formulation of elastostatic hybrid trefftz stress elements abstract the hybrid trefftz stress hts element is derived from the fundamental conditions of linear elastostatics the stress and boundary displacement fields are directly approximated consistent finite element variables are so defined as to obtain boundary integral expressions for the finite element equilibrium and compatibility conditions encoded by dual transformations independent of the constitutive relations reciprocity is preserved in the finite element elasticity relations which are also found to be described by boundary integral arrays when the trefftz constraint is enforced consequent upon duality and reciprocity the resulting boundary integral solving system is symmetric and sparse the energy statements associated with the hybrid trefftz stress element are established and the conditions for the existence uniqueness multiplicity and static and kinematic admissibility of the finite element solutions are presented the hts clement is compared with the displacement frame based hybrid trefftz ht d element of jirousek it is shown that the ht d element is in fact an hybrid stress element the numerical tests performed to assess the relative performances of the hts and ht d elements show that better accuracy levels are obtained using discontinuous boundary approximation bases

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

predictor selection for positive autoregressive processes let observations y1 yn be generated from a first order autoregressive ar model with positive errors in both the stationary and unit root cases we derive moment bounds and limiting distributions of an extreme value estimator of the ar coefficient these results enable us to provide asymptotic expressions for the mean squared error mse of and the mean squared prediction error mspe of the corresponding predictor of yn 1 based on these expressions we compare the relative performance of and the least squares predictor estimator from the mspe mse point of view our comparison reveals that the better predictor estimator is determined not only by whether a unit root exists but also by the behavior of the underlying error distribution near the origin and hence is difficult to identify in practice to circumvent this difficulty we suggest choosing the predictor estimator with the smaller accumulated prediction error and show that the predictor estimator chosen in this way is as

construction of hermite subdivision schemes reproducing polynomials abstract the aim of this study is to present a new class of quasi interpolatory hermite subdivision schemes of order two with tension parameters this class extends and unifies some of well known hermite subdivision schemes including the interpolatory hermite schemes acting on a function and the associated first derivative values each scheme in this class reproduces polynomials up to a certain degree depending on the size of stencil this is desirable property since the reproduction of polynomials up to degree d leads to the approximation order d 1 the smoothness analysis has been performed by using the factorization framework of subdivision operators lastly we present some numerical examples to demonstrate the performance of the proposed hermite schemes

c 1 interpolatory subdivision with shape constraints for curves we derive two reformulations of the c1 hermite subdivision scheme introduced by merrien one where we separate computation of values and derivatives and one based of refinement of a control polygon we show that the latter leads to a subdivision matrix which is totally positive based on this we give algorithms for constructing subdivision curves that preserve positivity monotonicity and convexity

structure selection for bounded parameter models consistency conditions and selection criterion strong consistency conditions for structure selection in bounded parameter models are studied a certain robust selection criterion based on the volume of the exact parameter bounding polytope is proposed for linear regression models the effectiveness of the polytope volume criterion is demonstrated on a model nonlinear in its variables but linear in its parameters its strong consistency is proved for a large class of noise distributions the usual assumptions on the noise namely independence constant variance or martingale difference properties need not be made but asymptotic independence is assumed

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

density estimation by stochastic complexity the results by p hall and e j hannan 1988 on optimization of histogram density estimators with equal bin widths by minimization of the stochastic complexity are extended and sharpened in two separate ways as the first contribution two generalized histogram estimators are constructed the first has unequal bin widths which together with the number of the bins are determined by minimization of the stochastic complexity using dynamic programming the other estimator consists of a mixture of equal bin width estimators each of which is defined by the associated stochastic complexity as the main contribution in the present work two theorems are proved which together extend the universal coding theorems to a large class of data generating densities the first gives an asymptotic upper bound for the code redundancy in the order of magnitude achieved with a special predictive type of histogram estimator which sharpens a related bound the second theorem states that this bound cannot be improved upon by any code whatsoever

relations between information criteria for model structure selection part 2 modelling by shortest data description the structure selection of stochastic dynamic models by the predictive least squares pls criterion introduced by rissanen and wertz 1985 is investigated in this paper in view of its relations to the bayesian estimator and to information criteria with a penalty term it is shown that in the autoregressive case the strong consistency of the pls criterion can be derived from the strong consistency of the bayesian order estimator

postbuckling analysis of thin plates by a hybrid trefftz finite element method abstract a hybrid trefftz finite element ht fe model for postbuckling analysis of thin plates is presented based on a modified variational principle with the aid of incremental form of equilibrium equations all non linear terms may be viewed as pseudo loads and then the in plane and out of plane equations can be uncoupled as a consequence the solution procedure for the postbuckling plate becomes very simple two numerical examples are considered to illustrate the correctness and accuracy of the proposed formulation

shape constraints and optimal bases for c 1 hermite interpolatory subdivision schemes we address the problem of constructing c 1 subdivision interpolating curves preserving salient shape properties of the data we use the hermite subdivision scheme introduced by merrien in numer algorithms 2 1992 pp 187 200 which depends on two parameters to be chosen in an appropriate region of convergence with shape preservation in view we try and find normalized totally positive bases in the associated four dimensional limit spaces we actually exhibit two subregions when choosing the pair of parameters in the first one no totally positive basis exists in the limit space on the opposite when choosing it in the second one the limit space does possess normalized totally positive bases and we explicitly describe the optimal one as special cases we recover the bases presented in adv comput math 20 2004 pp 229 246 siam j numer anal 44 2006 pp 1095 1121 j comput appl math 220 2008 pp 686 711 of which we thus obtain the optimality

will the pls criterion for order estimation work with aml and a posteriori prediction error the general formula of the pls predictive least squares criterion for order estimation is worked out under the assumption that the parameter estimates are calculated via the aml approximate maximum likelihood a particular case is then carefully analysed and it is shown that depending on the system generating the data the pls critetion using the a posteriori prediction error can surprisingly almost surely overestimate the true order

error exponents for ar order testing this paper is concerned with error exponents in testing problems raised by autoregressive ar modeling the tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving average arma modeling estimation in several related problems such as markov order or hidden markov model order estimation optimal error exponents have been determined thanks to large deviations theory ar order testing is specially challenging since the natural tests rely on quadratic forms of gaussian processes in sharp contrast with empirical measures of markov chains the large deviation principles ldps satisfied by gaussian quadratic forms do not always admit an information theoretic representation despite this impediment we prove the existence of nontrivial error exponents for gaussian ar order testing and furthermore we exhibit situations where the exponents are optimal these results are obtained by showing that the log likelihood process indexed by ar models of a given order satisfy an ldp upper bound with a weakened information theoretic representation

hybrid finite element formulations for elastodynamic analysis in the frequency domain abstract three alternative sets of hybrid formulations to solve linear elastodynamic problems by the finite element method are presented they are termed hybrid mixed hybrid and hybrid trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally two models namely the displacement and the stress models are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement interelement continuity a fourier time discretization is used to uncouple the solving system in the frequency domain the basic space discretization criterion is implemented directly on the fundamental relations of elastodynamics and used to derive the stress and displacement models of the hybrid mixed formulation the hybrid and hybrid trefftz formulations are presented in sequence as the variants of the hybrid mixed formulation obtained by progressively increasing the constraints on the approximation bases numerical implementation aspects are briefly discussed and the performance of the finite element models is illustrated with numerical applications

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

strongly consistent estimation of the order of stochastic control systems carma model abstract in this paper a variant of the plc predictive least squares criterion for order estimation is combined with an adaptive control strategy and applied to multidimensional carma systems it is shown that with this combination we can estimate recursively and in a strongly consistent way both the order and coefficients of the controlled system while achieving asymptotically optimal cost

ar infinity estimation and nonparametric stochastic complexity let h be the transfer function of a linear stochastic system such that h and its inverse are in h sup infinity d writing the system as an ar infinity system the best ar k approximation of the system is estimated using the method of least squares a useful representation theorem for the parameter estimation error is presented the effect of undermodeling and parameter uncertainty due to estimation on honest prediction and the optimal choice of k are questioned this question is answered and the result is applied to the ar approximation of arma systems the excess of the mean of the nonparametric stochastic complexity with respect to the ar class of an arma system with zeros less than 1 beta in moduli is found asymptotically to be less than sigma sup 2 log sup 2 n log beta after n samples

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

accumulated prediction errors information criteria and optimal forecasting for autoregressive time series the predictive capability of a modification of rissanen s accumulated prediction error ape criterion ape n is investigated in infinite order autoregressive ar oo models instead of accumulating squares of sequential prediction errors from the beginning ape n is obtained by summing these squared errors from stage n n where n is the sample size and 1 n n 1 1 n may depend on n under certain regularity conditions an asymptotic expression is derived for the mean squared prediction error mspe of an ar predictor with order determined by ape n this expression shows that the prediction performance of ape n can vary dramatically depending on the choice of n another interesting finding is that when n approaches 1 at a certain rate ape n can achieve asymptotic efficiency in most practical situations an asymptotic equivalence between ape n and an information criterion with a suitable penalty term is also established from the mspe point of view this offers new perspectives for understanding the information and prediction based model selection criteria finally we provide the first asymptotic efficiency result for the case when the underlying ar oo model is allowed to degenerate to a finite autoregression

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

a rate of convergence result for a universal d semifaithful code the problem of optimal rate universal coding is considered in the context of rate distortion theory a d semifaithful universal coding scheme for discrete memoryless sources is given the main result is a refined covering lemma based on the random coding argument and the method of types the average codelength of the code is shown to approach its lower bound the rate distortion function at a rate o n sup 1 log n and this is conjectured to be optimal based on a result of a j pilc 1968 issues of constructiveness and universality are addressed

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

ellipse preserving hermite interpolation and subdivision we introduce a family of piecewise exponential functions that have the hermite interpolation property our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses we prove that the proposed hermite functions form a riesz basis and that they reproduce prescribed exponential polynomials we present a method based on green s functions to unravel their multi resolution and approximation theoretic properties finally we derive the corresponding vector and scalar subdivision schemes which lend themselves to a fast implementation the proposed vector scheme is interpolatory and level dependent but its asymptotic behavioris the same as the classical cubic hermite spline algorithm the same convergence properties i e fourth order of approximation are hence ensured

structure identification of parameter bounding models by use of noise structure bounds the selection of a model for a given set of records is discussed for the case when the output error is required to be always within some specified bounds a new selection criterion is proposed the model should be the simplest giving sufficiently unstructured output errors the lack of structure is specified by requiring the candidate model to be capable of giving sufficiently small sample autocorrelations of the model output errors over a range of lags computational algorithms to apply the criterion are presented and illustrated by tests on ar processes

the redundancy of source coding with a fidelity criterion 1 known statistics the problem of redundancy of source coding with respect to a fidelity criterion is considered for any fixed rate r 0 and any memoryless source with finite source and reproduction alphabets and a common distribution p the nth order distortion redundancy d sub n r of fixed rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate r and the distortion rate function d p r of the source p it is demonstrated that for sufficiently large n d sub n r is equal to spl part spl part r d p r ln n 2n o ln n n where spl part spl part r d p r is the partial derivative of d p r evaluated at r and assumed to exist for any fixed distortion level d 0 and any memoryless source p the nth order rate redundancy r sub n d of coding at fixed distortion level d or by using d semifaithful codes is defined as the minimum of the difference between the expected rate per symbol of any d semifaithful code of length n and the rate distortion function r p d of p evaluated at d it is proved that for sufficiently large n r sub n d is upper bounded by ln n n o ln n n and lower bounded by in n 2n o in n n as a by product the lower bound of r sub n d derived in this paper gives a positive answer to a conjecture proposed by yu and speed 1993

selecting optimal multistep predictors for autoregressive processes of unknown order we consider the problem of choosing the optimal in the sense of mean squared prediction error multistep predictor for an autoregressive ar process of finite but unknown order if a working ar model which is possibly misspecified is adopted for multistep predictions then two competing types of multistep predictors i e plug in and direct predictors can be obtained from this model we provide some interesting examples to show that when both plug in and direct predictors are considered the optimal multistep prediction results cannot be guaranteed by correctly identifying the underlying model s order this finding challenges the traditional model order selection criteria which usually aim to choose the order of the true model a new prediction selection criterion which attempts to seek the best combination of the prediction order and the prediction method is proposed to rectify this difficulty when the underlying model is stationary the validity of the proposed criterion is justified theoretically to obtain this result asymptotic properties of accumulated squares of multistep prediction errors are investigated in addition to overcoming the above difficulty some other advantages of the proposed criterion are also mentioned

a new particular solution strategy for hyperbolic boundary value problems using hybrid trefftz displacement elements summary a new indirect approach to the problem of approximating the particular solution of non homogeneous hyperbolic boundary value problems is presented unlike the dual reciprocity method which constructs approximate particular solutions using radial basis functions polynomials or trigonometric functions the method reported here uses the homogeneous solutions of the problem obtained by discarding all time derivative terms from the governing equation nevertheless what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the trefftz compliant homogeneous solution basis the particular solution trial basis is capable of significantly improving the trefftz solution even when the original equation is genuinely homogeneous an advantage that is lost if the basis is used exclusively for the recovery of the source terms similarly a sufficiently refined trefftz compliant basis is able to compensate for possible weaknesses of the particular solution approximation the method is implemented using the displacement model of the hybrid trefftz finite element method the functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the hermitian sparse and localized structure of the solving system that typifies hybrid trefftz formulations even when domain integrals are present they are generally easy to handle because the integrand presents no singularity copyright 2015 john wiley sons ltd

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

a new order estimation technique for time series modeling a new approach to estimating the order of the autoregressive moving average model is proposed which is based on the approximate stochastic realization introduced in davis and fotopoulos 1991 the present approach is attractive because overparameterization a very common problem in order determination is avoided successfully simulation results are included to illustrate the effectiveness of the proposed order estimation approach