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

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

maximum entropy and the lottery abstract the distribution on m tuples of the first m integers is estimated from marginals of the distribution this problem is of interest in determining unpopular numbers in lotto games in canada s lotto 6 49 the proportion of tickets purchased in previous games containing each number is available under certain conditions the limiting distribution subject to the observed marginals is the constrained maximum entropy distribution this distribution is estimated and monte carlo methods are used to estimate the expected return of various lottery strategies tickets consisting of unpopular numbers may have expected return greater than their cost when the weekly sales are large or there are large carryover prizes prizes not won in earlier games the maximum entropy distribution is a rough approximation of the true distribution of tickets purchased certain aspects of the empirical distribution are not consistent with the maximum entropy distribution alternative methods which attempt to model the behavio

bayesian methods and maximum entropy for ill posed inverse problems in this paper we study linear inverse problems where some generalized moments of an unknown positive measure are observed we introduce a new construction called the maximum entropy on the mean method mem which relies on a suitable sequence of finite dimensional discretized inverse problems its advantage is threefold it allows us to interpret all usual deterministic methods as bayesian methods it gives a very convenient way of taking into account prior information it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena the key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure

maximum entropy principles for disordered spins we transform nonstationary independent random fields with exponential radon nikodym factors and study the asymptotics of the transformed processes as applications we deduce conditional limit theorems for such random fields and we study a curie weiss type mean field model of a quenched mixed magnetic crystal this model has quenched site disorder and frustration but non random coupling constants we find a continuous phase transition with critical exponents equal to those of the classical curie weiss theory

reconstruction sequences and equipartition measures an examination of the asymptotic equipartition property we consider a stationary source emitting letters from a finite alphabet a the source is described by a stationary probability measure spl alpha on the space spl omega a sup in of sequences of letters denote by spl omega sub n the set of words of length n and by spl alpha sub n the probability measure induced on spl omega sub n by spl alpha we consider sequences spl gamma sub n spl sub spl omega sub n n spl isin in having special properties call spl gamma sub n spl sub spl omega sub n n spl isin in a supporting sequence for spl alpha if lim sub n spl alpha sub n spl gamma sub n 1 it is well known that the exponential growth rate of a supporting sequence is bounded below by h sub sh spl alpha the shannon entropy of the source spl alpha for efficient simulation we require spl gamma sub n to be as large as possible subject to the condition that the measure spl alpha sub n is approximated by the equipartition measure spl beta sub n spl middot spl gamma sub n the probability measure on spl omega sub n which gives equal weight to the words in spl gamma sub n and zero weight to words outside it we say that a sequence spl gamma sub n spl sub spl omega sub n n spl isin in is a reconstruction sequence for spl alpha if each spl gamma sub n is invariant under cyclic permutations and lim sub n spl beta sub n spl middot spl gamma sub n spl alpha sub m for each m spl isin in we prove that the exponential growth rate of a reconstruction sequence is bounded above by h sub sh spl alpha we use a large deviation property of the cyclic empirical measure to give a constructive proof of an existence theorem if spl alpha is a stationary source then there exists a reconstruction sequence for spl alpha having maximal exponential growth rate if spl alpha is ergodic then the reconstruction sequence may be chosen so as to be supporting for spl alpha we prove also a characterization of ergodic measures which appears to be new

large sample properties of maximum entropy histograms the large sample properties of a new class of histogram estimators whose derivation is based on an information theory criterion the maximum entropy principle which preserves the observed mass and mean are studied the pointwise strong consistency the point wise asymptotic normality and the rate of convergence to normality are investigated the asymptotic mean square error mse of these estimates is also compared relative to the histogram based on spacings the classical k nearest neighbor the kernel estimator and the generalized k nearest neighbor density estimator

convergence analysis of meshfree approximation schemes this work is concerned with the formulation of a general framework for the analysis of meshfree approximation schemes and with the convergence analysis of the local maximum entropy lme scheme as a particular example we provide conditions for the convergence in sobolev spaces of schemes that are n consistent in the sense of exactly reproducing polynomials of degree less than or equal to n 1 and whose basis functions are of rapid decay the convergence of the lme in w 1 p loc follows as a direct application of the general theory the analysis shows that the convergence order is linear in h a measure of the density of the point set the analysis also shows how to parameterize the lme scheme for optimal convergence because of the convex approximation property of lme its behavior near the boundary is singular and requires additional analysis for the particular case of polyhedral domains we show that away from a small singular part of the boundary any sobolev function can be approximated by means of the lme scheme with the aid of a capacity argument we further obtain approximation results with truncated lme basis functions in h1 and for spatial dimension d 2

optimal order isogeometric collocation at galerkin superconvergent points abstract in this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least squares collocation method presented in anitescu et al 2015 and the variational collocation method presented in gomez and de lorenzis 2016 the focus is on smoothest b splines nurbs approximations i e having global c p 1 continuity for polynomial degree p within the framework of gomez and de lorenzis 2016 we select as collocation points a subset of those considered in anitescu et al 2015 which are related to the galerkin superconvergence theory with our choice that features local symmetry of the collocation stencil we improve the convergence behavior with respect to gomez and de lorenzis 2016 achieving optimal l 2 convergence for odd degree b splines nurbs approximations the same optimal order of convergence is seen in anitescu et al 2015 where however a least squares formulation is adopted further careful study is needed since the robustness of the method and its mathematical foundation are still unclear

maximum entropy and the method of moments in performance evaluation of digital communications systems the maximum entropy criterion for estimating an unknown probability density function from its moments is applied to the evaluation of the average error probability in digital communications accurate averages are obtained even when a few moments are available the method is stable and results compare well with those from the powerful and widely used gauss quadrature rules gqr method for test cases presented in this work the maximum entropy method achieved results with typically a few moments while the gqr method required many more moments to obtain the same as accurately the method requires about the same number of moments as techniques based on orthogonal expansions in addition it provides an estimate of the probability density function of the target variable in a digital communication application

a higher order method based on local maximum entropy approximation we present here a generalization of local maximum entropy max ent approximation for high orders of consistency i e quadratic cubic the method is based upon the application of the de boor s algorithm to the standard linear local max ent approximation the resulting approximation possesses some interesting properties such as non negativity smoothness exact interpolation on the boundary and variation diminishing no gibbs effect the resulting structure has many similarities with b spline surfaces but without the tensor product structure typical of that approximation examples are provided of its use in the framework of a galerkin method showing the potential of the proposed method in solving boundary value problems copyright 2010 john wiley sons ltd

pursuit and an evolutionary game pursuit is a familiar mechanical activity that humans and animals engage in athletes chasing balls predators seeking prey and insects manoeuvring in aerial territorial battles in this paper we discuss and compare strategies for pursuit the occurrence in nature of a strategy known as motion camouflage and some evolutionary arguments to support claims of prevalence of this strategy as opposed to alternatives we discuss feedback laws for a pursuer to realize motion camouflage as well as two alternative strategies we then set up a discrete time evolutionary game to model competition among these strategies this leads to a dynamics in the probability simplex in three dimensions which captures the mean field aspects of the evolutionary game the analysis of this dynamics as an ascent equation solving a linear programming problem is consistent with observed behaviour in monte carlo experiments and lends support to an evolutionary basis for prevalence of motion camouflage

derivatives of maximum entropy basis functions on the boundary theory and computations summary in this paper we obtain explicit expressions to evaluate the derivatives of maximum entropy max ent basis function on the boundary of a convex domain in the max ent formulation the basis functions are obtained by maximizing a concave functional subjected to linear constraints reproducing conditions in doing so it is found that the lagrange multipliers blow up when x and the expressions for the derivatives of the max ent basis functions in are of an indeterminate form for points on we appeal to l hopital s rule to derive expressions to determine the derivatives of the basis functions we consider the shannon entropy functional and the relative entropy functional with different choices of the prior weight function the first order derivatives of all basis functions are bounded in contrast on an irregular grid with a certain nodal spacing some of the second derivatives of the basis functions are unbounded on the boundary necessary and sufficient conditions on the priors to obtain bounded lagrange multipliers are established optimal convergence rates for fourth order problems are demonstrated for a galerkin approach with a quadratically complete partition of unity enriched max ent approximation copyright 2013 john wiley sons ltd

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

convergent meshfree approximation schemes of arbitrary order and smoothness local maximum entropy lme approximation schemes are meshfree approximation schemes that satisfy consistency conditions of order 1 i e they approximate affine functions exactly in addition lme approximation schemes converge in the sobolev space w 1 p i e they are c 0 continuous in the conventional terminology of finite element interpolation here we present a generalization of the local max ent approximation schemes that are consistent to arbitrary order i e interpolate polynomials of arbitrary degree exactly and which converge in w k p i e they are c k continuous to arbitrary order k we refer to these approximation schemes as high order local maximum entropy approximation schemes holmes we prove uniform error bounds for the holmes approximates and their derivatives up to order k moreover we show that the holmes of order k is dense in the sobolev space w k p for any 1 leq p infty the good performance of holmes relative to other meshfree schemes in selected test cases is also critically appraised

the polya information divergence extensions of sanov s theorem and the conditional limit theorem colt are established for a multicolor polya eggenberger pe urn sampling scheme giving the polya information divergence and a polya extension to the maximum relative entropy maxent method polya maxent includes the standard maxent as well as its variants used in bose einstein fermi dirac and intermediate acharya swamy statistics as special cases in the pe setting standard maxent is in general asymptotically inconsistent

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

local optima of the sherrington kirkpatrick hamiltonian we study local optima of the hamiltonian of the sherrington kirkpatrick model we compute the exponent of the expected number of local optima and determine the typical value of the hamiltonian we study local optima of the hamiltonian of the sherrington kirkpatrick model we compute the exponent of the expected number of local optima and determine the typical value of the hamiltonian

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

subset source coding this paper studies the fundamental limits of storage for structured data where statistics and structure are both critical to the application accordingly a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source dms for the lossless subset compression problem it turns out that the optimal source code may not index the conventional source typical sequences but rather index certain subset typical sequences consistent with the subset of interest building upon an achievability and a strong converse an analytic expression is given based on the shannon entropy relative entropy and subset entropy which identifies such subset typical sequences for a broad class of subsets of a dms intuitively subset typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy for the lossy subset compression problem an upper bound is derived on the subset rate distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset typical distribution and over a set of certain auxiliary subsets by proving a strong converse result this upper bound is shown to be tight for a class of symmetric subsets as shown in our numerical examples more often than not one achieves a gain in the fundamental limits in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate distortion function respectively although exceptions are also possible

minimization problems based on relative alpha entropy i forward projection minimization problems with respect to a one parameter family of generalized relative entropies are studied these relative entropies which we term relative alpha entropies denoted mathscr i alpha arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths these parametric relative entropies are a generalization of the usual relative entropy kullback leibler divergence just like relative entropy these relative alpha entropies behave like squared euclidean distance and satisfy the pythagorean property minimizers of these relative alpha entropies on closed and convex sets are shown to exist such minimizations generalize the maximum renyi or tsallis entropy principle vphantom int the minimizing probability distribution termed forward mathscr i alpha projection for a linear family is shown to obey a power law other results in connection with statistical inference namely subspace transitivity and iterated projections are also established in a companion paper a related minimization problem of interest in robust statistics that leads to a reverse mathscr i alpha projection is studied

minimum cross entropy estimation with inaccurate side information given a prior estimate of a probability q and a constraint spl sigma p sub i x sub i a one well known way of estimating p is to minimize the cross entropy i p q subject to the constraint a modification to this method is proposed for use when the value a is only approximately known the modification is based on the penalty function method in constrained optimization it has an interpretation in differential geometry methods in statistics and it sometimes gives a maximum likelihood estimate

local minima in disordered mean field ferromagnets we consider the complexity of random ferromagnetic landscapes on the hypercube pm 1 n given by ising models on the complete graph with i i d non negative edge weights this includes in particular the case of bernoulli disorder corresponding to the ising model on a dense random graph mathcal g n p previous results had shown that with high probability as n rightarrow infty the gradient search energy lowering algorithm initialized uniformly at random converges to one of the homogeneous global minima all plus or all minus here we devise two modified algorithms tailored to explore the landscape at near zero magnetizations where the effect of the ferromagnetic drift is minimized with these we numerically verify the landscape complexity of random ferromagnets finding a diverging number of 1 spin flip stable local minima as n rightarrow infty we then investigate some of the properties of these local minima e g typical energy and magnetization and compare to the situation where the edge weights are drawn from a heavy tailed distribution

arma models prewhitening and minimum cross entropy the problem of spectral estimation on the basis of observations from a finite stretch of a stationary time series is considered in connection with knowledge of a prior estimate of the spectral density a reasonable posterior spectral density estimate would be the density that is closest to the prior according to some measure of divergence while at the same time being compatible with the data the cross entropy has often been proposed to serve as such a measure of divergence a correction of the original minimum cross entropy spectral analysis mcesa method of j e shore see ieee trans acoust speech signal process vol 29 p 230 7 1981 to traditional prewhitening techniques and to autoregressive moving average arma models is pointed out and a fast approximate solution of the minimum cross entropy problem is proposed the solution is in a standard multiplicative form that is the posterior is equal to the prior multiplied by a correction factor

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

an information theoretic spectral density an information theoretic spectrum which minimizes the kullback leibler 1951 information number subject to the first p 1 autocovariance terms is proposed the kl spectrum includes the maximum entropy spectrum and the arma autoregressive moving average spectrum as special cases a method is proposed for modifying a spectral estimate on the basis of the results for the kl spectrum so that the revised spectral estimate is more loyal to the given observations than the primary estimate

on the existence and characterization of the maxent distribution under general moment inequality constraints a broad set of sufficient conditions that guarantees the existence of the maximum entropy maxent distribution consistent with specified bounds on certain generalized moments is derived most results in the literature are either focused on the minimum cross entropy distribution or apply only to distributions with a bounded volume support or address only equality constraints the results of this work hold for general moment inequality constraints for probability distributions with possibly unbounded support and the technical conditions are explicitly on the underlying generalized moment functions an analytical characterization of the maxent distribution is also derived using results from the theory of constrained optimization in infinite dimensional normed linear spaces several auxiliary results of independent interest pertaining to certain properties of convex coercive functions are also presented

nurbs enhanced maximum entropy schemes abstract in this paper we combine high order local maximum entropy schemes holmes with the integration framework developed in the nurbs enhanced finite element method nefem we focus on the two dimensional case where given a domain described by some nurbs curves a meshless formulation based on the holmes approximants is employed for the discretization and at the same time the geometric fidelity given by the nurbs boundary is preserved thanks to the nefem integration since holmes basis functions are not interpolatory on the boundary different techniques are considered for the imposition of essential boundary conditions the efficiency and the accuracy of the proposed methodology are confirmed with supportive numerical examples

generalized divergence measures information matrices amount of information asymptotic distribution and its applications to test statistical hypotheses abstract in this paper a transformation of csiszar s measures which generalizes the unified r s measures defined by sharma and mittal and taneja is presented for these transformations information matrices associated to a differential metric in the direction to the tangent space are obtained as well as the amount of information resulting from parameter perturbation in the direction of coordinate axes finally the asymptotic distribution of information matrices and the amount of information and its applications to test statistical hypotheses are obtained

information theory as a unifying statistical approach for use in marketing research abstract information theory is shown to provide a unified approach to a wide range of problems in marketing research for instance this approach can be used to obtain characterizations parallel to those of the hendry system and other entropic approaches with great economy of assumptions and with the added flexibility that constraints can be easily identified for explicit consideration and implemented as needed goodness of fit tests and decision modelling structures are supplied from these same stochastic models with a range of applications that include market segmentation and brand shifting choices a basic approach to these and other procedures is therefore obtainable from information theoretic methods these methods can be used to address stochastic model selection problems and other probabilistic models of marketing choice for example minimum discrimination information mdi estimation logit multiplicative competitive interaction mci and other important choice models are also shown in this paper to arise naturally from information theoretic formulations with duality relations developed by charnes and cooper providing additional simplifications and interpretations

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

bures and statistical distance for squeezed thermal states we compute the bures distance between two thermal squeezed states and deduce the statistical distance metric by computing the curvature of this metric we can identify regions of parameter space most sensitive to changes in these parameters and thus lead to optimum detection statistics

high order maximum entropy collocation methods abstract this paper considers the approximation of partial differential equations with a point collocation framework based on high order local maximum entropy schemes holmes in this approach smooth basis functions are computed through an optimization procedure and the strong form of the problem is directly imposed at the collocation points reducing significantly the computational times with respect to the galerkin formulation furthermore such a method is truly meshfree since no background integration grid is necessary the validity of holmes collocation is verified with supportive numerical examples where the expected convergence rates are obtained this includes the approximation of pdes on domains bounded by implicit and explicit nurbs curves illustrating a direct integration between geometric modeling and numerical analysis

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions joint confidence and likelihood regions for the parameters in nonlinear regression models can be defined using the geometric concepts of sample space and solution locus using a quadratic approximation to the solution locus instead of the usual linear approximation it is shown that these inference regions correspond to ellipsoids on the tangent plane at the least squares point accurate approximate inference regions can be obtained by mapping these ellipsoids into the parameter space and measures of the effect of intrinsic nonlinearity on inference can be based on the difference between the tangent plane ellipsoids and the sphere which would be obtained using a linear approximation

thin shell analysis from scattered points with maximum entropy approximants this is the accepted version of the following article millan d rosolen a and arroyo m 2011 thin shell analysis from scattered points with maximum entropy approximants int j numer meth engng 85 723 751 doi 10 1002 nme 2992 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 2992 abstract

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

fully probabilistic design of hierarchical bayesian models the minimum cross entropy principle is an established technique for design of an unknown distribution processing linear functional constraints on the distribution more generally fully probabilistic design fpd chooses the distribution within the knowledge constrained set of possible distributions for which the kullback leibler divergence to the designer s ideal distribution is minimized these principles treat the unknown distribution deterministically in this paper fully probabilistic design is applied to hierarchical bayesian models for the first time yielding optimal design of a possibly nonparametric stochastic model for the unknown distribution this equips minimum cross entropy and fpd distributional estimates with measures of uncertainty it enables robust choice of the optimal model as well as randomization of this choice the ability to process non linear functional constraints in the constructed distribution significantly extends the applicability of these principles currently available fpd procedures for a merging of external knowledge b approximate learning and stabilized forgetting c decision strategy design and d local adaptive control design are unified for the first time via the hierarchical fpd framework of this paper

robust line fitting in a noisy image by the method of moments the standard least squared distance method of fitting a line to a set of data points is known to be unreliable when the random noise in the input is significant compared with the data correlated to the line itself here we present a new statistical clustering method based on legendre moment theory and maximum entropy principle for line fitting in a noisy image we propose a new approach for estimating the underlying probability density function p d f of the data set the p d f is expanded in terms of legendre polynomials by means of the legendre moments the order of the expansion is selected according to the maximum entropy principle then the points corresponding to the maxima of the p d f will be the true points of the line to be extracted by a chaining algorithm this approach is directly generalized to multidimensional data the proposed algorithm was successfully applied to real and simulated noisy line images with comparison to some well known methods

maximum entropy principle and nonlinear stochastic oscillators abstract the principle of maximum entropy introduced first in physics and successfully applied in many other fields e g statistics reliability estimation simulation has recently been developed to analyze the systems governed by stochastic differential equations in this paper we extend the maximum entropy method to the general class of stochastic nonlinear systems and apply it to stochastic nonlinear oscillators to indicate the effectiveness of the method a comparison of the results with predictions of the known procedures is shown graphically the analysis presented in this paper indicates that the maximum entropy principle can serve as a general method for determining the probability distributions of a wide class of stochastic systems when we have only a partial information in the form of moments or equations for moments

the renyi redundancy of generalized huffman codes huffman s algorithm gives optimal codes as measured by average codeword length and the redundancy can be measured as the difference between the average codeword length and shannon s entropy if the objective function is replaced by an exponentially weighted average then a simple modification of huffman s algorithm gives optimal codes the redundancy can now be measured as the difference between this new average and a renyi s 1961 generalization of shannon s entropy by decreasing some of the codeword lengths in a shannon code the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved the lower bound is improved by randomizing between codeword lengths allowing linear programming techniques to be used on an integer programming problem these bounds are shown to be asymptotically equal the results are generalized to the renyi case and are related to r g gallager s 1978 bound on the redundancy of huffman codes

minimax compression and large alphabet approximation through poissonization and tilting this paper introduces a convenient strategy for coding and predicting sequences of independent identically distributed random variables generated from a large alphabet of size m in particular the size of the sample is allowed to be variable the employment of a poisson model and tilting method simplifies the implementation and analysis through independence the resulting strategy is optimal within the class of distributions satisfying a moment condition and it is close to optimal for the class of all i i d distributions on strings of a given length the method also can be used to code and predict strings with a condition on the tail of the ordered counts and it can be applied to distributions in an envelope class moreover we show that our model permits exact computation of the minimax optimal code for all alphabet sizes when conditioning on the size of the sample

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

smooth second order non negative meshfree approximants selected by maximum entropy we present a family of approximation schemes which we refer to as second order maximum entropy max ent approximation schemes that extends the first order local max ent approximation schemes to second order consistency this method retains the fundamental properties of first order max ent schemes namely the shape functions are smooth non negative and satisfy a weak kronecker delta property at the boundary this last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial the evaluation of the shape functions is not explicit but it is very efficient and robust to our knowledge the proposed method is the first higher order scheme for function approximation from unstructured data in arbitrary dimensions with non negative shape functions as a consequence the approximants exhibit variation diminishing properties as well as an excellent behavior in structural vibrations problems as compared with the lagrange finite elements mls based meshfree methods and even b spline approximations as shown through numerical experiments when compared with usual mls based second order meshfree methods the shape functions presented here are much easier to integrate in a galerkin approach as illustrated by the standard benchmark problems

a time domain interpretation of the kl spectrum an information theoretic spectrum minimizing the frequency domain kullback leibler kl information number has been proposed by b s choi ieee trans acoust speech and signal processing vol 38 p 717 21 1990 based on the kl spectrum a method of modifying a spectrum estimate was suggested some numerical examples illustrated that the kl spectrum estimate is superior to the initial estimate i e the autocovariances obtained by the inverse fourier transformation of the kl spectrum estimate are closer to the sample autocovariances of the given observations than those of the initial spectrum estimate it is shown that a gaussian autoregressive process of order p is the closest in the time domain kullback leibler sense to independently and identically distributed normal random variables subject to the first p 1 autocovariance constraints a rationale for the existence of the kl spectrum is given

approximation of density functions by sequences of exponential families 1 introduction consider the estimation of a probability density func tion p x defined on a bounded interval we approximate the logarithm of the density by a basis function expansion consisting of polynomials splines or trigonometric series the expansion yields a regular exponential family within which we estimate the density by the method of maximum likelihood this method of density estimation arises by application of the principle of maxi mum entropy or minimum relative entropy subject to empirical constraints we show that if the logarithm of the density has r square integrable deriva tives f idr log p12 ox then the sequence of density estimators p converges to p in the sense of relative entropy kullback leibler fp log p pn at rate opr 1 m2r m n as m x c and m2 n 0 in the spline and trigonometric cases and m3 n o 0 in the polynomial case where m is the dimension of the family and n is the sample size boundary conditions are assumed for the density in the trigonometric case this convergence rate specializes to op r n 2r 2r 1 by setting m nl 2r 1 when the log density is known to have degree of smoothness at least r analogous convergence results for the relative entropy are shown to hold in general for any class of log density functions and sequence of finite dimensional linear spaces having l2 and l approximation properties the approximation of log densities using polynomials has previously been considered by neyman 1937 to define alternatives for goodness of fit tests by good 1963 as an application of the method of maximum entropy or minimum relative entropy by crain 1974 1976a b 1977 who demonstrates existence and consistency of the maximum likelihood estimator and by mead and

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

maximum entropy methods for time harmonic acoustics abstract this paper explores the application of maximum entropy methods max ent to time harmonic acoustic problems max ent basis functions are mesh free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying jaynes s maximum entropy principle they are c continuous and therefore they are particularly suited for the resolution of helmholtz problems where classical finite element methods show a poor accuracy in the high frequency region in addition it was recently shown that max ent approximants can be blended with isogeometric basis functions on the boundary of the domain this preserves the correct representation of the boundary like in isogeometric analysis with the advantage that the discretization of the interior of the domain is straightforward in this paper the max ent mathematical formulation is reviewed and then some numerical applications are studied including a 2d car cavity geometry defined by b spline curves in all cases if the same nodal discretization is used finite elements results are significantly improved

isogeometric collocation for three dimensional geometrically exact shear deformable beams abstract we extend the isogeometric collocation method to the geometrically nonlinear beams an exact kinematic formulation able to represent three dimensional displacements and rotations without any restriction in magnitude is presented without the introduction of the moving frame concept a displacement based formulation is adopted full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold incremental rotations are parametrized through eulerian rotation vectors and configuration updates are performed by means of the exponential map numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy

object based estimation of dense motion fields motion estimation belongs to key techniques in image sequence processing segmentation of the motion fields such that ideally each independently moving object uniquely corresponds to one region is one of the essential elements in object based image processing this paper is concerned with unsupervised simultaneous estimation of dense motion fields and their segmentations it is based on a stochastic model relating image intensities to motion information based on the analysis of natural images a region based model of motion compensated prediction error is proposed in each region the error is modeled by a white stationary generalized gaussian random process the motion field and its segmentation are themselves modeled by a compound gibbs markov random field accounting for statistical bindings in spatial direction and along the direction of motion trajectories the a posteriori distribution of the motion field for a given image sequence is formulated as an objective function such that its maximization results in the map estimate a deterministic multiscale relaxation technique with regular structure is employed for optimization of the objective function simulation results are in a good agreement with human perception for both the motion fields and their segmentations

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

discretizing distributions with exact moments error estimate and convergence analysis the maximum entropy principle is a powerful tool for solving underdetermined inverse problems this paper considers the problem of discretizing a continuous distribution which arises in various applied fields we obtain the approximating distribution by minimizing the kullback leibler information relative entropy of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints we study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases we prove that i the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with and ii the approximate discrete distribution weakly converges to the given continuous distribution moreover we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem

connections of generalized divergence measures with fisher information matrix abstract in this paper parametric measures of information for the multiparameter case are obtained using csiszar s nonparametric measure of information as well as continuous and differentiable functions thereof the same idea is applied to some of the unified r s divergence measures presented in taneja relationships of the new parametric measures with fisher s information matrix are derived and their connections with the cramer rao inequality is established this work generalizes the results of kagan aggarwal and ferentinos and papioannou

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

isogeometric collocation for large deformation elasticity and frictional contact problems abstract isogeometric collocation methods have been recently proposed as an alternative to standard galerkin approaches as they provide a significant reduction in computational cost for higher order discretizations in this work we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems we first derive the non linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization some numerical examples demonstrate the performance of collocation in its basic and enhanced versions differing by the enforcement of neumann boundary conditions for problems with strong singularities enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to galerkin whereas for weaker or no singularities enhanced and basic collocation may give very similar results a large deformation contact formulation is subsequently developed and tested in the frictional setting where collocation confirms the excellent performance already obtained for the frictionless case finally it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three dimensional setting with arbitrarily inclined non matching discretizations thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints

isogeometric collocation for the reissner mindlin shell problem abstract we present an isogeometric collocation formulation for the reissner mindlin shell problem after recalling the necessary basics on differential geometry and the shell governing equations we show that the standard approach of expressing the equilibrium equations in terms of the primal variables is not a suitable way for shells due to the complexity of the underlying equations we then propose an alternative approach based on a stepwise formulation and show its numerical implementation within an isogeometric collocation framework the formulation is tested successfully on a set of benchmark examples which comprise important aspects like locking and boundary layers these test show that locking effects can be conveniently avoided by using high polynomial degrees an accompanying study on the computational time also confirms that high polynomial degrees are preferable in terms of computational efficiency

a class of statistical and spectral distance measures based on bose einstein statistics a class of statistical distance measures and their spectral counterparts are presented they have strong physical foundations since they are based on the combinatorial law leading to bose einstein statistics in statistical physics it is shown that these distance measures are very closely related to the recently introduced jensen shannon divergence measure the kullback leibler information 1951 number is found to be a limit case of this class

properties of cross entropy minimization the principle of minimum cross entropy minimum directed divergence minimum discrimination information is a general method of inference about an unknown probability density when there exists a prior estimate of the density and new information in the form of constraints on expected values various fundamental properties of cross entropy minimization are proven and collected in one place cross entropy s well known properties as an information measure are extended and strengthened when one of the densities involved is the result of cross entropy minimization the interplay between properties of cross entropy minimization as an inference procedure and properties of cross entropy as an information measure is pointed out examples are included and general analytic and computational methods of finding minimum cross entropy probability densities are discussed

maximum entropy and maximum likelihood in spectral estimation the power spectral measure an informative feature of a stationary time discrete stochastic process describes the relative strength of uncorrelated frequency components that compose the process in spectral estimation one wants to describe the spectral measures of processes having a prescribed initial block of autocorrelation coefficients in the continuum of possibilities two types are frequently distinguished the one parameter family of maximum likelihood spectral measures so named because they carry the largest possible weight at one specified frequency and the maximum entropy spectral measure which is often considered to be the most representative of the entire set of possible solutions although these choices are very different representing respectively the most and least predictable of the eligible processes we show that the maximum entropy measure is exactly the uniform average of the family of maximum likelihood measures over its parameter

on minimizing distortion and relative entropy a common approach for estimating a probability mass function w when given a prior q and moment constraints given by aw spl les b is to minimize the relative entropy between w and q subject to the set of linear constraints in such cases the solution w is known to have exponential form we consider the case in which the linear constraints are noisy uncertain infeasible or otherwise soft a solution can then be obtained by minimizing both the relative entropy and violation of the constraints aw spl les b a penalty parameter spl sigma weights the relative importance of these two objectives we show that this penalty formulation also yields a solution w with exponential form if the distortion is based on an spl lscr sub p norm then the exponential form of w is shown to have exponential decay parameters that are bounded as a function of spl sigma we also state conditions under which the solution w to the penalty formulation will result in zero distortion so that the moment constraints hold exactly these properties are useful in choosing penalty parameters evaluating the impact of chosen penalty parameters and proving properties about methods that use such penalty formulations

the relation between information theory and the differential geometry approach to statistics abstract cencov has shown that the riemannian metric on the probability simplex xi 1 defined by ds 2 dx i 2 x i has an invariance property under certain probabilistically natural mappings no other riemannian metric has the same property the geometry associated with this metric is shown to lead almost automatically to measures of divergence between probability distributions which are associated with kullback bhattacharyya and matusita certain vector fields are associated in a natural way with random variables the integral curves of these vector fields yield the maximum entropy or minimum divergence estimates of probabilities some other consequences of this geometric view are also explored

assessing risks through the determination of rare event probabilities we consider the problem of evaluating the probability of occurrence of rare but potentially catastrophic events the lack of historical data renders conventional statistical approaches inappropriate the problem is compounded by complex multivariate dependencies that may exist across potential event sites in order to evaluate the likelihood of one or more such catastrophic events occurring we provide an information theoretic model for merging a decision maker s opinion with expert judgment also provided is a methodology for the reconciling of conflicting expert judgments this merging approach is invariant to the decision maker s viewpoint in the limiting case of exceptionally rare events these methods are applied to likelihood assessment of liquid natural gas tanker spills and seismic induced light water nuclear reactor meltdowns

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

second order convex maximum entropy approximants with applications to high order pde we present a new approach for second order maximum entropy max ent meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non uniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants we examine the performance of the proposed approximation scheme in the numerical solution by a direct galerkin method of a number of partial differential equations pdes including structural vibrations elliptic second order pdes and fourth order pdes for kirchhoff love thin shells and for a phase field model describing the mechanics of biomembranes the examples highlight the ability of the method to deal with non uniform node distributions and the high accuracy of the solutions surprisingly the first order meshfree max ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples even in the higher order pdes

a general conditional large deviation principle given a sequence of borel probability measures on a hausdorff space which satisfy a large deviation principle ldp we consider the corresponding sequence of measures formed by conditioning on a set b if the large deviation rate function i is good and effectively continuous and the conditioning set has the property that 1 overline b circ overline b and 2 i x infty for all x in overline b then the sequence of conditional measures satisfies a ldp with the good effectively continuous rate function i b where i b x i x inf i b if x in overline b and i b x infty otherwise

explicit bounds for entropy concentration under linear constraints consider the set of all sequences of n outcomes each taking one of m values whose frequency vectors satisfy a set of linear constraints if m is fixed while n increases most sequences that satisfy the constraints result in frequency vectors whose entropy approaches that of the maximum entropy vector satisfying the constraints this well known entropy concentration phenomenon underlies the maximum entropy method existing proofs of the concentration phenomenon are based on limits or asymptotics and unrealistically assume that constraints hold precisely supporting maximum entropy inference more in principle than in practice we present for the first time non asymptotic explicit lower bounds on n for a number of variants of the concentration result to hold to any prescribed accuracies with the constraints holding up to any specified tolerance considering the fact that allocations of discrete units can satisfy constraints only approximately again unlike earlier results we measure concentration not by deviation from the maximum entropy value but by the ell 1 and ell 2 distances from the maximum entropy achieving frequency vector one of our results holds independently of the alphabet size m and is based on a novel proof technique using the multi dimensional berry esseen theorem we illustrate and compare our results using various detailed examples

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

extended truncated hierarchical catmull clark subdivision abstract in this paper we present an extended truncated hierarchical catmull clark subdivision ethccs method which improves the efficiency of local refinement in truncated hierarchical catmull clark subdivision thccs based on the stam s method we first build a set of basis functions over arbitrary quadrilateral meshes and apply them to thccs then a new basis function insertion scheme is developed with the aid of the truncation mechanism which refines one ring neighboring elements rather than two ring neighborhoods therefore ethccs significantly improves the efficiency of local refinement compared with thccs as demonstrated by one benchmark problem and several complex models moreover ethccs is also proved to preserve the input geometry and produce nested spaces

isogeometric collocation for kirchhoff love plates and shells abstract with the emergence of isogeometric analysis iga the galerkin rotation free discretization of kirchhoff love shells is facilitated enabling more efficient thin shell structural analysis high order shape functions used in iga also allow the collocation of partial differential equations avoiding the time consuming numerical integration of the galerkin technique the goal of the present work is to apply this method to nurbs based isogeometric kirchhoff love plates and shells under the assumption of small deformations since kirchhoff love plate theory yields a fourth order formulation two boundary conditions are required at each location on the contour generating some conflicts at the corners where there are more equations than needed to remedy this overdetermination we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions i e the clamped simply supported symmetric and free supports greville and alternative superconvergent points are used for nurbs basis of even and odd degrees respectively for square circular and annular flat plates convergence orders are found to be in agreement with a priori error estimates the proposed isogeometric collocation method is then validated and benchmarked against a galerkin implementation by studying a set of problems involving kirchhoff love shells