markov chain monte carlo methods for high dimensional inversion in remote sensing summary we discuss the inversion of the gas profiles ozone no3 no2 aerosols and neutral density in the upper atmosphere from the spectral occultation measurements the data are produced by the global ozone monitoring of occultation of stars instrument on board the envisat satellite that was launched in march 2002 the instrument measures the attenuation of light spectra at various horizontal paths from about 100 km down to 10 20 km the new feature is that these data allow the inversion of the gas concentration height profiles a short introduction is given to the present operational data management procedure with examples of the first real data inversion several solution options for a more comprehensive statistical inversion are presented a direct inversion leads to a non linear model with hundreds of parameters to be estimated the problem is solved with an adaptive single step markov chain monte carlo algorithm another approach is to divide the problem into several non linear smaller dimensional problems to run parallel adaptive markov chain monte carlo chains for them and to solve the gas profiles in repetitive linear steps the effect of grid size is discussed and we present how the prior regularization takes the grid size into account in a way that effectively leads to a grid independent inversion

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

quasi optimal convergence rate of an adaptive discontinuous galerkin method we analyze an adaptive discontinuous finite element method adfem for symmetric second order linear elliptic operators the method is formulated on nonconforming meshes made of simplices or quadrilaterals with any polynomial degree and in any dimension geq2 we prove that the adfem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops we design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent the geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the adfem we show that the adfem and the afem on nonconforming meshes yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

markov chain monte carlo method in bayesian reconstruction of dynamical systems from noisy chaotic time series the presence of a noise component in the time series generated by a dynamical system ds results in finite accuracy of reconstruction of the evolution operator eo any model that can be constructed will generally speaking differ from the original system one of the ways to take this fact into consideration is to describe parameters of the model eo as random quantities 1 two limiting cases may be distinguished 2 the perfect model scenario when the form of the operator describing the ds evolution is reliably known and the imperfect model scenario when eo is unknown the perfect model scenario is realized for instance when the problem of hidden information transmission is solved using chaotic time series ts generated by the known ds 3 in this case we are interested in the characteristics of the system perfect model parameters the most probable value mean dispersion distribution in investigations of many natural atmospheric oceanic tectonic biological systems the eo is unknown which corresponds to the imperfect model scenario then information about model parameters is of no value as the physical meaning of the parameters is unknown and we are interested in the properties of the model defined by these parameters and reflecting the properties of the reconstructed ds nevertheless in this case too evaluation of statistical characteristics of model parameters is the key element without which the problem of reconstructing a ds generating the original ts cannot be solved the mathematical body used to reconstruct the ds is determined by a specific application for example for finding the most probable set of model perfect or imperfect parameters it suffices to determine the maximum of posterior probability density of model parameters 4 6 however it should be remembered that in the presence of noise the inverse problem of reconstruction becomes ill posed i e it admits an infinite set of solutions selection of the most correct solutions demands regularization that is a physically justified constraint on admissible values of parameters based solely on a priori information 7 this approach is usually called bayesian within the bayesian approach there are methods which make it possible to estimate dispersions and mathematical expectations of the model parameters 8 a significant advantage of such methods is their relatively low computational resource requirements however due to the model nonlinearity the form of parameter distribution may differ strongly from the normal one in this case estimation of expectation and dispersion is insufficient to construct a correct model and one must use the bayesian approach in full in other words we must construct models which include distributions of parameters and take a priori information about the system into account correctly in this formulation the problem can be solved using the markov chain monte carlo mcmc algorithms 9 however in the case of reconstruction of dss by noisy chaotic time series computational resources required for these algorithms strongly depend on distribution dimensionality this limits applicability of the algorithms to a narrow class of problems e g the case of short ts 10 the method proposed in this paper makes it possible to expand the applicability domain of the mcmc algorithms to the case of arbitrarily long ts

new a posteriori error estimate and quasi optimal convergence of the adaptive nonconforming wilson element in this paper we establish the quasi optimal convergence of the adaptive nonconforming wilson element on the rectangular mesh the main ingredients are a new a posteriori error estimator and a crucial observation that there is some special orthogonality between the conforming part and the nonconforming part in the energy inner product which helps us to show the quasi orthogonality and the discrete reliability finally we integrate these components in a usual way to achieve the quasi optimal convergence

quasi optimal convergence rate for an adaptive boundary element method for the simple layer potential v associated with the three dimensional 3d laplacian we consider the weakly singular integral equation v phi f this equation is discretized by the lowest order galerkin boundary element method we prove convergence of an h adaptive algorithm that is driven by a weighted residual error estimator moreover we identify the approximation class for which the adaptive algorithm converges quasi optimally with respect to the number of elements in particular we prove that adaptive mesh refinement is superior to uniform mesh refinement

feedback and adaptive finite element solution of one dimensional boundary value problems this paper examines the concepts of feedback and adaptivity for the finite element method the model problem concernsc 0 elements of arbitrary fixed degree for a one dimensional two point boundary value problem three different feedback methods are introduced and a detailed analysis of their adaptivity is given

limiting performance of block fading channels with multiple antennas we derive the performance limits of a radio system consisting of a transmitter with t antennas and a receiver with r antennas a block fading channel with additive white gaussian noise awgn delay and transmit power constraints and perfect channel state information available at both the transmitter and the receiver because of a delay constraint the transmission of a codeword is assumed to span a finite and typically small number m of independent channel realizations therefore the relevant performance limits are the information outage probability and the delay limited or nonergodic capacity we derive the coding scheme that minimizes the information outage probability this scheme can be interpreted as the concatenation of an optimal code for the awgn channel without fading to an optimal beamformer for this optimal scheme we evaluate minimum outage probability and delay limited capacity among other results we prove that for the fairly general class of regular fading channels the asymptotic delay limited capacity slope expressed in bits per second per hertz b s hz per decibel of transmit signal to noise ratio snr is proportional to min t r and independent of the number of fading blocks m since m is a measure of the time diversity induced by interleaving or of the frequency diversity of the system this result shows that if channel state information is available also to the transmitter very high rates with asymptotically small error probabilities are achievable without the need of deep interleaving or high frequency diversity moreover for a large number of antennas delay limited capacity approaches ergodic capacity

outage theorems for mimo block fading channels the connection between the average codeword or frame error probability fep of space time codes and the outage probability over general block fading multiple input multiple output mimo channels is established three archetypal problems are considered under general fading distributions in a single framework wherein the receiver has channel state information whereas the transmitter knows a the fading distribution but not the channel realization b the channel realization but must follow a short term per codeword average power constraint and c the channel realization but is constrained only by a long term average power constraint three telescoping sets of space time codes are defined for a given rate and it is shown that average feps arbitrarily close to the respective outage probabilities for each of the three cases a c can be achieved by codes in each set for sufficiently large frame lengths for the smallest set among the three which contains codes with a spectral norm constraint that is stricter than the average or maximum energy constraints commonly assumed firm sphere packing lower bounds on the fep are obtained and consequently strong converse theorems are proved which assert that the respective outage probabilities also represent the best achievable fep in the large frame length limit moreover the set of spectral norm constrained codes are also shown to be large enough to contain universal codes that can communicate reliably over any channel realization for which the mutual information exceeds the information rate of the code

quasi optimal adaptive pseudostress approximation of the stokes equations the pseudostress velocity formulation of the stationary stokes problem allows some quasi optimal raviart thomas mixed finite element formulation for any polynomial degree the adaptive algorithm employs standard residual based explicit a posteriori error estimation from carstensen kim and park siam j numer anal 49 2011 pp 2501 2523 for the lowest order raviart thomas finite element functions in a simply connected lipschitz domain this paper proves optimal convergence rates in terms of the number of unknowns of the adaptive mesh refining algorithm based on the concept of approximation classes the proofs use some novel equivalence to first order nonconforming crouzeix raviart discretization plus a particular helmholtz decomposition of deviatoric tensors

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

an adaptive uzawa fem for the stokes problem convergence without the inf sup condition we introduce and study an adaptive finite element method fem for the stokes system based on an uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity we show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity whereas for pressure the elements can be either discontinuous of degree k 1 or continuous of degree k 1 and k the popular taylor hood family is the sole example of stable elements included in the theory which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf sup condition we discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi optimal

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

a unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

linear convergence of an adaptive finite element method for the p laplacian equation we study an adaptive finite element method for the p laplacian like pdes using piecewise linear continuous functions the error is measured by means of the quasi norm of barrett and liu we provide residual based error estimators without a gap between the upper and lower bound we show linear convergence of the algorithm which is similar to the one of morin nochetto and siebert all results are obtained without extra marking for the oscillation

convergence and optimal complexity of adaptive finite element eigenvalue computations in this paper an adaptive finite element method for elliptic eigenvalue problems is studied both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved the analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

convergence and quasi optimality of adaptive fem with inhomogeneous dirichlet data we consider the solution of a second order elliptic pde with inhomogeneous dirichlet data by means of adaptive lowest order fem as is usually done in practice the given dirichlet data are discretized by nodal interpolation as model example serves the poisson equation with mixed dirichlet neumann boundary conditions for error estimation we use an edge based residual error estimator which replaces the volume residual contributions by edge oscillations for 2d we prove convergence of the adaptive algorithm even with optimal convergence rate for 2d and 3d we show convergence if the nodal interpolation operator is replaced by the l 2 projection or the scott zhang quasi interpolation operator as a byproduct of the proof we show that the scott zhang operator converges pointwise to a limiting operator as the mesh is locally refined this property might be of independent interest besides the current application finally numerical experiments conclude the work

on performance limits of space time codes a sphere packing bound approach a sphere packing bound spb on average word error probability wep is derived to determine the performance limits of space time codes on rayleigh block fading channels under delay and maximum energy constraints two other explicit bounds a looser bound and a tight approximate bound are also derived to provide more intuition on how the system parameters affect the performance limits moreover it is shown that as the block length grows to infinity the spb converges to the outage probability and the asymptotic behavior of performance limits is determined by the outage probability

residual based a posteriori error estimate for hypersingular equation on surfaces the hypersingular integral equation of the first kind equivalently describes screen and neumann problems on an open surface piece the paper establishes a computable upper error bound for its galerkin approximation and so motivates adaptive mesh refining algorithms numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh refining the numerical realisation requires the evaluation of the hypersingular integral operator at a source point this and other details on the algorithm are included

a posteriori error estimate and h adaptive algorithm on surfaces for symm s integral equation a residual based a posteriori error estimate for boundary integral equations on surfaces is derived in this paper a localisation argument involves a lipschitz partition of unity such as nodal basis functions known from finite element methods the abstract estimate does not use any property of the discrete solution but simplifies for the galerkin discretisation of symm s integral equation if piecewise constants belong to the test space the estimate suggests an isotropic adaptive algorithm for automatic mesh refinement an alternative motivation from a two level error estimate is possible but then requires a saturation assumption the efficiency of an anisotropic version is discussed and supported by numerical experiments

better nonlinear models from noisy data attractors with maximum likelihood a new approach to nonlinear modelling is presented which by incorporating the global behaviour of the model lifts shortcomings of both least squares and total least squares parameter estimates although ubiquitous in practice a least squares approach is fundamentally flawed in that it assumes independent normally distributed ind forecast errors nonlinear models will not yield ind errors even if the noise is ind a new cost function is obtained via the maximum likelihood principle superior results are illustrated both for small data sets and infinitely long data streams

an optimal adaptive mixed finite element method various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods in order to save degrees of freedom not all but just some selected set of finite element domains are refined hence the fundamental question of convergence as well as the question of optimality require new mathematical arguments the presented adaptive algorithm for raviart thomas mixed finite element methods solves the poisson model problem with optimal convergence rate chen holst and xu presented convergence and optimality of adaptive mixed finite element methods 2008 following arguments of rob stevenson for the conforming finite element method their algorithm reduces oscillations separately before approximating the solution by some adaptive algorithm in the spirit of w dorfler 1996 the algorithm proposed here appears more natural in switching to either reduction of the edge error estimator or of the oscillations

quasi optimality of adaptive nonconforming finite element methods for the stokes equations we prove convergence and quasi optimal complexity of adaptive nonconforming low order finite element methods for the stokes equations covering the crouzeix raviart discretization on triangular and tetrahedral meshes as well as the rannacher turek discretization on two and three dimensional rectangular meshes hanging nodes are allowed in order to ease local mesh refinement the adaptive algorithm is based on standard a posteriori error estimators consisting of two parts a volume residual and an edge term measuring the nonconformity of the velocity approximation we use an adaptive marking strategies which in each step of the iteration takes only the dominant term into account this paper can be regarded as an extension of r becker s mao and z c shi siam j numer anal 47 2010 pp 4639 4659 to the stokes problem but the analysis here does not make use of any relationship between mixed and nonconforming finite element methods

statistical methods of parameter estimation for deterministically chaotic time series we discuss the possibility of applying some standard statistical methods the least square method the maximum likelihood method and the method of statistical moments for estimation of parameters to deterministically chaotic low dimensional dynamic system the logistic map containing an observational noise a segmentation fitting maximum likelihood ml method is suggested to estimate the structural parameter of the logistic map along with the initial value x 1 considered as an additional unknown parameter the segmentation fitting method called piece wise ml is similar in spirit but simpler and has smaller bias than the multiple shooting previously proposed comparisons with different previously proposed techniques on simulated numerical examples give favorable results at least for the investigated combinations of sample size n and noise level besides unlike some suggested techniques our method does not require the a priori knowledge of the noise variance we also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed bayesian approaches we note the trade off between the need of using a large number of data points in the ml analysis to decrease the bias to guarantee consistency of the estimation and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition the method of statistical moments for the estimation of the parameter of the logistic map is discussed this method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically not only numerically

performance of space time codes gallager bounds and weight enumeration since the standard union bound for space time codes may diverge in quasi static fading channels the limit before average lba technique has been exploited to derive tight performance bounds however it suffers from the computational burden arising from a multidimensional integral in this paper efficient bounding techniques for space time codes are developed in the framework of gallager bounds two closed form upper bounds the ellipsoidal bound and the spherical bound are proposed that come close to simulation results within a few tenths of a decibel in addition two novel methods of weight enumeration operating on a further reduced state diagram are presented which in conjunction with the bounding techniques give a thorough treatment of performance bounds for space time codes

space time coding techniques with bit interleaved coded modulations for mimo block fading channels the space time bit interleaved coded modulation st bicm is an efficient technique to obtain high diversity and coding gain on a block fading multiple input multiple output mimo channel its maximum likelihood ml performance is computed under ideal interleaving conditions which enables a global optimization taking into account channel coding thanks to a diversity upper bound derived from the singleton bound an appropriate choice of the time dimension of the space time coding is possible which maximizes diversity while minimizing complexity based on the analysis an optimized interleaver and a set of linear precoders called dispersive nucleo algebraic dna precoders are proposed the proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension with turbo codes they exhibit a frame error rate which does not increase with frame length

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

quantifying uncertainty in geoacoustic inversion i a fast gibbs sampler approach this paper develops a new approach to estimating seabed geoacoustic properties and their uncertainties based on a bayesian formulation of matched field inversion in bayesian inversion the solution is characterized by its posterior probability density ppd which combines prior information about the model with information from an observed data set to interpret the multi dimensional ppd requires calculation of its moments such as the mean covariance and marginal distributions which provide parameter estimates and uncertainties computation of these moments involves estimating multi dimensional integrals of the ppd which is typically carried out using a sampling procedure important goals for an effective bayesian algorithm are to obtain efficient unbiased sampling of these moments and to verify convergence of the sample this is accomplished here using a gibbs sampler gs approach based on the metropolis algorithm which also forms the basis for simulated annealing sa although gs can be computationally slow in its basic form just as modifications to sa have produced much faster optimization algorithms the gs is modified here to produce an efficient algorithm referred to as the fast gibbs sampler fgs an automated convergence criterion is employed based on monitoring the difference between two independent fgs samples collected in parallel comparison of fgs gs and monte carlo integration for noisy synthetic benchmark test cases indicates that fgs provides rigorous estimates of ppd moments while requiring orders of magnitude less computation time

efficiency of a posteriori bem error estimates for first kind integral equations on quasi uniform meshes in the numerical treatment of integral equations of the first kind using boundary element methods bem the author and e p stephan have derived a posteriori error estimates as tools for both reliable computation and self adaptive mesh refinement so far efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only this work affirms efficiency by proving the reverse inequality based on best approximation on inverse inequalities and on stability of the discretization and complementary to our previous work an abstract approach yields a converse estimate this estimate proves efficiency of an a posteriori error estimate in the bem on quasi uniform meshes for symm s integral equation for a hypersingular equation and for a transmission problem

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

a posteriori error estimates for boundary element methods this paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples namely symm s integral equation an integral equation with a hypersingular operator and a boundary integral equation for a transmission problem based on these estimates an analog of eriksson and johnson s adaptive finite element method is proposed for the h version of the galerkin boundary element method for integral equations of the first kind the efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities the construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations and we refer to the pioneering work of babuska and miller 3 and eriksson and johnson 10 11 whereas the main features ofadaptivity for finite element methods now seem to be visible and the door is open to implementation 15 comparably little is known for boundary element methods for integral equations see e g 1 13 18 19 24 in this paper a new adaptive version of the galerkin discretization for the boundary element method is presented based on a posteriori error estimates a general framework for these a posteriori error estimates is derived in 2 and three examples are discussed in 3 5 involving the dirichlet problem the neumann problem for a closed and an open surface and a transmission problem for the laplacian leading to integral equations with strongly elliptic pseudodifferential operators even for smooth data the lack of regularity of the solution near corners of a polygonal domain q leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used in practical problems such information is missing e g when we have singular or nearly singular data and the main problem is how to balance a graded mesh refinement towards singularities and a global

decay rates of adaptive finite elements with dorfler marking we investigate the decay rate for an adaptive finite element discretization of a second order linear symmetric elliptic pde we allow for any kind of estimator that is locally equivalent to the standard residual estimator this includes in particular hierarchical estimators estimators based on the solution of local problems estimators based on local averaging equilibrated residual estimators the zz estimator etc the adaptive method selects elements for refinement with dorfler marking and performs a minimal refinement in that no interior node property is needed based on the local equivalence to the residual estimator we prove an error reduction property in combination with minimal dorfler marking this yields an optimal decay rate in terms of degrees of freedom

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

sphere packing bounds in the grassmann and stiefel manifolds applying the riemann geometric machinery of volume estimates in terms of curvature bounds for the minimal distance of packings codes in the grassmann and stiefel manifolds will be derived and analyzed in the context of space time block codes this leads to a monotonically increasing minimal distance lower bound as a function of the block length this advocates large block lengths for the code design

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

adaptive boundary element methods for some first kind integral equations in this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the dirichlet problem and the neumann problem for the laplacian in a two dimensional lipschitz domain for the h version of the finite element galerkin discretization of the single layer potential and the hypersingular operator we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm up to a multiplicative constant following eriksson and johnson this yields adaptive algorithms steering the mesh refinement numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate

a convergent nonconforming adaptive finite element method with quasi optimal complexity in this paper we prove convergence and quasi optimal complexity of a simple adaptive nonconforming finite element method in each step of the algorithm the iterative solution of the discrete system is controlled by an adaptive stopping criterion and the local refinement is based on either a simple edge residual or a volume term depending on an adaptive marking strategy we prove that this marking strategy guarantees a strict reduction of the error augmented by the volume term and an additional oscillation term and quasi optimal complexity of the generated sequence of meshes

an oscillation free adaptive fem for symmetric eigenvalue problems a refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first order adaptive finite element method afem is presented the h 1 stability of the l 2 projection provides reliability and efficiency of the edge contribution of standard residual based error estimators for p 1 finite element methods in fact the volume contributions and even oscillations can be omitted for courant finite element methods this allows for a refined averaging scheme and so improves mao et al in adv comput math 25 1 3 135 160 2006 the proposed afem monitors the edge contributions in a bulk criterion and so enables a contraction property up to higher order terms and global convergence numerical experiments exploit the remaining l 2 error contributions and confirm our theoretical findings the averaging schemes show a high accuracy and the afem leads to optimal empirical convergence rates

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored

inhomogeneous dirichlet conditions in a priori and a posteriori finite element error analysis the numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given dirichlet data ud by functions ud h in the trace space of a finite element space on d in this paper quantitative a priori and a posteriori estimates are presented for two choices of ud h namely the nodal interpolation and the orthogonal projection in l2 d onto the trace space two corresponding extension operators allow for an estimate of the boundary data approximation in global h1 and l2 a priori and a posteriori error estimates the results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation