goal oriented error control in adaptive mixed fem for signorini s problem abstract this paper outlines goal oriented finite element error control for signorini s problem the discretization is based on a mixed formulation proposed by hlavacek et al which is extended to higher order polynomials a duality argument based on a variational inequality is applied which allows for the estimates in h as well as hp adaptivity numerical results confirm the applicability of the theoretical findings

effective postprocessing for equilibration a posteriori error estimators guaranteed error control via fully discrete a posteriori error estimators is possible with typical overestimation between 1 25 and 2 in simple computer benchmarks the equilibration techniques due to braess and that due to luce wohlmuth are efficient tools with an accuracy limited by the hyper circle threshold this motivates postprocessing strategies and the analysis of successive improvements of guaranteed upper error bounds with a few pcg iterations result in reduced overestimation factors between 1 and 1 25 numerical simulations for three classes of applications illustrate the efficiency for the poisson model problem with and without jumping coefficients or a simple obstacle problem

higher order fem for the obstacle problem of the p laplacian a variational inequality approach abstract we consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p laplacian differential operator for p 1 we prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity moreover we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions all our results contain the known results for the linear case of p 2 we present numerical results on the improved convergence rates of adaptive schemes mesh size adaptivity with and without polynomial degree adaptation for the singular case of p 1 5 and for the degenerated case of p 3

residual error estimators for coulomb friction this paper is concerned with residual error estimators for finite element approximations of coulomb frictional contact problems a recent uniqueness result by renard in siam j math anal 38 2006 pp 452 467 for the continuous problem allows us to perform an a posteriori error analysis we propose study and implement numerically two residual error estimators associated with two finite element discretizations in both cases the estimators permit us to obtain upper and lower bounds of the discretization error

the asymptotic distributions of incomplete u statistics an incomplete u statistic is obtained by sampling the terms of an u statistic this paper derives the asymptotic distribution if the variance is finite depending on the number of sampled terms the resulting distribution is either the same as for the u statistic a normal distribution or something intermediate also the case of a non random sampling of the terms is treated as an example a non parametric test of the independence of two circular random variables is studied the results are generalized to generalized u statistics

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

on the limiting distribution of pair summable potential functions in many particle systems for systems with finite phase space volume the density of states can be viewed as a multiple of the probability density of the energy when the phase space variables are independent uniformly distributed random variables we show that the distribution of a random variable proportional to the sum of pairwise interactions of independent identically distributed random variables converges to a limiting distribution as the number of variables goes to infinity when the interaction satisfies certain homogeneity requirements the moments of this distribution are simple combinations of cyclic integrals of the potential function the existence of this limit gives information about the structure function of some systems in statistical mechanics having pair summable interactions even in the absence of a thermodynamic limit the result is applied to several examples including systems of two dimensional point vortices

mesh and model adaptivity for frictional contact problems the article focuses on adaptive finite element methods for frictional contact problems the approach is based on a reformulation of the mixed form of the underlying signorini problem with friction as a nonlinear variational equation using nonlinear complimentarity functions the usual dual weighted residual framework for a posteriori error estimation is applied however we have to take into account the nonsmoothness of the problem formulation error identities for measuring the discretization as well as the model error with respect to a model hierarchy of friction laws are derived and a method for the numerical evaluation of them is proposed the estimates are utilized in an adaptive framework which balances the discretization and the model error several numerical examples substantiate the accuracy of the proposed estimates and the efficiency of the adaptive method

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

symmetric statistics poisson point processes and multiple wiener integrals the asymptotic behaviour of symmetric statistics of arbitrary order is studied as an application we describe all limit distributions of square integrable u statistics we use as a tool a randomization of the sample size a sample of poisson size n lambda with en lambda lambda can be interpreted as a poisson point process with intensity lambda and randomized symmetric statistics are its functionals as lambda rightarrow infty the probability distribution of these functionals tend to the distribution of multiple wiener integrals this can be considered as a stronger form of the following well known fact properly normalized a poisson point process with intensity lambda approaches a gaussian random measure as lambda rightarrow infty

asymptotic distributions for weighted u statistics we prove limit theorems for weighted u statistics and express the limit by means of multiple stochastic integrals this is a generalisation of the paper by o neil and redner in that paper the method of moments was applied which does not work in the general case hence we had to work out a different method in particular in theorem 4 we describe the limit of a model proposed by o neil and redner in this model the weight functions cause an intricate cancellation and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

limit theorems for a triangular scheme of u statistics with applications to inter point distances the annals of probability 1986 vol 14 no 4 1347 1358 limit theorems for a triangular scheme of u statistics with applications to inter point distances by s rao jammalamadaka1 and svante j anson university of california santa barbara and uppsala university the asymptotic distribution of a triangular scheme of u statistics is studied two limit theorems applicable in different situations are given one theorem yields convergence to a normal distribution the other includes poisson limits and other limit laws applications to statistics based on small interpoint distances in a sample are given 1 introduction there has been a renewed interest in the limit theory relating to u statistics hoe ding 1948 especially focussing on degenerate kernels and nonnormal limits see for instance rubin and vitale 1980 dynkin and mandelbaum 1983 and berman and eagleson 1983 in section 2 of this paper we study a triangular scheme of u statistics and establish their asymptotic normality under suitable conditions it is worth noting that one can obtain normal limit laws even with degenerate kernels other in nitely divisible limit laws for the triangular scheme including the poisson limits are examined in the next section applications to limit distributions of statistics based on interpoint distances are discussed in section 4 these applications which were the source of motivation for the results derived here were suggested by the studies one of us made about statistics based on spacings see for instance holst and rao 1981 they were also sparked by the recent papers of bickel and breiman 1983 and onoyama et al 1983 theorems yielding asymptotic normality in this setting have also been proved by weber 1983 using a martingale approach his theorems yield the same conclusion as theorem 2 1 but under di erent nonequivalent conditions a few words about notation we write gd to denote convergence in distribution an gc to denote complete convergence cf loeve 1963 page 178 we write po to denote the poisson distribution with mean a and np p 2 to denote a p dimensional normal distribution with mean vector p and covariance matrix 2 2 asymptotic normality of a triangular scheme of u statistics let x1 x2 be a sequence of independently and identically distributed i i d random variables we will use x and y to denote two independent random received october 1984 revised april 1985 formerly j s rao ams 1980 subject classi cations primary 60f05 60e07 secondary 62e20 key words and phrases u statistics triangular scheme multigraph limit laws in nitely divisible distributions inter point distances 1347 t institute of mathematical statistics is collaborating with j stor to digitize preserve and extend access to ef 73i the annals ofprobabiiity 5t0 r www jstor org

equilibration techniques for solving contact problems with coulomb friction abstract in this paper we consider residual and equilibrated error indicators for contact problems with coulomb friction the contact problem is handled within the abstract framework of saddle point problems more precisely the non penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual lagrange multiplier space thus from the displacement we can easily compute in a local post process the lagrange multiplier which acts as a neumann condition on the possible contact zone having computed the discrete lagrange multiplier we can apply standard error estimators by replacing the unknown neumann data by its approximation as it is shown in 1 this results in an error estimator for a one sided contact problem without friction here we consider more general situations and discuss two additional contact terms which measure the non conformity of the discrete lagrange multiplier numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process

first order least squares method for the obstacle problem we define and analyse a least squares finite element method for a first order reformulation of the obstacle problem moreover we derive variational inequalities that are based on similar but non symmetric bilinear forms a priori error estimates including the case of non conforming convex sets are given and optimal convergence rates are shown for the lowest order case we provide a posteriori bounds that can be used as error indicators in an adaptive algorithm numerical studies are presented

adaptive multilevel methods for obstacle problems the authors consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies the linear problems are solved iteratively by preconditioned conjugate gradient iterations the proposed preconditioners are treated theoretically as abstract additive schwarz methods and are implemented as truncated hierarchical basis preconditioners to allow for local mesh refinement semilocal and local a posteriors error estimates are derived providing lower and upper estimates for the discretization error the theoretical results are illustrated by numerical computations

the bayesian evidence scheme for regularizing probability density estimating neural networks training probability density estimating neural networks with the expectation maximization em algorithm aims to maximize the likelihood of the training set and therefore leads to overfitting for sparse data in this article a regularization method for mixture models with generalized linear kernel centers is proposed which adopts the bayesian evidence approach and optimizes the hyperparameters of the prior by type ii maximum likelihood this includes a marginalization over the parameters which is done by laplace approximation and requires the derivation of the hessian of the log likelihood function the incorporation of this approach into the standard training scheme leads to a modified form of the em algorithm which includes a regularization term and adapts the hyperparameters on line after each em cycle the article presents applications of this scheme to classification problems the prediction of stochastic time series and latent space models

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

a posteriori error control of h p finite elements for variational inequalities of the first and second kind in this paper residual based a posteriori error estimates for variational inequalities including those of the second kind are proposed the variational formulation and its discretizations are considered in an abstract framework of general banach spaces the main idea is to express the residual in terms of some lagrange multipliers which can be obtained for instance by some post processing the residual itself is defined for an arbitrary element of the trial space the error of this element and of the solution of the variational inequality is then estimated by the dual norm of the residual plus some computable remainder terms this concept is applied to a variety of frictional contact problems such as signorini and obstacle problems as well as to the bingham fluid problem the applicability of the estimates is confirmed by several numerical experiments in particular the general framework allows for the discretization with h p adaptivity which leads to nearly exponential convergence rates in most cases

variational learning for finite dirichlet mixture models and applications in this paper we focus on the variational learning of finite dirichlet mixture models compared to other algorithms that are commonly used for mixture models such as expectation maximization our approach has several advantages first the problem of over fitting is prevented furthermore the complexity of the mixture model i e the number of components can be determined automatically and simultaneously with the parameters estimation as part of the bayesian inference procedure finally since the whole inference process is analytically tractable with closed form solutions it may scale well to large applications both synthetic and real data generated from real life challenging applications namely image databases categorization and anomaly intrusion detection are experimented to verify the effectiveness of the proposed approach

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

a posteriori error control of discontinuous galerkin methods for elliptic obstacle problems in this article we derive an a posteriori error estimator for various discontinuous galerkin dg methods that are proposed in wang han and cheng siam j numer anal 48 708 733 2010 for an elliptic obstacle problem using a key property of dg methods we perform the analysis in a general framework the error estimator we have obtained for dg methods is comparable with the estimator for the conforming galerkin cg finite element method in the analysis we construct a non linear smoothing function mapping dg finite element space to cg finite element space and use it as a key tool the error estimator consists of a discrete lagrange multiplier associated with the obstacle constraint it is shown for non over penalized dg methods that the discrete lagrange multiplier is uniformly stable on non uniform meshes finally numerical results demonstrating the performance of the error estimator are presented

hierarchical error estimates for the energy functional in obstacle problems we present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems the main result is that the error in the energy minimum is up to oscillation terms equivalent to an appropriate hierarchical estimator the proof does not invoke any saturation assumption we even show that small oscillation implies a related saturation assumption in addition we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators finally we illustrate our theoretical results by numerical computations

adaptive optimal control of the obstacle problem this article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem to circumvent the nondifferentiability inherent to this type of problem we introduce a sequence of penalized but differentiable problems we show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization discretization and iterative solution of the discrete problems the effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples

a reduced basis method for parametrized variational inequalities reduced basis methods are an efficient tool for significantly reducing the computational complexity of solving parametrized pdes originally introduced for elliptic equations they have been generalized during the last decade to various types of elliptic parabolic and hyperbolic systems in this article we extend the reduction technique to parametrized variational inequalities first we propose a reduced basis variational inequality scheme in a saddle point form and prove existence and uniqueness of the solution we state some elementary analytical properties of the scheme such as reproduction of solutions a priori stability with respect to the data and lipschitz continuity with respect to the parameters an offline online decomposition guarantees an efficient assembling of the reduced scheme which can be solved by constrained quadratic programming second we provide rigorous a posteriori error bounds with a partial offline online decomposition the reduction scheme is applied to one dimensional o

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

convergence analysis of a conforming adaptive finite element method for an obstacle problem the adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual based a posteriori error estimator each cycle of the adaptive loop consists of the steps solve estimate mark and refine the techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of galerkin orthogonality we establish r linear convergence of the part of the energy above its minimal value if there is appropriate control of the data oscillations surprisingly the adaptive mesh refinement algorithm is the same as in the unconstrained case of a linear pde in fact there is no modification near the discrete free boundary necessary for r linear convergence the arguments are presented for a model obstacle problem with an affine obstacle and homogeneous dirichlet boundary conditions the proof of the discrete local efficiency is more involved than in the unconstrained case numerical results are given to illustrate the performance of the error estimator

a posteriori error estimators for obstacle problems another look we show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations the theory would be even simpler if the lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations

efficient and reliable a posteriori error estimators for elliptic obstacle problems a posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems these estimators yield global upper and local lower bounds for the discretization error here discretization error means the sum of two contributions the distance between continuous and discrete solution in the energy norm and some quantity that is related to the distance of continuous and discrete contact set moreover the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution

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

an efficient and reliable residual type a posteriori error estimator for the signorini problem we derive a new a posteriori error estimator for the signorini problem it generalizes the standard residual type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non linearity remarkably these additional contact related terms vanish in the case of so called full contact we prove reliability and efficiency for two and three dimensional simplicial meshes moreover we address the case of non discrete gap functions numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two and three dimensional case for simplicial as well as for unstructured mixed meshes

finite elements in space and time for dynamic contact problems dynamic obstacle and signorini problems are discretized by continuous and discontinuous finite elements in space and time the resulting discrete problems are attributed to a standard implicit time stepping scheme through relaxation of impact phenomena and suitable numerical integration the method can cope with dynamic contact problems which is shown by an analysis of a model problem moreover numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems which are not fully covered by the theory copyright 2008 john wiley sons ltd

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme