robust a posteriori error estimation for finite element approximation to h curl problem abstract in this paper we introduce a novel a posteriori error estimator for the conforming finite element approximation to the h curl problem with inhomogeneous media and with the right hand side only in l 2 the estimator is of the recovery type independent with the current approximation to the primary variable the electric field an auxiliary variable the magnetizing field is recovered in parallel by solving a similar h curl problem an alternate way of recovery is presented as well by localizing of the error flux the estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable it is proved that the estimator is approximately equal to the true error in the energy norm without the quasi monotonicity assumption finally we present numerical results for several h curl interface problems

instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

fault identification via nonparametric belief propagation we consider the problem of identifying a pattern of faults from a set of noisy linear measurements unfortunately maximum a posteriori map probability estimation of the fault pattern is computationally intractable to solve the fault identification problem we propose a nonparametric belief propagation nbp approach we show empirically that our belief propagation solver is more accurate than recent state of the art algorithms including interior point methods and semidefinite programming our superior performance is explained by the fact that we take into account both the binary nature of the individual faults and the sparsity of the fault pattern arising from their rarity

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

block locally optimal preconditioned eigenvalue xolvers blopex in hypre and petsc we describe our software package block locally optimal preconditioned eigenvalue xolvers blopex recently publicly released blopex is available as a stand alone serial library as an external package to petsc portable extensible toolkit for scientific computation a general purpose suite of tools developed by argonne national laboratory for the scalable solution of partial differential equations and related problems and is also built into hypre high performance preconditioners a scalable linear solvers package developed by lawrence livermore national laboratory the present blopex release includes only one solver the locally optimal block preconditioned conjugate gradient lobpcg method for symmetric eigenvalue problems hypre provides users with advanced high quality parallel multigrid preconditioners for linear systems with blopex the same preconditioners can now be efficiently used for symmetric eigenvalue problems petsc facilitates the integration of independently developed application modules with strict attention to component interoperability and makes blopex extremely easy to compile and use with preconditioners that are available via petsc we present the lobpcg algorithm in blopex for hypre and petsc we demonstrate numerically the scalability of blopex by testing it on a number of distributed and shared memory parallel systems including a beowulf system sun fire 880 an amd dual core opteron workstation and ibm bluegene l supercomputer using petsc domain decomposition and hypre multigrid preconditioning we test blopex on a model problem the standard 7 point finite difference approximation of the 3 d laplacian with the problem size in the range of 10 5 10 8

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

optimal two sided invariant similar tests for instrumental variables regression this paper considers tests of the parameter on an endogenous variable in an instrumental variables regression model the focus is on determining tests that have some optimal power properties we start by considering a model with normally distributed errors and known error covariance matrix we consider tests that are similar and satisfy a natural rotational invariance condition we determine a two sided power envelope for invariant similar tests this allows us to assess and compare the power properties of tests such as the conditional likelihood ratio clr the lagrange multiplier and the anderson rubin tests we find that the clr test is quite close to being uniformly most powerful invariant among a class of two sided tests the finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly nonnormal errors via weak instrument asymptotics strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

robust a posteriori error estimation for the maxwell equations we consider a robust a posteriori error estimator of residual type for the maxwell equations in a bounded two and three dimensional domain the continuous problem is approximated using conforming approximated spaces the main goal is to express the lower and upper bounds with respect to appropriate norms for that purpose a new interpolant of clement nedelec type is introduced and some interpolation error estimates are proved numerical tests are presented which confirm our theoretical results

a conditional likelihood ratio test for structural models this paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced form covariance matrix these tests are shown to be similar under weak instrument asymptotics when the reduced form covariance matrix is estimated and the errors are non normal the conditional test based on the likelihood ratio statistic is particularly simple and has good power properties like the score test it is optimal under the usual local to null asymptotics but it has better power when identification is weak

a recovery based a posteriori error estimator for h curl interface problems abstract this paper introduces a new recovery based a posteriori error estimator for the lowest order nedelec finite element approximation to the h curl interface problem the error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results under certain assumptions it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients

recovery based error estimator for interface problems conforming linear elements this paper studies a new recovery based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems instead of recovering the gradient in the continuous finite element space the flux is recovered through a weighted l 2 projection onto h mathrm div conforming finite element spaces the resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results this paper also proposes an adaptive finite element method based on either the recovery based estimators or the edge estimator through local mesh refinement and establishes its convergence in particular it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps

pivotal statistics for testing structural parameters in instrumental variables regression we propose a novel statistic for testing the structural parameters in instrumental variables regression the statistic is straightforward to compute and has a limiting distribution that is pivotal with a degrees of freedom parameter that is equal to the number of tested parameters it therefore differs from the anderson rubin statistic whose limiting distribution is pivotal but has a degrees of freedom parameter that is equal to the number of instruments and the likelihood based wald likelihood ratio and lagrange multiplier statistics whose limiting distributions are not pivotal we analyze the relationship between the statistic and the concentrated likelihood of the structural parameters and show that its limiting distribution is not affected by weak instruments we discuss examples of the non standard shapes of the asymptotically pivotal confidence sets that can be constructed using the statistic and investigate its power properties to show its importance for practical purposes we apply the statistic to the angrist krueger 1991 data and find similar results as in staiger and stock 1997 this discussion paper has resulted in a publication in econometrica 2002 70 5 1781 1803

structural inference from reduced forms with many instruments abstract this paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates particular attention is given to models with large numbers of instruments wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments comparisons are made with methods based on unrestricted reduced forms and numerical computations showing finite sample performance of the tests are reported some new results are obtained on inequalities between noncentral chi squared distributions with different degrees of freedom that assist in analytic power comparisons

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

recovery based error estimators for interface problems mixed and nonconforming finite elements in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 we introduced and analyzed a recovery based a posteriori error estimator for conforming linear finite element approximation to interface problems it was shown theoretically that the estimator is robust with respect to the size of jumps provided that the distribution of coefficients is locally monotone numerical examples showed that this condition is unnecessary this paper extends the idea in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to mixed and nonconforming finite element methods for developing and analyzing robust estimators numerical results on test problems are also presented moreover an a priori error estimate is obtained when the underlying problem has low regularity

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method