goal oriented adaptivity in control constrained optimal control of partial differential equations dual weighted goal oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived it is demonstrated that the constraints give rise to a primal dual weighted error term representing the mismatch in the complementarity system due to discretization the paper also contains a posteriori error estimators for the l 2 norm of the error in the state and in the adjoint state

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

diagnostics for use with regression recursive residuals conventional regression case diagnostics use the fit of the regression model to all cases to assess the deviation of individual cases from the full sample unfortunately as few as two bad cases can make these diagnostics completely unreliable an alternative approach is that based on recursive fitting in which the sample is stripped away case by case and the regression refitted after each deletion until there are only as many cases as parameters recursive residuals are residuals obtained from these successive recursive fits they are particularly effective for diagnosis when the assumptions of the regression model do not hold for the full data set but do hold after some bad cases are stripped away conventional regression case diagnostics provide not only residuals but also measures of leverage and influence and in this article i propose analogous single case diagnostics for use with recursive fitting furthermore since recursive fitting focuses on the compatibility of various subregressions with

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

convergence and quasi optimality of an adaptive finite element method for controlling l 2 errors in this paper a contraction property is proved for an adaptive finite element method for controlling the global l 2 error on convex polyhedral domains furthermore it is shown that the method converges in l 2 with the best possible rate the method that is analyzed is the standard adaptive method except that if necessary additional refinements are made to keep the meshes sufficiently mildly graded this modification does not compromise the quasi optimality of the resulting algorithm

wavelet transform domain adaptive fir filtering this paper presents and studies two new wavelet transform domain least mean square lms algorithms the algorithms exploit the special sparse structure of the wavelet transform of wide classes of correlation matrices and their cholesky factors in order to compute a whitening transformation of the input data in the wavelet domain and minimize computational complexity the procedures differ in the exact estimates they use and in the way they identify the data dependent whitening transformation the first approach explicitly computes a sparse estimate of the wavelet domain correlation matrix of the input process it then computes the cholesky factor of that matrix and uses its inverse to whiten the input the complexity of this first approach is o n log sup 2 n in contrast the second approach computes a sparse estimate of the cholesky factor of the wavelet domain correlation matrix of the input process directly this second approach has a computational complexity of o n log n floating point operations however it requires a more complex bookkeeping procedure both algorithms have a convergence rate that is faster than that of time domain lms and discrete fourier transform dft or discrete cosine transform dct based lms procedures the paper compares the two procedures and analyzes their mean and mean square performance

a posteriori error analysis of optimal control problems with control constraints we derive a unifying framework for the a posteriori error analysis of control constrained linear quadratic optimal control problems we consider finite element discretizations with discretized and nondiscretized control a fundamental error equivalence drastically simplifies the a posteriori error analysis for optimal control problems it basically remains to apply error estimators for the linear state and adjoint problem we give several examples including stabilized discretizations and investigate the quality of the estimators and the performance of the adaptive iteration by selected numerical experiments

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

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

using robust scale estimates in detecting multiple outliers in linear regression summary statistics for detecting outliers generally suffer from masking when multiple outliers are present one aspect of this masking is inflation by the outliers of estimates of scale this shrinks test statistics and results in loss of power to identify the outliers two familiar robust scale estimators are considered the interquartile range ir and the median absolute deviation from the median mad they are used here to scale statistics both for testing individual observations and for testing a no outliers hypothesis some of these statistics use ordinary least squares residuals others use recursive residuals calculated on adaptively ordered observations the more severe the masking problem the more advantageous robust scale estimation was found to be ir and mad worked equally well test statistics based on the recursive residuals were more powerful than those based on ordinary residuals

graphical exploratory data analysis 1 the role of graphics in data exploration 1 introduction 2 historical background 3 content of the book 4 central data sets 5 different types of data 6 computer programs 2 graphics for univariate and bivariate data 1 introduction 2 graphics for univariate data 3 stem and leaf plots 4 graphics for bivariate data 5 graphical perception 3 graphics for selecting a probability model 1 introduction 2 discrete models 3 continuous models 4 general 4 visual representation of multivariate data 1 introduction 2 scatterplots in more than two dimensions 3 profiles 4 star representations 5 glyphs 6 boxes 7 andrews curves 8 chernoff faces 9 general 5 cluster analysis 1 introduction 2 the probability approach 3 measures of distance and similarity 4 hierarchical cluster analysis 5 computer programs for hierarchical cluster analysis 6 digraphs 7 spanning trees 8 cluster analysis of variables 9 application of cluster analysis to fitness cholesterol data 10 other graphical techniques of cluster analysis 11 general 6 multidimensional scaling 1 introduction 2 the biplot 3 principal component analysis 4 correspondence analysis 5 classical metric scaling 6 non metric scaling 7 three way multidimensional scaling indscal 8 guttman s techniques 9 facet theory 10 partial order scalogram analysis 11 general 7 graphical representations in regression analysis 1 introduction 2 the scatterplot 3 residual plots 4 mallows q statistic 5 confidence and forecast bands 6 the ridge trace 7 general 8 chaid and xaid exploratory techniques for analyzing extensive data sets 1 introduction 2 chaid an exploratory technique for analyzing categorical data 3 applying a chaid analysis 4 xaid an exploratory technique for analyzing a quantitative dependent variable with categorical predictors 5 application of xaid analysis 6 general 9 control charts 1 introduction 2 process capability 3 control charts for items with quantitative characteristics 4 control charts for dichotomous measurements p chart 5 cumulative sum charts 6 cumulative sine charts 7 general 10 time series representations 1 representations in the time domain 2 representations in the frequency domain 11 further useful graphics 1 graphics for the two sample problem 2 graphical techniques in analysis of variance 3 four fold circular display of 2 x 2 contingency tables references inde

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate 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 extensions to higher order elements and applications to saddle point problems are discussed as well

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

linear unbiased approximators of the disturbances in the standard linear model abstract the least squares residuals from the standard linear model have a variance matrix which is a function of the n q matrix of observations on the regressors we examine two classes of residuals which do not suffer from this defect our first class of residuals luz residuals has a variance matrix which is a scalar multiple of an n n idempotent matrix of rank n q specified by the user and our second class of residuals lus residuals has a variance matrix which is a scalar multiple of the n q n q identity matrix

a review of the development and application of recursive residuals in linear models abstract recursive residuals have been shown to be useful in a variety of applications in linear models unlike the more familiar ordinary least squares residuals or studentized residuals recursive residuals are independent as well as homoscedastic under the model their independence is particularly appealing for use in developing test statistics they are not uniquely defined their values depend on the order in which they are calculated although their properties do not in some applications one can exploit this order dependence coupled with the fact that they are in clear one to one correspondence with the observations for which they are calculated uses for recursive residuals have been suggested in almost all areas of regression model validation regression diagnostics have been constructed from recursive residuals for detecting serial correlation heteroscedasticity functional misspecification and structural change other statistics based on recursive residuals have focused on detection of outli

adaptive finite element method for elliptic optimal control problems convergence and optimality in this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints we use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable based on the well established convergence theory of afem for elliptic boundary value problems we rigorously prove the convergence and quasi optimality of afem for optimal control problems with respect to the state and adjoint state variables by using the so called perturbation argument numerical experiments confirm our theoretical analysis

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

a posteriori error estimates for hp finite element solutions of convex optimal control problems in this paper we present a posteriori error analysis for hp finite element approximation of convex optimal control problems we derive a new quasi interpolation operator of clement type and a new quasi interpolation operator of scott zhang type that preserves homogeneous boundary condition the scott zhang type quasi interpolation is suitable for an application in bounding the errors in l 2 norm then hp a posteriori error estimators are obtained for the coupled state and control approximations such estimators can be used to construct reliable adaptive finite elements for the control problems

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

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

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

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

using recursive residuals calculated on adaptively ordered observations to identify outliers in linear regression a new procedure for identifying outliers or influential observations is proposed the procedure uses recursive residuals calculated on observations that have been ordered according to their studentized residuals values of cook s d or another regression diagnostic of the user s choice under the model these recursive residuals appropriately standardized have approximate student s t distributions thus convenient critical values are available for deciding which observations merit scrutiny and perhaps special treatment the power of the test procedure to identify one or more outliers is investigated through simulation and its dependence on the number and configuration of the outliers that is their placement with respect to the main body of the data explored the proposed procedure and two variations of it also based on these recursive residuals are compared with alternatives based on internally or externally studentized residuals the use of recursive residuals calculated on adaptively ordered observations increases power and helps to combat the masking of one outlier by another when multiple outliers are present in configurations that create masking

convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problem abstract in this paper we focus on the convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problems we use piecewise linear finite elements to approximate the state and the adjoint state variables and the variational discretization to approximate the control variable under mild assumption on the initial mesh we prove the contraction property for the sum of the energy errors of the state and adjoint state and the scaled error estimator on two consecutive adaptive loops the resulting linear convergence yields the quasi optimal convergence rate for the afem algorithm applied to our problem additionally some numerical results are provided to support our theoretical analysis