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

development of a genetic algorithm based lookup table approach for efficient numerical integration in the method of finite spheres with application to the solution of thin beam and plate problems it is observed that for the solution of thin beam and plate problems using the meshfree method of finite spheres gaussian and adaptive quadrature schemes are computationally inefficient in this paper we develop a novel technique in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline computational step during online computations this lookup table is used much like a table of gaussian integration points and weights in the finite element computations this technique offers significant reduction of computational time without sacrificing accuracy example problems are solved which demonstrate the effectiveness of the procedure copyright 2006 john wiley sons ltd

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

a nodal integration technique for meshfree radial point interpolation method ni rpim a novel nodal integration technique for the meshfree radial point interpolation method ni rpim is presented for solid mechanics problems in the ni rpim radial basis functions rbfs augmented with polynomials are used to construct shape functions that possess the delta function property galerkin weak form is adopted for creating discretized system equations in which nodal integration is used to compute system matrices a stable and simple nodal integration scheme is proposed to perform the nodal integration numerically the ni rpim is examined using a number of example problems including stress analysis of an automobile mechanical component the effect of shape parameters and dimension of local support domain on the results of the ni rpim is investigated in detail through these examples the numerical solutions show that the present method is a robust reliable stable meshfree method and possesses better computational properties compared with traditional linear fem and original rpim using gauss integration scheme

development of a formal likelihood function for improved bayesian inference of ephemeral catchments 1 the application of formal bayesian inferential approaches in hydrologic modeling is often criticized for requiring explicit assumptions to be made about the distribution of the errors via the likelihood function these assumptions can be adequate in some situations but often little attention is paid to the selection of an appropriate likelihood function this paper investigates the application of bayesian methods in modeling ephemeral catchments we consider two modeling case studies including a synthetically generated data set and a real data set of an arid australian catchment the case studies investigate some typical forms of the likelihood function which have been applied widely by previous researchers as well as introducing new implementations aimed at better addressing likelihood function assumptions in dry catchments the results of the case studies indicate the importance of explicitly accounting for model residuals that are highly positively skewed due to the presence of many zeros zero inflation arising from the dry spells experienced by the catchment specifically the form of the likelihood function was found to significantly impact the calibrated parameter posterior distributions which in turn have the potential to greatly affect the uncertainty estimates in each application the likelihood function that explicitly accounted for the nonconstant variance of the errors and the zero inflation of the errors resulted in 1 equivalent or better fits to the observed discharge in both timing and volume 2 superior estimation of the uncertainty as measured by the reliability and sharpness metrics and 3 more linear quantile quantile plots indicating the errors were more closely matched to the assumptions of this form of the likelihood function

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

normalized sph with stress points smoothed particle hydrodynamics is extended to a normalized staggered particle formulation with boundary conditions a companion set of interpolation points is introduced that carry the stress velocity gradient and other derived field variables the method is stable linearly consistent and has an explicit treatment of boundary conditions also a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement test problems show that these improvements lead to increased accuracy and stability published in 2000 by john wiley sons ltd

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

generalized likelihood uncertainty estimation glue and approximate bayesian computation what s the connection 1 there has been a recent debate in the hydrological community about the relative merits of the informal generalized likelihood uncertainty estimation glue approach to uncertainty assessment in hydrological modeling versus formal probabilistic approaches some recent literature has suggested that the methods can give similar results in practice when properly applied in this note we show that the connection between formal bayes and glue is not merely operational but goes deeper with glue corresponding to a certain approximate bayesian procedure even when the generalized likelihood is not a true likelihood the connection we describe relates to recent approximate bayesian computation abc methods originating in genetics abc algorithms involve the use of a kernel function and the generalized likelihood in glue can be thought of as relating to this kernel function rather than to the model likelihood two interpretations of glue emerge one as a computational approximation to a bayes procedure for a certain error free model and the second as an exact bayes procedure for a perturbation of that model in which the truncation of the generalized likelihood in glue plays a role the intent of this study is to encourage cross fertilization of ideas regarding glue and abc in hydrologic applications the connection we outline suggests the possibility of combining a formal likelihood with a kernel based on a generalized likelihood within the abc framework and also allows advanced abc computational methods to be used in glue applications the model based interpretation of glue may also be helpful in partially illuminating the implicit assumptions in different choices of generalized likelihood

an adaptive sparse grid high order stochastic collocation method for bayesian inference in groundwater reactive transport modeling 1 bayesian analysis has become vital to uncertainty quantification in groundwater modeling but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function ppdf of model parameters this is particularly the case when the ppdf is estimated using markov chain monte carlo mcmc sampling in this study a new approach is developed to improve the computational efficiency of bayesian inference by constructing a surrogate of the ppdf using an adaptive sparse grid high order stochastic collocation asg hsc method unlike previous works using first order hierarchical basis this paper utilizes a compactly supported higher order hierarchical basis to construct the surrogate system resulting in a significant reduction in the number of required model executions in addition using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space which further improves computational efficiency to efficiently build the surrogate system for the ppdf with multiple significant modes optimization techniques are used to identify the modes for which high probability regions are defined and components of the asg hsc approximation are constructed after the surrogate is determined the ppdf can be evaluated by sampling the surrogate system directly without model execution resulting in improved efficiency of the surrogate based mcmc compared with conventional mcmc the developed method is evaluated using two synthetic groundwater reactive transport models the first example involves coupled linear reactions and demonstrates the accuracy of our high order hierarchical basis approach in approximating high dimensional posteriori distribution the second example is highly nonlinear because of the reactions of uranium surface complexation and demonstrates how the iterative asg hsc method is able to capture multimodal and non gaussian features of ppdf caused by model nonlinearity both experiments show that asg hsc is an effective and efficient tool for bayesian inference

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

on the evaluation of the method of finite spheres for the solution of reissner mindlin plate problems using the numerical inf sup test in this paper we use the numerical inf sup test to evaluate both displacement based and mixed discretization schemes for the solution of reissner mindlin plate problems using the meshfree method of finite spheres while an analytical proof of whether a discretization scheme passes the inf sup condition is most desirable such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved the numerical inf sup test int j numer meth engng 1997 40 3639 3663 developed to test finite element discretization spaces has therefore been adopted in this paper tests have been performed for both regular and irregular nodal configurations while like linear finite elements pure displacement based approximation spaces with linear consistency do not pass the inf sup test and exhibit shear locking quadratic discretizations unlike quadratic finite elements pass the test pure displacement based and mixed approximation spaces that pass the numerical inf sup test exhibit optimal or near optimal convergence behaviour copyright 2007 john wiley sons ltd

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

a unified stability analysis of meshless particle methods a uni ed stability analysis of meshless methods with eulerian and lagrangian kernels is presented three types of instabilities were identi ed in one dimension an instability due to rank de ciency a tensile instability and a material instability which is also found in continua the stability of particle methods with eulerian and lagrangian kernels is markedly di erent lagrangian kernels do not exhibit the tensile instability in both kernels the instability due to rank de ciency can be suppressed by stress points in two dimensions the stabilizing e ect of stress points is dependent on their locations it was found that the best approach to stable particle discretizations is to use lagrangian kernels with stress points the stability of the least squares stabilization was also studied copyright 2000 john wiley sons ltd

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

finite element stabilization matrices a unification approach abstract in this paper we show that the stabilization vector can be obtained naturally by taking the partial derivatives with respect to the natural coordinates hence the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates with these definitions of stresses and strains a general form of the finite element stabilization matrices is then constructed for underintegrated irregular shape and anisotropic elements the explicit expressions of the stabilization matrices for the 4 9 and 8 node laplace and continuum elements and the 4 node plate element are then derived from the general form these are sufficiently general for the developments herein the generalization of the stabilization matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear finite deformation constitutive equations is also presented the computer implementation aspects and numerical evaluation of these stabilized elements are also considered numerical tests confirm the stability and accuracy characteristics of the resulting elements in particular the numerical experiments reported here also show that the rates of convergence in the l2 norm and in the energy norm agree well with the expected convergence rates

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

a four point integration scheme with quadratic exactness for three dimensional element free galerkin method based on variationally consistent formulation abstract the formulation of three dimensional element free galerkin efg method based on the hu washizu three field variational principle is described the orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions this leads to a variationally consistent formulation which has a similar form as the formulation of standard galerkin weak form based on this formulation an integration scheme which employs only four cubature points in each background tetrahedral element cell is rationally developed for three dimensional efg with quadratic approximation the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved extension of the proposed method to small strain elastoplasticity is also presented the proposed method can exactly pass quadratic patch test that is quadratic exactness is achieved and thus it is named as quadratically consistent 4 point qc4 integration method in contrast efg with standard tetrahedral cubature and the existing linearly consistent 1 point lc1 integration fail to exactly pass quadratic patch test numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy convergence efficiency and stability the capability of the proposed qc4 scheme in solving elastoplastic problems is also demonstrated by numerical examples

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

inference of reactive transport model parameters using a bayesian multivariate approach parameter estimation of subsurface transport models from multispecies data requires the definition of an objective function that includes different types of measurements common approaches are weighted least squares wls where weights are specified a priori for each measurement and weighted least squares with weight estimation wls we where weights are estimated from the data together with the parameters in this study we formulate the parameter estimation task as a multivariate bayesian inference problem the wls and wls we methods are special cases in this framework corresponding to specific prior assumptions about the residual covariance matrix the bayesian perspective allows for generalizations to cases where residual correlation is important and for efficient inference by analytically integrating out the variances weights and selected covariances from the joint posterior specifically the wls and wls we methods are compared to a multivariate mv approach that accounts for specific residual correlations without the need for explicit estimation of the error parameters when applied to inference of reactive transport model parameters from column scale data on dissolved species concentrations the following results were obtained 1 accounting for residual correlation between species provides more accurate parameter estimation for high residual correlation levels whereas its influence for predictive uncertainty is negligible 2 integrating out the co variances leads to an efficient estimation of the full joint posterior with a reduced computational effort compared to the wls we method and 3 in the presence of model structural errors none of the methods is able to identify the correct parameter values

a hyperspherical adaptive sparse grid method for high dimensional discontinuity detection this work proposes and analyzes a hyperspherical adaptive hierarchical sparse grid method for detecting jump discontinuities of functions in high dimensional spaces the method is motivated by the theoretical and computational inefficiencies of well known adaptive sparse grid methods for discontinuity detection our novel approach constructs a function representation of the discontinuity hypersurface of an n dimensional discontinuous quantity of interest by virtue of a hyperspherical transformation then a sparse grid approximation of the transformed function is built in the hyperspherical coordinate system whose value at each point is estimated by solving a one dimensional discontinuity detection problem due to the smoothness of the hypersurface the new technique can identify jump discontinuities with significantly reduced computational cost compared to existing methods moreover hierarchical acceleration techniques are also incorporated to further reduce the overall complexity rigorous complex

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 bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

consistent element free galerkin method an element free galerkin efg method with linear quadratic and cubic approximations which can exactly in a numerical sense pass the corresponding patch tests is proposed and is named as consistent efg cefg method the development of this method is based on the hu washizu three field variational principle numerical integration schemes with corrected nodal derivatives at quadrature points are proposed according to the satisfaction of the orthogonality condition between stress and strain difference thus the method is variationally consistent the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions are theoretically proved and also numerically validated numerical results show that the proposed cefg method greatly improves the numerical performance of the efg method in terms of accuracy convergence efficiency and stability especially the proposed cubic cefg method which shows exceptional accuracy and convergence copyright 2014 john wiley sons ltd

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

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

floating stress point integration meshfree method for large deformation analysis of elastic and elastoplastic materials summary a new meshfree formulation of stress point integration called the floating stress point integration meshfree method is proposed for the large deformation analysis of elastic and elastoplastic materials this method is a galerkin meshfree method with an updated lagrangian procedure and a quasi implicit time advancing scheme without any background cell for domain integration its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation which is not generally used in meshfree formulations hence this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved the details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method copyright 2012 john wiley sons ltd

uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data we present a computational approach to quantify predictive uncertainty in complex phenomena which is modeled by partial differential equations with uncertain parameters exhibiting multi scale variability the approach is motivated by flow in random composites whose internal architecture spatial arrangement of constitutive materials and spatial variability of properties of each material are both uncertain the proposed two scale framework combines a random domain decomposition rdd and a probabilistic collocation method pcm on sparse grids to quantify these two sources of uncertainty respectively the use of sparse grid points significantly reduces the overall computational cost especially for random processes with small correlation lengths a series of one two and three dimensional computational examples demonstrate that the combined rdd pcm approach yields efficient robust and non intrusive approximations for the statistics of diffusion in random composites

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

quadratically consistent one point qc1 quadrature for meshfree galerkin methods abstract a robust and efficient integration method named quadratically consistent one point qc1 scheme which evaluates the galerkin weak form only at the centers of background triangle elements cells is proposed for meshfree methods using quadratic basis the strain at the evaluation points is approximated by corrected smoothed nodal derivatives which are determined by a discrete form of the divergence theorem between nodal shape functions and their derivatives in taylor s expansion we prove that such smoothed nodal derivatives also meet the differentiation of the approximation consistency dac the same taylor s expansion is applied to the weak form and the smoothed nodal derivatives are used to compute the stiffness matrix the proposed qc1 scheme can pass both the linear and the quadratic patch tests exactly in a numerical sense several examples are provided to demonstrate its better numerical performance in terms of convergence accuracy efficiency and stability over other one point integration methods in the meshfree literature especially its superiority over the existing linearly consistent one point lc1 quadratures

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

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

meshfree co rotational formulation for two dimensional continua in this paper a meshfree co rotational formulation for two dimensional continua is proposed in a co rotational formulation the motion of a body is separated into rigid motion and strain producing deformation traditionally this has been done in the setting of finite elements for beams and shell type elements in the present work every node in a meshfree discretized domain has its own co rotating coordinate system three key ingredients are established in order to apply the co rotational formulation i the relationship between global and local variables ii the angle of rotation of a typical co rotating coordinate system and iii a variationally consistent tangent stiffness matrix an algorithm for the co rotational formulation based on load control is provided maximum entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations numerical examples are presented to demonstrate the validity of the meshfree co rotational formulation copyright 2009 john wiley sons ltd

convergence and stabilization of stress point integration in mesh free and particle methods stress point integration provides significant reductions in the computational effort of mesh free galerkin methods by using fewer integration points and thus facilitates the use of mesh free methods in applications where full integration would be prohibitively expensive the influence of stress point integration on the convergence and stability properties of mesh free methods is studied it is shown by numerical examples that for regular nodal arrangements good rates of convergence can be achieved for non uniform nodal arrangements stress point integration is associated with a mild instability which is manifested by small oscillations addition of stabilization improves the rates of convergence significantly the stability properties are investigated by an eigenvalue study of the laplace operator it is found that the eigenvalues of the stress point quadrature models are between those of full integration and nodal integration stabilized stress point integration is proposed in order to improve convergence and stability properties copyright 2007 john wiley sons ltd

adaptive element free galerkin method applied to the limit analysis of plates the implementation of an h adaptive element free galerkin efg method in the framework of limit analysis is described the naturally conforming property of meshfree approximations with no nodal connectivity required facilitates the implementation of h adaptivity nodes may be moved discarded or introduced without the need for complex manipulation of the data structures involved with the use of the taylor expansion technique the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy a stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method to demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

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

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

stabilized conforming nodal integration in the natural element method a stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non sibsonian interpolation in this approach both the shape functions and the integration scheme are defined through use of first order voronoi diagrams the method illustrates improved performance and significant advantages over previous natural neighbour formulations the method also shows substantial promise for problems with large deformations and for the computation of higher order gradients copyright 2004 john wiley sons ltd

gradient and dilatational stabilizations for stress point integration in the element free galerkin method stabilized stress point integration schemes based on gradient stabilization and dilatational stabilization methods are presented for linear elastostaticity problems in the framework of element free galerkin efg method the instability in stress fields associated with the stress point integration is treated by the addition to the galerkin weak form of stabilization terms which contain product of the gradient of the residual or the trace of the gradient of the residual the latter is called dilatational stabilization numerical results show that the oscillations in the stress fields are successfully removed by the presented stabilization methods and that the convergence and stability properties of direct stress point integration are greatly improved these stabilization methods are particularly suitable for the solution of non linear continua with explicit time integration methods copyright 2008 john wiley sons ltd

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

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

a new support integration scheme for the weakform in mesh free methods a new support integration technique is proposed which is similar to those used in truly mesh free methods the contribution of this paper is to exploit the divergence free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions numerical examples show that the proposed integration method can achieve results that agree with manufactured closed form solutions copyright 2009 john wiley sons ltd