an affine invariant bivariate version of the sign test abstract the generalized median of h oja yields a notion of bivariate quantile and in turn an affine invariant bivariate analogue of the sign test its properties include a simple null covariance formula facilitating a permutation or sign change test in the case of bivariate symmetry normal efficiency coinciding with that of the oja median and bounded influence hence strong robustness author

on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

rank based optimal tests of the adequacy of an elliptic varma model we are deriving optimal rank based tests for the adequacy of a vector autoregressive moving average varma model with elliptically contoured innovation density these tests are based on the ranks of pseudo mahalanobis distances and on narmed residuals computed from tyler s ann statist 15 1987 234 251 scatter matrix they generalize the univariate signed rank procedures proposed by hallin and puri j multivariate anal 39 1991 1 29 two types of optimality properties are considered both in the local and asymptotic sense a la le cam a fixed score procedures local asymptotic minimaxity at selected radial densities and b estimated score procedures local asymptotic minimaxity uniform over a class f of radial densities contrary to their classical counterparts based on cross covariance matrices these tests remain valid under arbitrary elliptically symmetric innovation densities including those with infinite variance and heavy tails we show that the ares of our fixed score procedures with respect to traditional gaussian methods are the same as for the tests of randomness proposed in hallin and paindaveine bernoulli 8 2002b 787 815 the multivariate serial extensions of the classical chernoff savage and hodges lehmann results obtained there thus also hold here in particular the van der waerden versions of our tests are uniformly more powerful than those based on cross covariances as for our estimated score procedures they are fully adaptive hence uniformly optimal over the class of innovation densities satisfying the required technical assomptions

optimal tests for multivariate location based on interdirections and pseudo mahalanobis ranks we propose a family of tests based on randies 1989 concept of inter directions and the ranks of pseudo mahalanobis distances computed with respect to a multivariate m estimator of scatter due to tyler 1987 for the multivariate one sample problem under elliptical symmetry these tests which generalize the univariate signed rank tests are affine invariant depending on the score function considered van der waerden laplace they allow for locally asymptotically maximin tests at selected densities multivariate normal multivariate double exponential local powers and asymptotic relative efficiencies are derived with respect to hotelling s test randies 1989 multivariate sign test peters and randies 1990 wilcoxon type test and with respect to the oja median tests we moreover extend to the multivariate setting two famous univariate results the traditional chemoff savage 1958 property showing that hotelling s traditional procedure is uniformly dominated in the pitman sense by the van der waerden version of our tests and the celebrated hodges lehmann 1956 864 result providing for any fixed space dimension k the lower bound for the asymptotic relative efficiency of wilcoxon type tests with respect to hotelling s these asymptotic results are confirmed by a monte carlo investigation and application to a real data set

a unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations piecewise linear finite element approximations to two dimensional poisson problems are treated for simplicity consideration is restricted to problems having dirichlet boundary conditions and defined on rectangular domains which are partitioned by a uniform triangular mesh it is also required that the solutions u h3 a method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of o h2 both at element edge mid points and element vertices using simple averaging schemes over adjacent elements piecewise linear interpolants respectively discontinuous and continuous are then fitted to these recovered gradients and are shown to be o h2 estimates for u in the l2 norm and thus superconvergent a discussion is given of the extension of the results to problems with more general region and mesh geometries boundary conditions and with solutions of lower regularity and also to other second order elliptic boundary value problems e g the problem of planar linear elasticity

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

a weighted multivariate sign test for cluster correlated data we consider the multivariate location problem with cluster correlated data a family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights under weak assumptions the test statistic is asymptotically distributed as a chi squared random variable as the number of clusters goes to infinity the asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality optimal weights maximizing pitman asymptotic efficiency are provided these weights depend on the cluster sizes and on the intracluster correlation several approaches for estimating these weights are presented using pitman asymptotic efficiency we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster a multivariate weighted t test is also introduced the finite sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive a real data example illustrates the practical application of the methodology

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

averaging technique for fe a posteriori error control in elasticity part i conforming fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper the reliability of any averaging estimator is shown for low order finite element methods in elasticity theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides and independent of the structure of a shape regular mesh

η superconvergence of finite element approximations in the interior of general meshes of triangles abstract in this paper we introduce a new definition of superconvergence tne superconvergence which generalizes the classical idea of superconvergence to general meshes we show that this new definition can be employed to determine the regions of least error in any element in the interior of any grid by using a computer based approach we present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat conduction for meshes of conforming triangles of degree p 1 p 5 and elements in the interior of the mesh the results demonstrate that unlike classical superconvergence superconvergence is applicable to the complex grids which are employed in practical engineering computations

optimal rank based tests for homogeneity of scatter we propose a class of locally and asymptotically optimal tests based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations contrary to the existing parametric procedures these tests remain valid without any moment assumptions and thus are perfectly robust against heavy tailed distributions validity robustness nevertheless they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all efficiency robustness in particular their normal score version outperforms traditional gaussian likelihood ratio tests and their pseudo gaussian robustifications under a very broad range of non gaussian densities including for instance all multivariate student and power exponential distributions

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

on multivariate runs tests for randomness this paper proposes several extensions of the concept of runs to the multivariate setup and studies the resulting tests of multivariate randomness against serial dependence two types of multivariate runs are defined i an elliptical extension of the spherical runs proposed by marden 1999 and ii an original concept of matrix valued runs the resulting runs tests themselves exist in various versions one of which is a function of the number of data based hyperplanes separating pairs of observations only all proposed multivariate runs tests are affine invariant and highly robust in particular they allow for heteroscedasticity and do not require any moment assumption their limiting distributions are derived under the null hypothesis and under sequences of local vector arma alternatives asymptotic relative efficiencies with respect to gaussian portmanteau tests are computed and show that while marden type runs tests suffer severe consistency problems tests based on matrix valued runs perform u

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

on the asymptotic exactness of bank weiser s estimator in this paper we analyze an error estimator introduced by bank and weiser we prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes by considering a simple example we show that this is not true for general meshes

a posteriori error analysis of finite element solutions for one dimensional problems the paper develops a theory of a posteriors error estimates under the l p energy norm for 2 leqq p leqq infty the theory is based on a general concept of error indicators and error estimators several specific examples of these quantities are introduced and analyzed in detail the results provide a variety of easily computable error estimates under all these important norms for finite element solutions of one dimensional problems

superconvergence and a posteriori error estimation in triangular mixed finite elements in this paper we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of raviart thomas type 17 on uniform triangulations are used i e that the h mbox div omega distance between the approximate solution and a suitable projection of the real solution is of higher order than the h mbox div omega error we prove results for both dirichlet and neumann boundary conditions recently duran 9 proved similar results for rectangular mixed finite elements and superconvergence along the gauss lines for rectangular mixed finite elements was considered by douglas ewing lazarov and wang in 11 8 and 18 the triangular case however needs some extra effort using the superconvergence results a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the l 2 omega error in the approximation of the vector variable

a supraconvergent scheme for the korteweg de vries equation a finite difference method for the integration of the korteweg de vries equation on irregular grids is analyzed under periodic boundary conditions the method is shown to be supraconvergent in the sense that though being inconsistent it is second order convergent however such a convergence only takes place on grids with an odd number of points per period when a grid with an even number of points is used the inconsistency of the method leads to divergence numerical results backing the analysis are presented

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

a distribution free multivariate sign test based on interdirections abstract distribution free tests are investigated for the one sample multivariate location problem counts called interdirections which measure the angular distance between two observation vectors relative to the positions of the other observations are introduced these counts are invariant under nonsingular linear transformations and have a small sample distribution free property over a broad class of population models called distributions with elliptical directions which includes all elliptically symmetric populations and many skewed populations a sign test based on interdirections is described including as special cases the two sided univariate sign test and blumen s bivariate sign test the statistic is shown to have a limiting 2 p null distribution and because it is based on interdirections it is also seen to be invariant and to have a small sample distribution free property pitman asymptotic relative efficiencies and a monte carlo study show the test to perform well compared with hotell

evolution galerkin methods and their supraconvergence evolution galerkin methods for partial differential equations of the form u t l u 0 are characterised by i the use of some form of approximation to the corresponding evolution operator e t and ii projection onto an approximation space s h to obtain u n in this paper we concentrate on characteristic galerkin and lagrange galerkin methods to derive basic error estimates for multidimensional convection problems methods covered include those using recovery techniques to improve accuracy many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given together with a number of particular examples

opinion formation games with dynamic social influences abstract we investigate opinion formation games with dynamic social influences where opinion formation and social relationships co evolve in a cross influencing manner we show that these games always admit an ordinal potential and so pure nash equilibria and we design a polynomial time algorithm for computing the set of all pure nash equilibria and the set of all social optima of a given game we also derive non tight upper and lower bounds on the price of anarchy and stability which only depend on the players stubbornness that is on the scaling factor used to counterbalance the cost that a player incurs for disagreeing with the society and the cost she incurs for breaking away from her innate beliefs

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

multivariate sign based high dimensional tests for the two sample location problem abstractthis article concerns tests for the two sample location problem when data dimension is larger than the sample size existing multivariate sign based procedures are not robust against high dimensionality producing tests with type i error rates far away from nominal levels this is mainly due to the biases from estimating location parameters we propose a novel test to overcome this issue by using the leave one out idea the proposed test statistic is scalar invariant and thus is particularly useful when different components have different scales in high dimensional data asymptotic properties of the test statistic are studied compared with other existing approaches simulation studies show that the proposed method behaves well in terms of sizes and power supplementary materials for this article are available online

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

semiparametrically efficient rank based inference for shape ii optimal r estimation of shape a class of r estimators based on the concepts of multivariate signed ranks and the optimal rank based tests developed in hallin and paindaveine 2006a is proposed for the estimation of the shape matrix of an elliptical distribution these r estimators are root n consistent under any radial density g without any moment assumptions and semiparametrically efficient at some prespecified density f when based on normal scores they are uniformly more efficient thanthe traditional normal theory estimator based on empirical covariance matrices the asymptotic normality of which moreover requires finite moments of order four irrespective of the actual underlying elliptical density they rely on an original rank based version of le cam s one step methodology which avoids the unpleasant nonparametric estimation of cross information quantities that is generally required in the context of r estimation although they are not strictly equivariant they are shown to be equivariant in a weak asymptotic sense simulations confirm their feasability and excellent finite sample performances

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

interior estimates for nonconforming finite element methods interior error estimates are derived for a wide class of nonconforming finite element methods for second order scalar elliptic boundary value problems it is shown that the error in an interior domain can be estimated by three terms the first one measures the local approximability of the finite element space to the exact solution the second one measures the degree of continuity of the finite element space the consistency error and the last one expresses the global effect through the error in an arbitrarily weak sobolev norm over a slightly larger domain as an application interior superconvergences of some difference quotients of the finite element solution are obtained for the derivatives of the exact solution when the mesh satisfies some translation invariant condition

asymptotically exact functional error estimators based on superconvergent gradient recovery the use of dual adjoint problems for approximating functionals of solutions of pdes with great accuracy or to merely drive a goal oriented adaptive refinement scheme has become well accepted and it continues to be an active area of research the traditional approach involves dual residual weighting drw in this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact the first is of drw type and is derived for meshes in which most triangles satisfy an inlinemediaobject not available see fulltext approximate parallelogram property the second functional estimator involves dual error estimate weighting dew using any superconvergent gradient recovery technique for the primal and dual solutions several experiments are done which demonstrate the asymptotic exactness of a dew estimator which uses a gradient recovery scheme proposed by bank and xu and the effectiveness of refinement done with respect to the corresponding local error indicators

decentralized dynamics for finite opinion games game theory studies situations in which strategic players can modify the state of a given system in the absence of a central authority solution concepts such as nash equilibrium have been defined in order to predict the outcome of such situations in multi player settings it has been pointed out that to be realistic a solution concept should be obtainable via processes that are decentralized and reasonably simple accordingly we look at the computation of solution concepts by means of decentralized dynamics these are algorithms in which players move in turns to decrease their own cost and the hope is that the system reaches an equilibrium quickly we study these dynamics for the class of opinion games recently introduced by bindel et al 10 these are games important in economics and sociology that model the formation of an opinion in a social network we study best response dynamics and show upper and lower bounds on the convergence to nash equilibria we also study a noisy version of best response dynamics called logit dynamics and prove a host of results about its convergence rate as the noise in the system varies to get these results we use a variety of techniques developed to bound the mixing time of markov chains including coupling spectral characterizations and bottleneck ratio

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

on polynomial approximation in sobolev spaces we obtain new bounds for the error in the polynomial approximation in sobolev spaces in an open bounded set which is star shaped with respect to every point in a set of positive measure b subset omega this estimate follows by applying the hardy littlewood maximal function

asymptotically exact error estimators for rectangular finite elements this paper deals with a posteriors error estimates for finite element approximations of elliptic problems based on superconvergence results a general approach is introduced to construct under reasonable assumptions asymptotically exact error estimators finally several examples are presented and it is shown that other well known estimators can be analyzed in this framework the estimators introduced are of a local character and therefore inexpensive to compute

a practical affine equivariant multivariate median summary a robust affine equivariant estimator of location for multivariate data is proposed which becomes the univariate median for data of dimension one the estimator is robust in the sense that it has a bounded influence function a positive breakdown value and has high efficiency compared to the sample mean for heavy tailed distributions perhaps its greatest strength is that unlike other affine equivariant multivariate medians it is easily computed for data in any practical dimension

affine invariant multivariate multisample sign tests summary invariant sign type tests for the multivariate multisample location models are developed the definition of sign is based on the oja criterion function emphasis is on the twosample location model and the associated equivariant estimate of shift asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency tests are then developed for the one way and two way lay outs

a finite difference formula for the discretization of d 3 dx 3 on nonuniform grids we analyze the use of a five point difference formula for the dis cretization of the third derivative operator on nonuniform grids the formula was derived so as to coincide with the standard five point formula on regular grids and to lead to skew symmetric schemes it is shown that under periodic boundary conditions the formula is supraconvergent in the sense that in spite of being inconsistent it gives rise to schemes with second order convergence however such a supraconvergence only takes place when the number of points in the grid is odd for grids with an even number of points the inconsistency of the formula results in lack of convergence both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments

a superconvergent linear fe approximation for the solution of an elliptic system of pdes the aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations this nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem the method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

on a global superconvergence of the gradient of linear triangular elements abstract we study a simple superconvergent scheme which recovers the gradient when solving a second order elliptic problem in the plane by the usual linear elements the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the l 2 norm than the piecewise constant gradient of the ritz galerkin solution a superconvergent approximation to the boundary flux is presented as well

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

on some new recovery based a posteriori error estimators the paper is concerned with the formulation of recovery based a posteriori error estimators at first we analyze a variant of the well known zienkiewicz zhu method which is here formulated so as to be exact in one dimension for quadratic solutions on non uniform grids next we discuss two methods which operate directly on the solution rather than its gradient one is based on a solution enrichment using the zienkiewicz zhu recovered gradient while the other consists of a roughening of the solution followed by a zienkiewicz zhu like recovery the three new proposed methods are compared in terms of their effectivity indices and solution accuracy to the zienkiewicz zhu estimator and are applied to representative two and three dimensional problems

a posteriori error estimates for some model boundary control problems in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal boundary control problems we derive a posteriori error estimates for both the state and the control approximation on polygonal domains such estimates which are apparently not available in the literature can be used to construct reliable adaptive finite element approximation schemes for the control problems

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

bivariate sign tests abstract in this article the bivariate location problem is treated new appealing bivariate analogs of the univariate sign tests are proposed for testing the null hypothesis concerning the unknown symmetry center these tests remain unaltered under any nonsingular linear transformation from these promising findings a whole family of locally most powerful invariant sign tests is introduced the tests proposed earlier blumen 1958 hodges 1955 are specific members of this family for example blumen s test appears to be optimal against bivariate normal or any other elliptic alternatives the limiting distributions are derived both under the null hypothesis and under the contiguous alternatives these limiting distributions are then used to derive asymptotic relative efficiencies it is found that blumen s test has the efficiency 785 relative to hotelling s test against bivariate normal alternatives for other locally most powerful sign tests the corresponding efficiency depends on the significance leve

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

a simpler affine invariant multivariate distribution free sign test abstract a simpler multivariate sign test is proposed that uses the transformation retransformation approach of chakraborty chaudhuri and oja together with a directional transformation due to tyler this produces a multivariate sign test that is practical to apply to data of any dimension makes minimal assumptions about the underlying distribution and has a small sample distribution free property over a broad class of population models it is shown to perform very well in comparison to hotelling s t 2 and other multivariate sign tests for heavy tailed and skewed distributions