ma estimation in polynomial time the parameter estimation of moving average ma signals from second order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible we show how the problem of ma parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the ma order two methods are proposed that rely on two specific over parametrizations of the ma covariance sequence whose use makes the minimization of a covariance fitting criterion a convex problem the ma estimation algorithms proposed here are computationally fast statistically accurate and reliable none of the previously available algorithms for ma estimation methods based on higher order statistics included shares all these desirable properties our methods can also be used to obtain the optimal least squares approximant of an invalid estimated ma spectrum that takes on negative values at some frequencies which was another long standing problem in the signal processing literature awaiting a satisfactory solution

root n consistent semiparametric regression on considere une variable aleatoire x z dans 7b r p 7b r q on construit un estimateur generalisant l estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur z

optimization of the higher density discrete wavelet transform and of its dual tree we propose a parameterization of the finite impulse response fir filters that generate the higher density discrete wavelet transform hd dwt the parameterization has the form of a linear system whose variables are the coefficients of two positive trigonometric polynomials the parameterization allows the easy separation of independent variables and can be used to transform the optimization of hd dwt into semidefinite programming sdp form using the parameterization we are able to optimize filters whose length is greater than the minimal one imposed by regularity constraints we also propose two optimization methods for the dual tree hd dwt the first generalizes the design based on allpass approximation of the half sample delay the second is a brute force optimization based on generating values for the independent variables by using successive sections through the admissible set as optimization criteria we use an analyticity measure in the latter method and stopband energy in the former ones for each type of design we present examples and compare them with previous work

best possible constant for bandwidth selection for the data based choice of the bandwidth of a kernel density estimator several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth in particular the relative rate of convergence is the square root of the sample size which is known to be the best possible the point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient that is an analog of the usual fisher information in this convergence this establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform it is seen that some existing methods attain the bound others do not

a uniqueness result on the solution for a landau lifshitz system with penalization abstract the author proves the uniqueness of the solution to an evolution landau lifshitz type problem when the parameter tends to zero in addition the unique solution is just a heat flow of a harmonic map this uniqueness result is derived by establishing a uniform estimate for the solution

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

on the parameterization of positive real sequences and ma parameter estimation an algorithm for moving average ma parameter estimation was proposed by stoica et al see ibid vol 48 p 1999 2012 2000 its key step covariance fitting is a semidefinite programming sdp problem with two convex constraints one reflecting the real positiveness of the desired covariance sequence and the other having a second order cone form we analyze two parameterizations of a positive real sequence and show that there is a one to one correspondence between them we also show that the dual of the covariance fitting problem has a significantly smaller number of variables and thus a much reduced computational complexity we discuss in detail the formulations that are best suited for the currently available semidefinite quadratic programming packages experimental results show that the execution times of the newly proposed algorithms scale well with the ma order which are therefore convenient for large order ma signals

efficient large scale filter filterbank design via lmi characterization of trigonometric curves many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semi infinite constraints recent advances in design methodology are based on various linear matrix inequality lmi characterizations of the semi infinite constraints and semidefinite programming sdp solutions despite these advances the design of filters of several hundredth order which typically arise in multicarrier communication and signal compression cannot be accommodated this hurdle is due mainly to the large number of additional variables incurred in the lmi characterizations this paper proposes a novel lmi characterization of the semi infinite constraints that involves additional variables of miminal dimensions consequently the design of high order filters required in practical applications can be achieved examples of designs of up to 1200 tap filters are presented to verify the viability of the proposed approach

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

far field multicast beamforming for uniform linear antenna arrays the problem of transmit beamforming to multiple cochannel multicast groups is considered for the important special case when the channel vectors are vandermonde this arises when a uniform linear antenna antenna ula array is used at the transmitter under far field line of sight propagation conditions as provisioned in 802 16e and related wireless backhaul scenarios two design approaches are pursued i minimizing the total transmitted power subject to providing at least a prescribed received signal to interference plus noise ratio sinr to each intended receiver and ii maximizing the minimum received sinr under a total transmit power budget whereas these problems have been recently shown to be np hard in general it is proven here that for vandermonde channel vectors it is possible to recast the optimization in terms of the autocorrelation sequences of the sought beam vectors yielding an equivalent convex reformulation this affords efficient optimal solution using modern interior point methods the optimal beam vectors can then be recovered using spectral factorization robust extensions for the case of partial channel state information where the direction of each receiver is known to lie in an interval are also developed interestingly these also admit convex reformulation the various optimal designs are illustrated and contrasted in a suite of pertinent numerical experiments

semiparametric models for longitudinal data with application to cd4 cell numbers in hiv seroconverters summary the paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of cd4 cell numbers in hiv seroconverters the essential ingredients of the model are a parametric linear model for covariate adjustment a nonparametric estimation of a smooth time trend serial correlation between measurements on an individual subject and random measurement error a back fitting algorithm is used in conjunction with a cross validation prescription to fit the model a notable feature in the application is that the onset of hiv infection is associated with a sudden drop in cd4 cells followed by a longer term slower decay the model is also used to estimate an individual s curve by combining his data with the population curve shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data

convex optimization problems involving finite autocorrelation sequences we discuss convex optimization problems in which some of the variables are constrained to be finite autocorrelation sequences problems of this form arise in signal processing and communications and we describe applications in filter design and system identification autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density which results in a set of linear inequalities they can also be cast as linear matrix inequalities via the kalman yakubovich popov lemma the linear matrix inequality formulation is exact and results in convex optimization problems that can be solved using interior point methods for semidefinite programming however it has an important drawback to represent an autocorrelation sequence of length n it requires the introduction of a large number n n 1 2 of auxiliary variables this results in a high computational cost when general purpose semidefinite programming solvers are used we present a more efficient implementation based on duality and on interior point methods for convex problems with generalized linear inequalities

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

multistage iir filter design using convex stability domains defined by positive realness in this paper we consider infinite impulse response iir filter design where both magnitude and phase are optimized using a weighted and sampled least squares criterion we propose a new convex stability domain defined by positive realness for ensuring the stability of the filter and adapt the steiglitz mcbride sm gauss newton gn and classical descent methods to the new stability domain we show how to describe the stability domain such that the description is suited to semidefinite programming and is implementable exactly in addition we prove that this domain contains the domain given by rouche spl acute s theorem finally we give experimental evidence that the best designs are usually obtained with a multistage algorithm where the three above methods are used in succession each one being initialized with the result of the previous and where the positive realness stability domain is used instead of that defined by rouche spl acute s theorem

semiparametric estimates of the relation between weather and electricity sales abstract a nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data this accommodates introduction of covariates timing adjustments due to the actual billing schedules and serial correlation the procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross validation criterion introduced by craven and wahba 1979 estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines

additive partial linear models with measurement errors we consider statistical inference for additive partial linear models when the linear covariate is measured with error we propose attenuation to correction and simex estimators of the parameter of interest it is shown that the first resulting estimator is asymptotically normal and requires no undersmoothing this is an advantage of our estimator over existing backfitting based estimators for semiparametric additive models which require undersmoothing of the nonparametric component in order for the estimator of the parametric component be root n consistent this feature stems from a decrease of the bias of the resulting estimator which is appropriately derived using a profile procedure a similar characteristic in semiparametric partially linear models was obtained by wang et al 2005 we also discuss the asymptotics of the proposed simex approach finite sample performance of the proposed estimators is assessed by simulation experiments the proposed methods are applied to a dataset from a semen study

asymptotics for the simex estimator in nonlinear measurement error models abstract cook and stefanski have described a computer intensive method the simex method for approximately consistent estimation in regression problems with additive measurement error in this article we derive the asymptotic distribution of their estimators and show how to compute estimated standard errors these standard error estimators can either be used alone or as prepivoting devices in a bootstrap analysis we also give theoretical justification to some of the phenomena observed by cook and stefanski in their simulations

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

efficient semiparametric marginal estimation for longitudinal clustered data we consider marginal generalized semiparametric partially linear models for clustered data lin and carroll derived the semiparametric efficient score function for this problem in the multivariate gaussian case but they were unable to construct a semiparametric efficient estimator that actually achieved the semiparametric information bound here we propose such an estimator and generalize the work to marginal generalized partially linear models we investigate asymptotic relative efficiencies of the estimators that ignore the within cluster correlation structure either in nonparametric curve estimation or throughout we evaluate the finite sample performance of these estimators through simulations and illustrate it using a longitudinal cd4 cell count dataset both theoretical and numerical results indicate that properly taking into account the within subject correlation among the responses can substantially improve efficiency

comparison of data driven bandwidth selectors abstract this article compares several promising data driven methods for selecting the bandwidth of a kernel density estimator the methods compared are least squares cross validation biased cross validation and a plug in rule the comparison is done by asymptotic rate of convergence to the optimum and a simulation study it is seen that the plug in bandwidth is usually most efficient when the underlying density is sufficiently smooth but is less robust when there is not enough smoothness present we believe the plug in rule is the best of those currently available but there is still room for improvement

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

sdp approximation of a fractional delay and the design of dual tree complex wavelet transform we show that an hinfin optimization problem related to fractional delay approximation can be formulated as a semideflnite programming sdp problem and thus solved reliably particularly given the finite impulse response fir filter h z we find the fir filter g z of given degree such that g z z 1 2h z infin is minimum this half sample delay approximation problem is used in the design of filters generating orthogonal dual tree complex wavelet transforms since the solution does not conform to the orthogonality constraints exactly we propose their enforcement in a second stage of optimization in which an analyticity criterion is optimized the proposed designs compare favorably with previous ones

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

focused information criterion and model averaging for generalized additive partial linear models we study model selection and model averaging in generalized additive partial linear models gaplms polynomial spline is used to approximate nonparametric functions the corresponding estimators of the linear parameters are shown to be asymptotically normal we then develop a focused information criterion fic and a frequentist model average fma estimator on the basis of the quasi likelihood principle and examine theoretical properties of the fic and fma the major advantages of the proposed procedures over the existing ones are their computational expediency and theoretical reliability simulation experiments have provided evidence of the superiority of the proposed procedures the approach is further applied to a real world data example

asymptotic analysis of penalized likelihood and related estimators a general approach to the first order asymptotic analysis ofpenalized likelihood and related estimators is described the method gives expansions for the systematic and random error asymptotic convergence rates in a family of spectral norms are obtained the theory applies to a broad range of function estimation prob erns including non paxametric dellsity hazard and generalized regression curve estimation some examples are provided ams 1980 subject classifications primary 62 g05 secondary 62j05 41 a35 41 a25 47 a53 45 llo 45 m05

a structure exploiting preprocessor for semidefinite programs derived from the kalman yakubovich popov lemma semidefinite programs derived from the kalman yakubovich popov kyp lemma are quite common in control and signal processing applications the programs are often of high dimension which makes them hard or impossible to solve with general purpose solvers here we present a customized preprocessor kypd that utilizes the inherent structure of this particular optimization problem the key to an efficient implementation is to transform the optimization problem into an equivalent semidefinite program this equivalent problem has much fewer variables and the matrices in the linear matrix inequality constraints are of low rank kypd can use any primal dual solver for semidefinite programs as an underlying solver

optimized analog filter designs with flat responses by semidefinite programming analog filters constitute indispensable components of analog circuits inspired by recent advances in digital filter design this paper provides a flexible design for analog filters all pole filters have maximally flat passband so our design minimizes their passband distortion analogously maximally flat filters have maximally flat passband so our design maximizes their stopband attenuation its particular cases provide flexible alternatives to the classical counterparts semidefinite program sdp formulations for the posed filter design problems are presented which are efficiently solved by existing optimization software several examples and comparisons are provided to validate the viability of our design

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

simulation extrapolation the measurement error jackknife abstract this article provides theoretical support for our simulation based estimation procedure simex for measurement error models we do so by establishing a strong relationship between simex estimation and jackknife estimation a result of our investigation is the identification of a variance estimation method for simex that parallels jackknife variance estimation data from the framingham heart study are used to illustrate the variance estimation procedure in logistic regression measurement error models

quasi likelihood estimation in semiparametric models abstract suppose the expected value of a response variable y may be written h x t where x and t are covariates each of which may be vector valued is an unknown parameter vector is an unknown smooth function and h is a known function in this article we outline a method for estimating the parameter of this type of semiparametric model using a quasi likelihood function algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed the generalization of this approach to the case in which y is a multivariate response is also considered the methodology is illustrated on two data sets and the results of a small monte carlo study are presented

on the new barrier function and specialized algorithms for a class of semidefinite programs semidefinite programs sdps arising from the kalman yakubovich popov kyp lemma are frequently encountered in systems robustness analysis filter design and other control signal processing related applications these programs possess a special structure that can be exploited to construct specialized algorithms that substantially outperform general purpose sdp solvers in this paper a new interior path following algorithm that utilizes this structure is proposed the main idea behind the algorithm is a new barrier function for these specially structured sdps convergence of the new algorithm is shown and a measure of the accuracy of suboptimal solutions produced by the algorithm is provided the algorithm is tested in numerical experiments and the results indicate that the new algorithm is indeed favorable against general purpose sdp solvers in many circumstances

design of orthogonal pulse shapes for communications via semidefinite programming in digital communications orthogonal pulse shapes are often used to represent message symbols for transmission through a channel in this paper the design of such pulse shapes is formulated as a convex semidefinite programming problem from which a globally optimal pulse shape can be efficiently found the formulation is used to design filters that achieve a the minimal bandwidth for a given filter length b the minimal filter length for a given bandwidth c the maximal robustness to timing error for a given bandwidth and filter length bandwidth is measured either in spectral energy concentration terms or with respect to a spectral mask the effectiveness of the method is demonstrated by the design of waveforms with substantially improved performance over the chip waveforms specified in standards for digital mobile telecommunications

the weak solutions to the evolution problems of harmonic maps on demontre l existence de solutions faibles globales a des problemes d evolution des applications harmoniques de varietes de riemann compactes sans bord dans des spheres

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

simulation extrapolation estimation in parametric measurement error models we describe a simulation based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated the method entails adding add

semidefinite descriptions of cones defining spectral mask constraints we discuss in detail an additive structure of cones of trigonometric polynomials nonnegative on the union of a finite number of pairwise disjoint segments of the unit circle we derive new descriptions of these cones in terms of semidefinite constraints we explain the results of m krein and a nudelman m krein a nudelman the markov moment problem and extremal problems american mathematical society providence 1977 providing a description of dual cones in terms of the same type of constraints

positive polynomials and robust stabilization with fixed order controllers recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial an application to the design of fixed order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed based on linear matrix inequality optimization the key ingredient in the design procedure resides in the choice of the central polynomial several numerical examples illustrate the relevance of the approach

optimal nonnegative definite approximations of estimated moving average covariance sequences the problem of finding the closest nonnegative definite moving average covariance sequence to a given estimate which may not be nonnegative definite is considered an algorithm is developed which is based on a set of constrained minimization problems each parameterized by the zero frequencies of the spectral density function corresponding to the optimal solution the algorithm entails first solving a simple minimization problem with linear constraints whose closed form solution is given by a projection onto a subspace these solutions lie either outside the set of nonnegative definite sequences or on its boundary if the solution lies on the boundary it is the optimal solution the problem is considered directly in the space of covariance sequence elements as a result the nonlinear maximization step is performed on sets of low dimension by considering the minimization problem in this space it is possible to characterize some of the geometrical properties of the optimal solution in terms of the locations of its zero frequencies

sampling method for semidefinite programmes with non negative popov function constraints an important class of optimisation problems in control and signal processing involves the constraint that a popov function is non negative on the unit circle or the imaginary axis such a constrain

trigonometric polynomials positive on frequency domains and applications to 2 d fir filter design we propose a characterization of multivariate trigonometric polynomials that are positive on a given frequency domain the positive polynomials are parameterized as a linear function of sum of squares polynomials and so semidefinite programming sdp is applicable the frequency domain is expressed via the positivity of some trigonometric polynomials we also give a bounded real lemma brl in which a bounding condition on the magnitude of the frequency response of a multidimensional finite impulse response fir filter is expressed as a linear matrix inequality lmi this brl avoids the problem of a lack of spectral factorization in the multidimensional case all the proposed theoretical contributions can be implemented only as sufficient conditions due to degree limitations on the sum of square polynomials however the two dimensional 2 d fir filter designs we study numerically suggest that these limitations have negligible impact on the optimality

estimation in a semiparametric partially linear errors in variables model we consider the partially linear model relating a response y to predictors x t with mean function x t g t when the x s are measured with additive error the semiparametric likelihood estimate of severini and staniswalis leads to biased estimates of both the parameter and the function g when measurement error is ignored we derive a simple modification of their estimator which is a semiparametric version of the usual parametric correction for attenuation the resulting estimator of is shown to be consistent and its asymptotic distribution theory is derived consistent standard error estimates using sandwich type ideas are also developed

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

peak constrained least squares optimization we presented the basic concepts for peak constrained least squares pcls optimization in previous papers we present advanced pcls optimization concepts in this paper this paper discusses only a few of the many ways that pcls design problems can be formulated we believe that the pcls optimality criterion can be customized to fit virtually any practical application for example pcls optimization can readily be used to design equalizers in the prefilter equalizer or interpolated fir implementations of filters

convergence of the parabolic ginzburg landau equation to motion by mean curvature for the complex parabolic ginzburg landau equation we prove that asymptotically vorticity evolves according to motion by mean curvature in brakke s weak formulation the only assumption is a natural energy bound on the initial data in some cases we also prove convergence to enhanced motion in the sense of ilmanen