of the boussinesq equations in two dimensional domains and its numerical approximation the paper deals first with existence and uniqueness results as well as the description of the regularity of any solution the problem is then approximated by a mixed finite element method where the gradient of the velocity and the gradient of the temperature quantities of practical importance are introduced as new unknowns an existence result for the finite element solution and convergence results are proved near a nonsingular solution quasi optimal error estimates are finally presented

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

development of objective oriented groundwater models 1 robust parameter identification 1 when the structure of an aquifer is complex and unknown determining the complexity of model structure and identifying its spatially distributed parameters become very difficult this two paper series describes an approach for objective oriented model construction that can identify a model with simplified structure and assure its reliability for predetermined model applications the theoretical basis of the methodology includes the definition of structure error the conditions of structure reducibility the calculation of the worst case parameter wcp and the solution of a generalized inverse problem in the first paper of the series we extend the basic theory established in our previous works from zonation parameterization to any kind of parameterization the extension makes the methodology more general and complete we present an effective direct search algorithm that can form a nonnested structure series with an optimized pattern for solving the generalized inverse problem we use numerical examples to show how the wcp depends on the structure complexity prior information boundary conditions and the objectives of model application generally speaking with more prior information simpler boundary conditions fewer objectives and lower accuracy requirement the wcp will have a simpler shape as a result a reliable model can be identified with less observation data in part 2 of this series we will give the conditions of structure reducibility and the method for robust experimental design a robust experimental design is independent of the true parameter and can provide sufficient information for identifying an objective oriented model