analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

optimal control in fluid mechanics by finite elements with symmetric stabilization there are two main possibilities for the numerical computation of optimal control problems with constraints given by partial differential equations one may consider first the discretized problem and then build the optimality condition the other possibility is to formulate first the optimality condition on the continuous level and then discretize both approaches may lead to different discrete adjoint equations because discretization and building the adjoint do not commute in general this type of inconsistency takes place when conventional stabilized finite elements for flow problems as for instance streamline diffusion supg are used due to its nonsymmetry consequently the computed control is significantly affected by the way of defining the discrete optimality condition hence there is a need for symmetric stabilization so that discretization and building the adjoint commute we formulate the use of this kind of stabilization and give a quasi optimal a priori estimate in the context of optimal control problems for the oseen system in particular we show that local projection stabilization and edge oriented stabilization result to be quasi optimal for optimal control problems

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

robust a posteriori estimator for advection diffusion reaction problems we propose an almost robust residual based a posteriori estimator for the advection diffusion reaction model problem the theory is developed in the one dimensional setting the numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators in particular the mentioned norm possesses features that allow us to obtain a meaningful a posteriori estimator robust up to a log pe factor where pe is the global peclet number of the problem various numerical tests are performed in one dimension to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature we also consider a possible two dimensional extension of our result and only present a few basic numerical tests indicating that the estimator seems to preserve the good features of the one dimensional setting

a posteriori error estimation for control constrained linear quadratic optimal control problems we derive a posteriori error estimates for control constrained linear quadratic optimal control problems the error is measured in a norm which is motivated by the objective our abstract error estimator is separated into three contributions the error in the variational inequality i e in the optimality condition for the control and the errors in the state and adjoint equation hence one can use well established estimators for the differential equations we show that the abstract error estimator is reliable and efficient if the utilized estimators for the differential equations have these properties we apply the error estimator to two distributed optimal control problems with distributed and boundary observation respectively numerical examples exhibit a good error reduction if we use the local error contributions for an adaptive mesh refinement

fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements we obtain a fully computable a posteriori error bound on the broken energy norm of the error in the nonconforming finite element approximation on triangles of arbitrary order of a linear second order elliptic problem with variable permeability the estimator is completely free of unknown constants and provides a guaranteed numerical bound on the broken energy norm of the error this estimator is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms

optimal detection using bilinear time frequency and time scale representations bilinear time frequency representations tfrs and time scale representations tsrs are potentially very useful for detecting a nonstationary signal in the presence of nonstationary noise or interference as quadratic signal representations they are promising for situations in which the optimal detector is a quadratic function of the observations all existing time frequency formulations of quadratic detection either implement classical optimal detectors equivalently in the time frequency domain without fully exploiting the structure of the tfr or attempt to exploit the nonstationary structure of the signal in an ad hoc manner we identify several important nonstationary composite hypothesis testing scenarios for which tfr tsr based detectors provide a natural framework that is in which tfr tsr based detectors are both optimal and exploit the many degrees of freedom available in the tfr tsr we also derive explicit expressions for the corresponding optimal tfr tsr kernels as practical examples we show that the proposed tfr tsr detectors are directly applicable to many important radar sonar detection problems finally we also derive optimal tfr tsr based detectors which exploit only partial information available about the nonstationary structure of the signal

guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity we obtain fully computable a posteriori error estimators for the energy norm of the error in second order conforming and nonconforming finite element approximations in planar elasticity these estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error the estimators are shown to also provide local lower bounds up to a constant and higher order data oscillation terms numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator

local error estimates for supg solutions of advection dominated elliptic linear quadratic optimal control problems we derive local error estimates for the discretization of optimal control problems governed by linear advection diffusion partial differential equations pdes using the streamline upwind petrov galerkin supg stabilized finite element method we show that if the supg method is used to solve optimization problems governed by an advection dominated pde the convergence properties of the supg method is substantially different from the convergence properties of the supg method applied for the solution of an advection dominated pde the reason is that the solution of the optimal control problem involves another advection dominated pde the so called adjoint equation whose advection field is just the negative of the advection of the governing pdes for the solution of the optimal control problem a coupled system involving both the original governing pde as well as the adjoint pde must be solved we show that in the presence of a boundary layer the local error between the solution of the supg discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer in the presence of interior layers we prove optimal convergence rates for the local error in a region away from the layer between the solution of the supg discretized optimal control problems and the solution of the infinite dimensional problem numerical examples are presented to illustrate some of the theoretical results

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

time frequency projection filters and time frequency signal expansions we consider the problems of designing a linear time varying filter with a specified time frequency tf pass region and of constructing an orthonormal basis for the parsimonious expansion of signals located in a given tf support region these problems of tf filtering and tf signal expansion are reduced to the problem of designing a tf subspace i e a linear signal space comprising all signals located in a given tf legion specifically the tf filter is taken to be the orthogonal projection operator on the tf subspace we present an optimum design of tf subspaces that is based on the wigner distribution of a linear signal space and is an extension of the well known signal synthesis problem the optimum tf subspace is shown to be an eigenspace of the tf region and some properties of eigenspaces are discussed the performance of tf projection filters and tf signal expansions is studied both analytically and via computer simulation

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

time frequency formulation design and implementation of time varying optimal filters for signal estimation this paper presents a time frequency framework for optimal linear filters signal estimators in nonstationary environments we develop time frequency formulations for the optimal linear filter time varying wiener filter and the optimal linear time varying filter under a projection side constraint these time frequency formulations extend the simple and intuitive spectral representations that are valid in the stationary case to the practically important case of underspread nonstationary processes furthermore we propose an approximate time frequency design of both optimal filters and we present bounds that show that for underspread processes the time frequency designed filters are nearly optimal we also introduce extended filter design schemes using a weighted error criterion and we discuss an efficient time frequency implementation of optimal filters using multiwindow short time fourier transforms our theoretical results are illustrated by numerical simulations

scattering symmetry and bias analysis of source position estimates in euler deconvolution and its practical implications despite being widely used euler deconvolution has received little attention in its theoretical aspects we obtain analytical estimators x 0 and z 0 for the horizontal x0 and the vertical z0 source positions which are the parameters to be estimated when the structural index is assigned a tentative value by analyzing the estimators properties we show two things 1 scattering of the alternative solutions associated with different data window positions has two components data noise and the use of a wrong tentative value for the structural index 2 the differences x 0 x0 and z 0 z0 as a function of the data window position exhibit antisymmetric and symmetric behaviors respectively about x0 when the magnetization and the geomagnetic field inclinations are 90 0 or 45 for intermediate inclinations there is a moderate departure from the symmetry properties this analysis shows that the criterion for determining the structural index as the tentative value producing the smallest solution scat

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

a time frequency formulation of optimum detection a time frequency formulation is proposed for the optimum detection of gaussian signals in white gaussian noise by choosing the wigner ville distribution as the basic time frequency tool it is shown that the corresponding receivers generally take the form of a correlation between time frequency structures matching mathematical optimality with a physically meaningful interpretation the case of low snr is examined and various examples are considered deterministic signal rayleigh fading signal random jitter and random time varying channel a general class of time frequency receivers is proposed which admits as limiting cases different known structures and its suboptimum performance is evaluated possible extensions to more elaborate situations including parameter estimation are mentioned

polynomial wigner ville distributions and their relationship to time varying higher order spectra the wigner ville distribution wvd has optimal energy concentration for linear frequency modulated fm signals this paper presents a generalization of the wvd in order to effectively process nonlinear polynomial fm signals a class of polynomial wvd s pwvd s that give optimal concentration in the time frequency plane for fm signals with a modulation law of arbitrary polynomial form are defined a class of polynomial time frequency distributions ptfd s are also defined based on the class of pwvd s the optimal energy concentration of the pwvd enables it to be used for estimation of the instantaneous frequency if of polynomial fm signals finally a link between pwvd s and time varying higher order spectra tvhos is established just as the expected value of the wvd of a nonstationary random signal is the time varying power spectrum the expected values of the pwvd s have interpretations as reduced tvhos

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

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

on the transfer function calculus for underspread ltv channels the freedom in the definition of a time varying transfer function is formalized by introducing the generalized weyl correspondence which includes zadeh s 1950 function and the weyl symbol for slowly time varying linear systems with properly limited memory underspread ltv systems we prove the asymptotic validity of the multiplicative transfer function calculus

generalized evolutionary spectral analysis and the weyl spectrum of nonstationary random processes the evolutionary spectrum es is a time varying power spectrum of nonstationary random processes starting from an innovations system interpretation of the es we introduce the generalized evolutionary spectrum ges as a novel family of time varying power spectra the ges contains the es and the transitory evolutionary spectrum as special cases we consider the problem of finding an innovations system for a process characterized by its correlation function and we discuss the connection between ges analysis and the class of underspread processes furthermore we show that another special case of the ges a novel time varying power spectrum that we call the weyl spectrum has substantial advantages over all other members of the ges family the properties of the weyl spectrum are discussed and its superior performance is verified experimentally for synthetic and real data processes

nonstationary spectral analysis based on time frequency operator symbols and underspread approximations we present a unified framework for time varying or time frequency tf spectra of nonstationary random processes in terms of tf operator symbols we provide axiomatic definitions and tf operator symbol formulations for two broad classes of tf spectra one of which is new these classes contain all major existing tf spectra such as the wigner ville evolutionary instantaneous power and physical spectrum our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high lag tf correlations so called underspread processes we demonstrate that for underspread processes all tf spectra yield effectively identical results and satisfy several desirable properties at least approximately we also show that gabor frames provide approximate karhunen loeve kl functions of underspread processes and tf spectra provide a corresponding approximate kl spectrum finally we formulate simple approximate input output relations for the tf spectra of underspread processes that are passed through underspread linear time varying systems all approximations are substantiated mathematically by upper bounds on the associated approximation errors our results establish a tf calculus for the second order analysis and time varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

toeplitz and hankel kernels for estimating time varying spectra of discrete time random processes for a nonstationary random process the dual time correlation function and the dual frequency loeve spectrum are complete theoretical descriptions of second order behavior that is each may be used to synthesize the random process itself according to the cramer loeve spectral representation when suitably transformed on one of its two variables each of these descriptions produces a time varying spectrum this spectrum is in fact the expected value of the rihaczek distribution in this paper we derive two large families of estimators for this spectrum one based on a diagonal toeplitz diagonal dtd factorization of smoothing kernels and the other based on a diagonal hankel diagonal dhd factorization the dtd factorization produces noncoherent averages of the time varying spectrogram and the dhd factorization produces coherent averages some of the dtd estimators may be called time varying power spectrum estimators and some of the dhd estimators may be called time varying wigner ville wv estimators the former may always be implemented as multiwindow spectrum estimators and in some casts they are true time variations on the blackman tukey rosenblatt grenander btgr spectrogram the latter are variations on the stankovic class of wv estimators

time frequency transfer function calculus symbolic calculus of linear time varying systems linear operators based on a generalized underspread theory we introduce a generalized concept of underspread linear time varying systems linear operators which contains a previous definition as a special case we show that an existing approximate transfer function calculus symbolic calculus can be extended to this wider and practically more relevant class of underspread operators as a mathematical underpinning of this calculus we establish explicit bounds on various error quantities associated with it the transfer function calculus provides a theoretical basis for various methods recently proposed for nonstationary signal processing and it has important implications in the theory of time varying power spectra

a framework for obtaining guaranteed error bounds for finite element approximations we give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming non conforming mixed and discontinuous finite element schemes whilst the details of the actual estimator are rather different for each particular scheme there is nonetheless a common underlying structure at work in all cases we aim to illustrate this structure by treating conforming non conforming and discontinuous finite element schemes in a single framework in taking a rather general viewpoint some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but in return we hope to allow the reader to see the wood despite the trees

on the adaptive selection of the parameter in stabilized finite element approximations a systematic approach is developed for the selection of the stabilization parameter for stabilized finite element approximation of the stokes problem whereby the parameter is chosen to minimize a computable upper bound for the error in the approximation the approach is applied in the context of both a single fixed mesh and an adaptive mesh refinement procedure the optimization is carried out by a derivative free optimization algorithm and is based on minimizing a new fully computable error estimator numerical results are presented illustrating the theory and the performance of the estimator together with the optimization algorithm

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

cross spectral analysis of nonstationary processes consideration is given to the generalization of stationary cross spectral analysis methods to a class of nonstationary processes specifically the class of semistationary finite energy processes possessing sample functions that are of finite energy almost surely a new quantity called the time frequency coherence tfc is defined and it is demonstrated that its properties are analogous to those possessed by the stationary coherence function the problem of estimating the tfc by using elements of l cohen s 1966 class of joint time frequency representations is investigated it is shown that the only admissible estimators are those based on the class of time frequency smoothed periodograms thus the familiar procedure of segmentation and smoothed short time fourier analysis cannot be improved upon within the framework considered by the use of the higher resolution nonparametric time frequency methods procedures for selection of the appropriate estimators and a possible application are suggested

explicit error bounds in a conforming finite element method the goal of this paper is to define a procedure for bounding the error in a conforming finite element method the new point is that this upper bound is fully explicit and can be computed locally numerical tests prove the efficiency of the method it is presented here for the case of the poisson equation and a first order finite element approximation

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

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

a theory of polyspectra for nonstationary stochastic processes harmonizable processes constitute an important class of nonstationary stochastic processes we present a theory of polyspectra higher order moment spectra for the harmonizable class we define and discuss four basic quantities the nth order moment function the nth order time frequency polyspectrum the nth order ambiguity function and the nth order frequency frequency polyspectrum the latter generalizes the conventional polyspectrum to nonstationary stochastic processes these four functions are related to one another by fourier transforms we show that the frequency and time marginals of the time frequency polyspectrum are the instantaneous nth order moment and the conventional nth order stationary polyspectrum respectively all quantities except the nth order ambiguity function allow for insightful interpretations in terms of hilbert space inner products the inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process the polycoherences are objective measures of stationarity to order n which can be used to construct various statistical tests finally we give some specific examples and apply the theory to linear time varying systems which are popular models for fading multipath communication channels

a posteriori error estimators for stabilized finite element approximations of an optimal control problem abstract we derive a posteriori error estimators for an optimal control problem governed by a convection reaction diffusion equation control constraints are also considered we consider a family of low order stabilized finite element methods to approximate the solutions to the state and adjoint equations we obtain a fully computable a posteriori error estimator for the optimal control problem all the constants that appear in the upper bound for the error are fully specified therefore the proposed estimator can be used as a stopping criterion in adaptive algorithms we also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative numerical examples in two and three dimensions are presented to illustrate the theory

a robust supg norm a posteriori error estimator for stationary convection diffusion equations a robust residual based a posteriori estimator is proposed for the supg finite element method applied to stationary convection diffusion reaction equations the error in the natural supg norm is estimated the main concern of this paper is the consideration of the convection dominated regime a global upper bound and a local lower bound for the error are derived where the global upper estimate relies on some hypotheses numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses a comparison to other residual based estimators with respect to the adaptive grid refinement is also provided

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

adaptive covariance estimation of locally stationary processes it is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions we model locally stationary processes with pseudo differential operators that are time varying convolutions an adaptive covariance estimation is calculated by searching first for a best local cosine basis which approximates the covariance by a band or a diagonal matrix the estimation is obtained from regularized versions of the diagonal coefficients in the best basis

a priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection dominated diffusion equation in this paper we study an edge stabilization galerkin approximation scheme for the constrained optimal control problem governed by convection dominated diffusion equation the method uses least square stabilization of the gradient jumps across element edges a priori and a posteriori error estimates are derived for both the state co state and the control the theoretical results are illustrated by two numerical experiments

instantaneous power spectra conventional spectral analysis methods do not describe the distribution of signal power in both time and frequency simultaneously time dependent fourier transforms are defined to extend the conventional theory to include instantaneous energy and power spectra for the past and future signal these spectra are combined to define an instantaneous power spectrum ips of the entire signal the ips is a noncausal mathematical property of the signal from which observable properties can be derived e g the lagged window fourier transform estimate of the time frequency variations of the signal the ips for a nonstationary process is shown to be given by the fourier transform of loeve s generalized power spectrum of the process and to be a generalization of the wiener kintchine theorem it is also shown to represent the special cases of the stationary process the locally stationary process and the deterministic process

adaptive finite elements for elliptic optimization problems with control constraints in this paper we develop a posteriori error estimates for finite element discretization of elliptic optimization problems with pointwise inequality constraints on the control variable we derive error estimators for assessing the discretization error with respect to the cost functional as well as with respect to a given quantity of interest these error estimators provide quantitative information about the discretization error and guide an adaptive mesh refinement algorithm allowing for substantial saving in degrees of freedom the behavior of the method is demonstrated on numerical examples

realistic computable error bounds for three dimensional finite element analyses in linear elasticity we obtain a computable estimator for the energy norm of the error in piecewise affine and piecewise quadratic finite element approximations of linear elasticity in three dimensions we show that the estimator provides guaranteed upper bounds on the energy norm of the error as well as up to a constant and data oscillation terms local lower bounds

robust a posteriori error estimates for stationary convection diffusion equations we analyze a posteriori error estimators for finite element discretizations of convection dominated stationary convection diffusion equations using locally refined isotropic meshes the estimators are based on either the evaluation of local residuals or the solution of discrete local problems with dirichlet or neumann boundary conditions all estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative they are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection the estimates are also uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100