a full bayesian approach to the groundwater inverse problem for steady state flow a full bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two dimensional steady state groundwater flow the approach combines both bayesian and maximum entropy viewpoints of probability in the first phase log transmissivity measurements are incorporated into bayes theorem and the prior probability density function is updated yielding posterior estimates of the mean value of the log transmissivity field and covariance the two central moments are generated assuming that the prior mean variance and integral scales are hyperparameters that is they are treated as random variables in themselves which is contrary to classical statistical approaches the probability density functions pdfs of these hyperparameters are in turn determined from maximum entropy considerations in other words pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information this methodology is quite general and provides an alternative to kriging for spatial interpolation the final step consists of updating the conditioned natural logarithm transmissivity ln t field with hydraulic head measurements utilizing a linearized aquifer equation it is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain at each step uncertainties in all pertinent hyperparameters are removed by marginalization finally what is produced is a ln t field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln t field in addition we can also produce resolution matrices confidence credibility limits and the like for the ln t field it is shown that the application of the methodology yields good estimates of transmissivities even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln t variance of ln t and integral scales

transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

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

three dimensional analysis of piezocomposite plates with arbitrary geometry and boundary conditions abstract in this paper an accurate series solution in conjunction with an energy formulation for the treatment of piezocomposite plates with arbitrary geometry and aspect ratio under both electrical and mechanical loadings are proposed a remedy for dealing with nonhomogeneous boundary conditions is also presented through introduction of amending polynomials of order p k for the k th layer the accuracy and convergence rate are dramatically improved these polynomials ensure continuity of the generalized displacement fields across the interfaces while their derivatives can have the required discontinuities up to a desired order moreover depending on the nature of the physical problem under consideration incorporation of the appropriate functions result in greater convergence rate and precision of the solution

efficient geostatistical inverse methods for structured and unstructured grids 1 the geostatistical inverse approach of kitanidis 1995 requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix the explicit computation of the full covariance matrix is prohibitive for large scale problems discrete spectral methods which help to reduce computational costs dramatically are restricted to regular grids to make the geostatistical inverse approach applicable to unstructured grids which may be of great interest for practical applications we use the functional parameterization of the spatial random field by the karhunen loeve kl expansion in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function in the inverse approach the estimation of the spatial variable becomes estimating the weights of base functions the prior covariance matrix of the weights is the identity matrix the base functions are continuous in space and can be discretized in any fashion for the derivation of the base functions we embed the domain into a larger unit cell of a periodic domain then the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling for illustration we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids the results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the kl expansion we also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

statistical inverse problems discretization model reduction and inverse crimes the article discusses the discretization of linear inverse problems when an inverse problem is formulated in terms of infinite dimensional function spaces and then discretized for computational purposes a discretization error appears since inverse problems are typically ill posed neglecting this error may have serious consequences to the quality of the reconstruction the bayesian paradigm provides tools to estimate the statistics of the discretization error that is made part of the measurement and modelling errors of the estimation problem this approach also provides tools to reduce the dimensionality of inverse problems in a controlled manner the ideas are demonstrated with a computed example

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

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

probabilistic inference of multi gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction 2015 american geophysical union all rights reserved we present a bayesian inversion method for the joint inference of high dimensional multi gaussian hydraulic conductivity fields and associated geostatistical parameters from indirect hydrological data we combine gaussian process generation via circulant embedding to decouple the variogram from grid cell specific values with dimensionality reduction by interpolation to enable markov chain monte carlo mcmc simulation using the matern variogram model this formulation allows inferring the conductivity values simultaneously with the field smoothness also called matern shape parameter and other geostatistical parameters such as the mean sill integral scales and anisotropy direction s and ratio s the proposed dimensionality reduction method systematically honors the underlying variogram and is demonstrated to achieve better performance than the karhunen loeve expansion we illustrate our inversion approach using synthetic error corrupted data from a tracer experiment in a fairly heterogeneous 10 000 dimensional 2 d conductivity field a 40 times reduction of the size of the parameter space did not prevent the posterior simulations to appropriately fit the measurement data and the posterior parameter distributions to include the true geostatistical parameter values overall the posterior field realizations covered a wide range of geostatistical models questioning the common practice of assuming a fixed variogram prior to inference of the hydraulic conductivity values our method is shown to be more efficient than sequential gibbs sampling sgs for the considered case study particularly when implemented on a distributed computing cluster it is also found to outperform the method of anchored distributions mad for the same computational budget key points joint bayesian inference of gaussian conductivity fields and their variograms a dimensionality reduction that systematically honors the underlying variogram distributed multiprocessor implementation is straightforward

the subsequence composition of a string words that appear as constrained subsequences in a text string are considered as possible indicators of the host string structure hence also as a possible means of sequence comparison and classification the constraint consists of imposing a bound on the number w of positions in the text that may intervene between any two consecutive characters of a subsequence a subset of such w sequences is then characterized that consists in intuitive terms of sequences that could not be enriched with more characters without losing some occurrence in the text a compact spatial representation is then proposed for these representative sequences within which a number of parameters can be defined and measured in the final part of the paper such parameters are empirically analyzed on a small collection of text strings endowed with various degrees of structure

characterization of reservoir simulation models using a polynomial chaos based ensemble kalman filter 1 model based predictions of flow in porous media are critically dependent on assumptions and hypotheses that are not always based on first principles and that cannot necessarily be justified on the basis of known prevalent physics constitutive models for instance fall under this category while these predictive tools are usually calibrated using observational data the scatter in the resulting parameters has typically been ignored in this paper this scatter is used to construct stochastic process models of the parameters which are then used as the cornerstone in a novel model validation methodology useful for ascertaining the confidence in model based predictions the uncertainties are first quantified by representing the unknown model parameters via their polynomial chaos decompositions these are descriptions of stochastic processes in terms of their coordinates with respect to an orthogonal basis this is followed by a filtering step to update these representations with measurements as they become available in order to account for the non gaussian nature of model parameters and model errors an adaptation of the ensemble kalman filter is developed instead of propagating an ensemble of model states forward in time as is suggested within the framework of the ensemble kalman filter the proposed approach allows the propagation of a stochastic representation of unknown variables using their respective polynomial chaos decompositions the model is propagated forward in time by solving the system of partial differential equations using a stochastic projection method whenever measurements are available the proposed data assimilation technique is used to update the stochastic parameters of the numerical model the proposed method is applied to a black oil reservoir simulation model where measurements are used to stochastically characterize the flow medium and to verify the model validity with specified confidence bounds the updated model can then be employed to forecast future flow behavior

matching a set of strings with variable length don t cares given an alphabet a a pattern p is a word 1 m where i a and a is a distinguished symbol called a variable length don t care symbol pattern p matches a text t a if t u0 1u1 um 1 mum for some u0 um a we address the following problem given a set p of patterns and a text t test whether one of the patterns of p matches t we describe an algorithm that solves the problem in time o t p log p in contrast to most of the existing string matching algorithms such as that of aho corasick our algorithm is not composed of two successive stages preprocessing the pattern resp the text and reading through the text resp the pattern but has these two stages essentially interleaved our approach is based on using the dawg directed acyclic word graph a data structure studied by a blumer j blumer haussler ehrenfeucht crochemore chen seiferas

recognition of the factors of a finite language in a linear time text resume nous donnons ici premierement une construction en ligne d un transducteur l qui reconnait tous les facteurs d un langage fini l et place chacun d eux comme facteur d un des mots de l l peut etre deux fois plus petit que l automate partiel introduit par a blumer j blumer d haussier r mcconnell et a ehrenfeucht 6 8 et qui reconnait les memes mots par contre la construction de l est en o l a min l lgmax ou l et a sont les cardinaux respectifs de l et de son alphabet a l est la somme des longueurs des mots de l et lgmax est la longueur maximum des mots de l et non en o l comme le leur nous construisons ensuite un second transducteur l qui admet les m mes etats que l et qui pour tout facteur u de l et pour toute lettre a de a tels que ua ne soit pas facteur de l determine et place le plus grand facteur droit de ua qui soit facteur de l ce transducteur generalise celui que nous avons introduit en 20 pour un mot unique et sa construction est en o l a a l aide des transducteurs l et l nous obtenons algorithme qui determine toutes les occurences de facteurs de l dans un texte lineairement par rapport a la longueur du texte et independamment de l alphabet du texte cet algorithme peut etre utilise par un informaticien retrouver et modifier toute une famille d identificateurs dans un programme un linguiste peut aussi determiner tous les mots d une meme famille ou relatifs a un meme concept en eliminant eventuellement les mots paronymes

a method for detecting structure in polygons abstract in order to efficiently represent any data in a computer it is desirable to have available information about any inherent structure in the data when the data is visual the structural information may also be useful in analyzing a scene or object here an inductive methodology for finding structure is presented and applied to the problem of shape analysis for polygons the polygon structure graph is introduced as a tool for shape analysis based on angles and lengths of sides it not only aids in finding repeated substructures but makes it easy to match parts of the polygon with parts of others an algorithm for computing the polygon structure graph is given and examples are presented

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

three dimensional free vibration of arbitrarily shaped laminated micro plates with sliding interfaces within couple stress theory abstract free vibration of laminated micro plates with arbitrary geometry and boundary conditions consisting of several micro layers with free sliding frictional sliding perfect interfaces is of interest the inter layer bond in the direction normal to the interfaces is perfect and thus the corresponding displacement component is continuous across the interfaces the mentioned in plane interface conditions may be realized by varying the stiffness of the sliding inter layer spring of vanishing thickness for free sliding and perfect interface conditions the stiffness 0 and respectively couple stress theory as a useful higher order continuum theory is utilized to formulate the problem subsequently the corresponding hamiltonian is presented the three components of the displacement field in each layer are given by three distinct expressions each expression consists of the product of a base function and three dimensional polynomials with unknown coefficients the base functions are chosen according to the geometries of the system and the corresponding layer and to satisfy the homogeneous essential boundary conditions for the enforcement of the displacement continuity in the direction normal to the interfaces an appropriate series with unknown coefficients pertinent to each layer is added the influences of the level of inter layer imperfection and couple stress on the dynamical characteristics including the angular frequencies of free vibration and the corresponding mode shapes are addressed

bayesian geostatistical design task driven optimal site investigation when the geostatistical model is uncertain 1 geostatistical optimal design optimizes subsurface exploration for maximum information toward task specific prediction goals until recently most geostatistical design studies have assumed that the geostatistical description i e the mean trends covariance models and their parameters is given a priori this contradicts as emphasized by rubin and dagan 1987a the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort we believe that geostatistical design should 1 avoid unjustified a priori assumptions on the geostatistical description 2 instead reduce geostatistical model uncertainty as secondary design objective 3 rate this secondary objective optimal for the overall prediction goal and 4 be robust even under inaccurate geostatistical assumptions bayesian geostatistical design follows these guidelines by considering uncertain covariance model parameters we transfer this concept from kriging like applications to geostatistical inverse problems we also deem it inappropriate to consider parametric uncertainty only within a single covariance model the matern family of covariance functions has an additional shape parameter controlling model shape by a parameter converts covariance model selection to parameter identification and resembles bayesian model averaging over a continuous spectrum of covariance models this is appealing since it generalizes bayesian model averaging from a finite number to an infinite number of models we illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases the underlying scenarios are to minimize the prediction variance of 1 contaminant concentration or 2 arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

estimating effective model parameters for heterogeneous unsaturated flow using error models for bias correction 1 estimates of effective parameters for unsaturated flow models are typically based on observations taken on length scales smaller than the modeling scale this complicates parameter estimation for heterogeneous soil structures in this paper we attempt to account for soil structure not present in the flow model by using so called external error models which correct for bias in the likelihood function of a parameter estimation algorithm the performance of external error models are investigated using data from three virtual reality experiments and one real world experiment all experiments are multistep outflow and inflow experiments in columns packed with two sand types with different structures first effective parameters for equivalent homogeneous models for the different columns were estimated using soil moisture measurements taken at a few locations this resulted in parameters that had a low predictive power for the averaged states of the soil moisture if the measurements did not adequately capture a representative elementary volume of the heterogeneous soil column second parameter estimation was performed using error models that attempted to correct for bias introduced by soil structure not taken into account in the first estimation three different error models that required different amounts of prior knowledge about the heterogeneous structure were considered the results showed that the introduction of an error model can help to obtain effective parameters with more predictive power with respect to the average soil water content in the system this was especially true when the dynamic behavior of the flow process was analyzed

a hybrid regularized inversion methodology for highly parameterized environmental models 1 a hybrid approach to the regularized inversion of highly parameterized environmental models is described the method is based on constructing a highly parameterized base model calculating base parameter sensitivities and decomposing the base parameter normal matrix into eigenvectors representing principal orthogonal directions in parameter space the decomposition is used to construct super parameters super parameters are factors by which principal eigenvectors of the base parameter normal matrix are multiplied in order to minimize a composite least squares objective function these eigenvectors define orthogonal axes of a parameter subspace for which information is available from the calibration data the coordinates of the solution are sought within this subspace super parameters are estimated using a regularized nonlinear gauss marquardt levenberg scheme though super parameters are estimated tikhonov regularization constraints are imposed on base parameters tikhonov regularization mitigates over fitting and promotes the estimation of reasonable base parameters use of a large number of base parameters enables the inversion process to be receptive to the information content of the calibration data including aspects pertaining to small scale parameter variations because the number of super parameters sustainable by the calibration data may be far less than the number of base parameters used to define the original problem the computational burden for solution of the inverse problem is reduced the hybrid methodology is described and applied to a simple synthetic groundwater flow model it is then applied to a real world groundwater flow and contaminant transport model the approach and programs described are applicable to a range of modeling disciplines copyright 2005 by the american geophysical union

time optimal top k document retrieval let mathcal d be a collection of d documents which are strings over an alphabet of size sigma of total length n we describe a data structure that uses linear space and reports k most relevant documents that contain a query pattern p which is a string of length p packed in p log sigma n words in time o p log sigma n k this is optimal in the ram model in the general case where log d theta log n and involves a novel ram optimal suffix tree search our construction supports an ample set of important relevance measures such as the number of times p appears in a document called term frequency a fixed document importance and the minimal distance between two occurrences of p in a document when log d o log n we show how to reduce the space of the data structure from o n log n to o n log sigma log d log log n bits and to o n log sigma log d bits in the case of the popular term frequency measure of relevance at the price of an additive term o log varepsilon sigma n in the query time for any constant varepsilon 0 we also consider the dynamic scenario where documents can be inserted and deleted from the collection we obtain linear space and query time o p log log n 2 log sigma n log n k log log k whereas insertions and deletions require o log 1 varepsilon n time per symbol for any constant varepsilon 0 finally we consider an extended static scenario where an extra parameter mathtt par p d is defined and the query must retrieve only documents d such that mathtt par p d in tau 1 tau 2 where this range is specified at query time we solve these queries using linear space and o p log sigma n log 1 varepsilon n k log varepsilon n time for any constant varepsilon 0 our technique is to translate these top k problems into multidimensional geometric search problems as a bonus we describe some improvements to those problems

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

approximate string matching using factor automata given a text t over alphabet and a complete index for t constructed using the finite automaton called the factor automaton or dawg accepting all the substrings factors of text t an answer to the query whether a pattern p occurs in text t with k differences is discussed to be done by an algorithm having the time complexity independent on the length of text t the algorithm searches for the final state of the finite automaton accepting the intersection of languages fac t the set of all factors of t and lk p the set of all strings having at most k differences with respect to pattern p

probabilistic collocation method for flow in porous media comparisons with other stochastic methods 1 an efficient method for uncertainty analysis of flow in random porous media is explored in this study on the basis of combination of karhunen loeve expansion and probabilistic collocation method pcm the random log transformed hydraulic conductivity field is represented by the karhunen loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points the procedure is straightforward and analogous to the monte carlo method but the number of simulations required in pcm is significantly reduced steady state flows in saturated random porous media are simulated with the probabilistic collocation method and comparisons are made with other stochastic methods monte carlo method the traditional polynomial chaos expansion pce approach based on galerkin scheme and the moment equation approach based on karhunen loeve expansion klme this study reveals that pcm and klme are more efficient than the galerkin pce approach while the computational efforts are greatly reduced compared to the direct sampling monte carlo method the pcm and klme approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head

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

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

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

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

inverse modeling of geochemical and mechanical compaction in sedimentary basins through polynomial chaos expansion we present an inverse modeling procedure for the estimation of model parameters of sedimentary basins subject to compaction driven by mechanical and geochemical processes we consider a sandstone basin whose dynamics are governed by a set of unknown key quantities these include geophysical and geochemical system attributes as well as pressure and temperature boundary conditions we derive a reduced or surrogate model of the system behavior based on generalized polynomial chaos expansion gpce approximations which are directly linked to the variance based sobol indices associated with the selected uncertain model parameters parameter estimation is then performed within a maximum likelihood ml framework we then study the way the ml inversion procedure can benefit from the adoption of anisotropic polynomial approximations a gpce in which the surrogate model is refined only with respect to selected parameters according to an analysis of the nonlinearity of the input output mapping as quantified through the sobol sensitivity indices results are illustrated for a one dimensional setting involving quartz cementation and mechanical compaction in sandstones the reliability of gpce and a gpce approximations in the context of the inverse modeling framework is assessed the effects of a the strategy employed to build the surrogate model leading either to a gpce or a gpce representation and b the type and quality of calibration data on the goodness of the parameter estimates is then explored

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

general suffix automaton construction algorithm and space bounds suffix automata and factor automata are efficient data structures for representing the full index of a set of strings they are minimal deterministic automata representing the set of all suffixes or substrings of a set of strings this paper presents a novel analysis of the size of the suffix automaton or factor automaton of a set of strings it shows that the suffix automaton or factor automaton of a set of strings u has at most 2q 2 states where q is the number of nodes of a prefix tree representing the strings in u this bound significantly improves over email protected email protected 1 the bound given by blumer et al a blumer j blumer d haussler r m mcconnell a ehrenfeucht complete inverted files for efficient text retrieval and analysis journal of the acm 34 1987 578 589 where email protected is the sum of the lengths of all strings in u more generally we give novel and general bounds for the size of the suffix or factor automaton of an automaton as a function of the size of the original automaton and the maximal length of a suffix shared by the strings it accepts we also describe in detail a linear time algorithm for constructing the suffix automaton s or factor automaton f of u in time o s our algorithm applies in fact to any input suffix unique automaton and strictly generalizes the standard on line construction of a suffix automaton for a single input string our algorithm can also be used straightforwardly to generate the suffix oracle or factor oracle of a set of strings which has been shown to have various useful properties in string matching our analysis suggests that the use of factor automata of automata can be practical for large scale applications a fact that is further supported by the results of our experiments applying factor automata to a music identification task with more than 15 000 songs

optimization and uncertainty assessment of strongly nonlinear groundwater models with high parameter dimensionality 1 highly parameterized and cpu intensive groundwater models are increasingly being used to understand and predict flow and transport through aquifers despite their frequent use these models pose significant challenges for parameter estimation and predictive uncertainty analysis algorithms particularly global methods which usually require very large numbers of forward runs here we present a general methodology for parameter estimation and uncertainty analysis that can be utilized in these situations our proposed method includes extraction of a surrogate model that mimics key characteristics of a full process model followed by testing and implementation of a pragmatic uncertainty analysis technique called null space monte carlo nsmc that merges the strengths of gradient based search and parameter dimensionality reduction as part of the surrogate model analysis the results of nsmc are compared with a formal bayesian approach using the differential evolution adaptive metropolis dream algorithm such a comparison has never been accomplished before especially in the context of high parameter dimensionality despite the highly nonlinear nature of the inverse problem the existence of multiple local minima and the relatively large parameter dimensionality both methods performed well and results compare favorably with each other experiences gained from the surrogate model analysis are then transferred to calibrate the full highly parameterized and cpu intensive groundwater model and to explore predictive uncertainty of predictions made by that model the methodology presented here is generally applicable to any highly parameterized and cpu intensive environmental model where efficient methods such as nsmc provide the only practical means for conducting predictive uncertainty analysis

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

bayesian inverse problem and optimization with iterative spatial resampling 1 measurements are often unable to uniquely characterize the subsurface at a desired modeling resolution in particular inverse problems involving the characterization of hydraulic properties are typically ill posed since they generally present more unknowns than data in a bayesian context solutions to such problems consist of a posterior ensemble of models that fit the data up to a certain precision specified by a likelihood function and that are a subset of a prior distribution two possible approaches for this problem are markov chain monte carlo mcmc techniques and optimization calibration methods both frameworks rely on a perturbation mechanism to steer the search for solutions when the model parameters are spatially dependent variable fields obtained using geostatistical realizations such as hydraulic conductivity or porosity it is not trivial to incur perturbations that respect the prior spatial model to overcome this problem we propose a general transition kernel iterative spatial resampling isr that preserves any spatial model produced by conditional simulation we also present a stochastic stopping criterion for the optimizations inspired from importance sampling in the studied cases this yields posterior distributions reasonably close to the ones obtained by a rejection sampler but with a greatly reduced number of forward model runs the technique is general in the sense that it can be used with any conditional geostatistical simulation method whether it generates continuous or discrete variables therefore it allows sampling of different priors and conditioning to a variety of data types several examples are provided based on either multi gaussian or multiple point statistics

inverse stochastic moment analysis of steady state flow in randomly heterogeneous media 1 nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties we present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables our approach accounts for all scales of spatial variability resolvable by the computational grid it is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete pilot points prior estimates of pilot point values are obtained optionally by generalized kriging posterior estimates at pilot points and optionally at measurement points are obtained by calibrating mean flow against measured values of head the parameters are projected onto a computational grid via kriging maximum likelihood calibration allows estimating not only hydraulic but also optionally unknown variogram parameters with or without prior information about the former the approach yields covariance matrices for parameter estimation as well as head and flux prediction errors the latter being obtained a posteriori we implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two dimensional domain our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty we also evaluate the functional form of the log conductivity variogram and its parameters using likelihood based model discrimination criteria

fully functional suffix trees and optimal text searching in bwt runs bounded space indexing highly repetitive texts such as genomic databases software repositories and versioned text collections has become an important problem since the turn of the millennium a relevant compressibility measure for repetitive texts is r the number of runs in their burrows wheeler transforms bwts one of the earliest indexes for repetitive collections the run length fm index used o r space and was able to efficiently count the number of occurrences of a pattern of length m in a text of length n in o m log log n time with current techniques however it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of r in this article we close this long standing problem showing how to extend the run length fm index so that it can locate the occ occurrences efficiently in o occ log log n time within o r space by raising the space to o r log log n our index counts the occurrences in optimal time o m and locates them in optimal time as well o m occ by further raising the space by an o w log factor where is the alphabet size and w e log n is the ram machine size in bits we support count and locate in o m log w and o m log w occ time which is optimal in the packed setting and had not been obtained before in compressed space we also describe a structure using o r log n r space that replaces the text and extracts any text substring of length e in the almost optimal time o log n r e log w within that space we similarly provide access to arbitrary suffix array inverse suffix array and longest common prefix array cells in time o log n r and extend these capabilities to full suffix tree functionality typically in o log n r time per operation our experiments show that our o r space index outperforms the space competitive alternatives by 1 2 orders of magnitude in time competitive implementations of the original fm index are outperformed by 1 2 orders of magnitude in space and or 2 3 in time

evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods evaluating marginal likelihood is the most critical and computationally expensive task when conducting bayesian model averaging to quantify parametric and model uncertainties the evaluation is commonly done by using laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using monte carlo mc methods to evaluate arithmetic or harmonic mean of a joint likelihood function this study introduces a new mc method i e thermodynamic integration which has not been attempted in environmental modeling instead of using samples only from prior parameter space as in arithmetic mean evaluation or posterior parameter space as in harmonic mean evaluation the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space this is done through a path sampling that conducts markov chain monte carlo simulation with different power coefficient values applied to the joint likelihood function the thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the laplace approximation method and three mc methods including the nested sampling method that is recently introduced into environmental modeling the thermodynamic integration method outperforms the other methods in terms of their accuracy convergence and consistency the thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models the application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of bayesian model averaging the thermodynamic integration method is mathematically rigorous and its mc implementation is computationally general for a wide range of environmental problems

complete inverted files for efficient text retrieval and analysis given a finite set of texts s w 1 w k over some fixed finite alphabet s a complete inverted file for s is an abstract data type that provides the functions find w which returns the longest prefix of w that occurs as a subword of a word in s freq w which returns the number of times w occurs in s and locations w which returns the set of positions where w occurs in s a data structure that implements a complete inverted file for s that occupies linear space and can be built in linear time using the uniform cost ram model is given using this data structure the time for each of the above query functions is optimal to accomplish this techniques from the theory of finite automata and the work on suffix trees are used to build a deterministic finite automaton that recognizes the set of all subwords of the set s this automaton is then annotated with additional information and compacted to facilitate the desired query functions the result is a data structure that is smaller and more flexible than the suffix tree

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

the minimum structure solution to the inverse problem the inverse problem of estimating the conductivity function from head observations is generally ill posed many conductivity functions are consistent with the data it is widely accepted now that a well defined estimate can be obtained only if additional information about the function structure is introduced into the problem formulation this work presents a method to obtain a stable and reasonable estimate that utilizes only the data and the flow or transport model with the minimum possible suppositions about the unknown function or its structure the motivation is to develop a solution that has only characteristics that are traced directly to the data and the flow or transport model without taking advantage of spatial continuity or other prior information the solution is obtained by minimizing the upper bound to the error or in a stochastic conceptual framework as the most likely solution given the data this solution although generally not the most accurate since it neglects to utilize structural information that may be available is of fundamental importance and may be useful as a benchmark for example by comparing this solution with other solutions one can become aware of how prior information or the model of spatial structure affects the solution to the inverse problem

a semi analytical method for piezocomposite structures with arbitrary interfaces abstract a general three dimensional semi analytical solution of piezocomposite media consisting of various domains of distinct electro mechanical properties is proposed the interfaces between the domains may be non planar the geometry and boundary conditions may be arbitrary and the applied loading can be in the form of traction displacement voltage or any combination of them in this approach the unknown displacement field u1 u2 u3 and the electric potential are taken as appropriate functions in each domain such that the continuity of the displacement field and the electric potential are exactly satisfied at the interfaces also the continuities of traction stresses and normal electric displacement across the interfaces are resolved with high accuracy the homogeneous and non homogeneous kinematic boundary conditions associated with the electric potential and the displacement field are incorporated at the boundaries exactly

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

efficient dynamic dictionary matching with dawgs and ac automata abstract the dictionary matching is a task to find all occurrences of pattern strings in a set d called a dictionary on a text string t the aho corasick automaton ac automaton which is built on d is a fundamental data structure which enables us to solve the dictionary matching problem in o d log preprocessing time and o n log occ matching time where d is the total length of the patterns in the dictionary d n is the length of the text is the alphabet size and occ is the total number of occurrences of all the patterns in the text the dynamic dictionary matching is a variant where patterns may dynamically be inserted into and deleted from the dictionary d this problem is called semi dynamic dictionary matching if only insertions are allowed in this paper we propose two efficient algorithms that can solve both problems with some modifications for a pattern of length m our first algorithm supports insertions in o m log log d log log d time and pattern matching in o n log occ for the semi dynamic setting this algorithm also supports both insertions and deletions in o m log d log log d time and pattern matching in o n log d log log d log occ log d log log d time for the dynamic dictionary matching problem by some modifications this algorithm is based on the directed acyclic word graph dawg of blumer et al jacm 1987 our second algorithm which is based on the ac automaton supports insertions in o m log u f u o time for the semi dynamic setting and supports both insertions and deletions in o m u f u o time for the dynamic setting where u f and u o respectively denote the numbers of states in which the failure function and the output function need to be updated this algorithm performs pattern matching in o n log occ time for both settings our algorithm achieves optimal update time for ac automaton based methods over constant size alphabets since any algorithm which explicitly maintains the ac automaton requires m u f u o update time

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

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

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

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

coarse gradient langevin algorithms for dynamic data integration and uncertainty quantification the main goal of this paper is to design an efficient sampling technique for dynamic data integration using the langevin algorithms based on a coarse scale model of the problem we compute the proposals of the langevin algorithms using the coarse scale gradient of the target distribution to guarantee a correct and efficient sampling each proposal is first tested by a metropolis acceptance rejection step with a coarse scale distribution if the proposal is accepted in the first stage then a fine scale simulation is performed at the second stage to determine the acceptance probability comparing with the direct langevin algorithm the new method generates a modified markov chain by incorporating the coarse scale information of the problem under some mild technical conditions we prove that the modified markov chain converges to the correct posterior distribution we would like to note that the coarse scale models used in the simulations need to be inexpensive but not necessarily very accurate as our analysis and numerical simulations demonstrate we present numerical examples for sampling permeability fields using two point geostatistics karhunen loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data such as the production data as well as the static data the numerical examples show that the coarse gradient langevin algorithms are much faster than the direct langevin algorithms but have similar acceptance rates

the smallest automaton recognizing the subwords of a text let a partial deterministic finite automaton be a dfa in which each state need not have a transition edge for each letter of the alphabet we demonstrate that the smallest partial dfa for the set of all subwords of a given word w iwl 2 has at most 21w 2 states and 3 wl 4 transition edges independently of the alphabet size we give an algorithm to build this smallest partial dfa from the input w on line in linear time

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

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

nonlocal and localized analyses of conditional mean transient flow in bounded randomly heterogeneous porous media 1 we consider the numerical prediction of transient flow in bounded randomly heterogeneous porous media driven by random sources initial heads and boundary conditions without resorting to monte carlo simulation after applying the laplace transform to the governing stochastic flow equations we derive exact nonlocal integro differential equations for the mean and variance covariance of transformed head and flux conditioned on measured values of log conductivity y ln k approximating these conditional moment equations recursively to second order in the standard deviation y of y we solve them by finite elements for superimposed mean uniform and convergent flows in a two dimensional domain an alternative conditional mean solution is obtained through localization of the exact moment expressions the nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain a comparison with monte carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with y2 as large as 4 the nonlocal solution is only slightly more accurate than the much simpler localized solution but the latter does not yield information about predictive uncertainty the accuracy of each solution improves markedly with conditioning a preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for monte carlo statistics to stabilize when the same direct matrix solver is used for all three we do not presently know how using iterative solvers would have affected this conclusion this is true whether the laplace inversion and monte carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size the underlying exact and recursive moment equations as well as the proposed computational algorithm are valid in both two and three dimensions only the numerical implementation of our algorithm is two dimensional

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

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

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

erratum transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations water resources research 1998 34 1 1 12

use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory scale porous media in this work we show how the use of global sensitivity analysis gsa in conjunction with the polynomial chaos expansion pce methodology can provide relevant information for the interpretation of transport experiments in laboratory scale heterogeneous porous media we perform gsa by calculating the sobol indices which provide a variance based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model the choice of pce has the following two benefits 1 it provides the global sensitivity indices in a straightforward manner and 2 pce can serve as a surrogate model for the calibration of parameters the coefficients of the pce are computed by probabilistic collocation the methodology is applied to two nonreactive transport experiments available in the literature while considering both transient and pseudo steady state transport regimes this method allows a rigorous investigation of the relative effects and importance of different uncertain quantities which include boundary conditions as well as porous medium hydraulic and dispersive parameters the parameters that are most relevant to depicting the system s behavior can then be evaluated in addition one can assess the space time distribution of measurement points which is the most influential factor for the identifiability of parameters our work indicates that these methods can be valuable tools in the proper design of model based transport experiments

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

a formal likelihood function for parameter and predictive inference of hydrologic models with correlated heteroscedastic and non gaussian errors estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions residual errors are often assumed to be independent and to be adequately described by a gaussian probability distribution with a mean of zero and a constant variance here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed a formal generalized likelihood function is presented which extends the applicability of previously used likelihood functions to situations where residual errors are correlated heteroscedastic and non gaussian with varying degrees of kurtosis and skewness the approach focuses on a correct statistical description of the data and the total model residuals without separating out various error sources application to bayesian uncertainty analysis of a conceptual rainfall runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors when applied to daily rainfall runoff data from a humid basin we find that 1 residual errors are much better described by a heteroscedastic first order auto correlated error model with a laplacian distribution function characterized by heavier tails than a gaussian distribution and 2 compared to a standard least squares approach proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference application to daily rainfall runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that 1 multiplicative bias factors can be used to compensate for some of the largest errors and 2 a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near zero flows we conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well

stochastic analysis of unsaturated flow with probabilistic collocation method 1 in this study we present an efficient approach called the probabilistic collocation method pcm for uncertainty analysis of flow in unsaturated zones in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van genuchten mualem model spatial variability of soil parameters leads to uncertainty in predicting flow behaviors the aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation in the proposed approach input random fields i e the soil parameters are represented via the karhunen loeve expansion and the flow quantities are expressed by polynomial chaos expansions pces the coefficients in the pces are determined by solving the equations for a set of carefully selected collocation points in the probability space to illustrate this approach we use two dimensional examples with different input variances and correlation scales and under steady state and transient conditions we also demonstrate how to deal with multiple input random parameters to validate the pcm we compare the resulting mean and variance of the flow quantities with those from monte carlo mc simulations the comparison reveals that the pcm can accurately estimate the flow statistics with a much smaller computational effort than the mc

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 2 synthetic experiments paper 2 of this three part series uses synthetic data to investigate the properties of the adjoint state maximum likelihood cross validation asmlcv method presented in paper 1 samper and neuman this issue a more than 40 synthetic experiments are performed to compare various conjugate gradient algorithms investigate the manner in which computer time varies with asmlcv parameters study the effect of sample size and choice of kriging points on asmlcv estimates evaluate the ability of various model structure identification criteria to help select the most appropriate semivariogram model among given alternatives study the conditions required for parameter identifiability uniqueness and stability quantify the statistics of cross validation errors test hypotheses concerning the distribution and autocorrelation of these errors and illustrate the computation of approximate quality indicators for asmlcv parameter estimates