optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

weak convergence of linear and quadratic forms and related statements on lp approximability abstract in this paper we obtain central limit theorems for quadratic forms of non causal short memory linear processes with independent identically distributed innovations nabeya and tanaka 1988 suggested the format which links the asymptotic distribution to integral operators in their approach integral operators had to have continuous symmetric kernels mynbaev 2001 employed the theory of approximations to get rid of the continuity requirement here we go one step further by lifting the kernel symmetry condition also we establish l p approximability of the special sequences which arise in the theory of regressions with slowly varying regressors

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

central limit theorem for linear processes iences mixing sequences or associated sequences the results are important in analyzing the asymptotical properties of some estimators as well as of linear processes 4 1 introduction let be a centered sequence of random variables k 4 and let a 1 i n be a triangular array of numbers many statistical ni procedures produce estimators of the type n 1 1 s a nn i i i1 to give an example let us consider the nonlinear regression model yx gx x where gx is an unknown function and x is the noise now we fix the design points x and we get ni

combination technique based k th moment analysis of elliptic problems with random diffusion we consider the efficient deterministic solution of elliptic boundary value problems with random diffusion matrix assuming random perturbations with known k moments we derive to leading order in the random perturbation s amplitude deterministic equations for k moments of the random solution the solution s k th moment satisfies a k fold tensor product boundary value problem on the k fold product domain which can efficiently be discretized in sparse tensor product spaces by defining the complement spaces via galerkin projections the related system of linear equations decouples and can be solved by standard multilevel finite element solvers numerical results for k 2 are presented to validate and quantify our theoretical findings

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

efficient hierarchical approximation of high dimensional option pricing problems a major challenge in computational finance is the pricing of options that depend on a large number of risk factors prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation the objective of this article is to show how an efficient discretization can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem the relation to a number of state of the art methods is highlighted

central limit theorem for stationary linear processes we establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in mcleish ann probab 5 1977 616 621 and motivated by gordin soviet math dokl 10 1969 1174 1176 in doing so we shall preserve the generality of the coefficients including the long range dependence case and we shall express the variance of partial sums in a form easy to apply ergodicity is not required

on the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique abstract we introduce the combination technique for the numerical solution of d dimensional eigenproblems on sparse grids here o d log n d 1 different problems each of size o n have to be solved independently this is in contrast to the one problem of size o n d for a conventional finite element discretization where n denotes the number of grid points in one coordinate direction therefore also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach we apply this method to solve the three dimensional schrodinger equation for hydrogen one electron problem and the six dimensional schrodinger equation for helium two electron problem in strong magnetic and electric fields

error analysis of the combination technique the combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations in this paper we present a new technique to analyze the convergence rate of the combination technique this technique is applied to general second order elliptic differential equations in two dimensions furthermore it is proved that the combination technique for poisson s equation convergences in arbitrary dimensions

invariance principles for self similar set indexed random fields for a stationary random field x j j in z d and some measure m on r d we consider the set indexed weighted sum process s n a sum j in z d m na cap r j frac12 x j where r j is the unit cube with lower corner j we establish a general invariance principle under a p stability assumption on the x j s and an entropy condition on the class of sets a the limit processes are self similar set indexed gaussian processes with continuous sample paths using chentsov s type representations to choose appropriate measures m and particular sets a we show that these limits can be levy fractional brownian fields or fractional brownian sheets

the functional central limit theorem for linear processes with strong near epoch dependent innovations abstract this paper discusses linear processes with innovations exhibiting asymptotic weak dependence by being strong near epoch dependent functions of mixing processes the functional central limit theorem for the normalized partial sum process is established the conditions given essentially improve on existing results in the literature in terms of the size requirement for the amount of dependence it is also shown that two important econometric models arma and garch models are strong near epoch dependent sequences

integral operators on sparse grids in this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions the spaces are based on biorthogonal anisotropic tensor product wavelets we introduce sparse grid hyperbolic cross approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured furthermore we introduce compression schemes for the corresponding discretizations numerical examples for the laplace equation with dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

necessary and sufficient conditions for the conditional central limit theorem following lindeberg s approach we obtain a new condition for a stationary sequence of square integrable and real valued random variables to satisfy the central limit theorem in the adapted case this condition is weaker than any projective criterion derived from gordin s theorem dokl akad nauk sssr 188 1969 739 741 about approximating martingales moreover our criterion is equivalent to the conditional central limit theorem which implies stable convergence in the sense of renyi to a mixture of normal distributions we also establish functional and triangular versions of this theorem from these general results we derive sufficient conditions which are easier to verify and may be compared to other results in the literature to be complete we present an application to kernel density estimators for some classes of discrete time processes

two scale boolean galerkin discretizations for fredholm integral equations of the second kind in this paper some two scale boolean galerkin discretizations are proposed and analyzed for a class of fredholm integral equations of the second kind in multidimensions it is shown by both theory and numerics that this type of multiscale discretization algorithm not only significantly reduces the number of degrees of freedom but also produces very accurate approximations

postprocessed two scale finite element discretizations part i this paper studies a simple postprocessing of two scale finite element discretizations of elliptic partial differential operators where boundary value problems and eigenvalue problems on tensor product domains omega subset mathbb r d are both examined and conforming piecewise d linear elements are used for d geq2 suppose that the two scale method is applied using a coarse mesh and some univariate fine meshes satisfying h mathcal o h 2 3 where h and h are the fine and coarse mesh widths it is shown for both classes of problem that the postprocessed two scale finite element solution achieves the same order of accuracy in the h 1 omega norm as the postprocessed standard finite element solution but the former solution uses only mathcal o h 2d 1 3 degrees of freedom compared with the mathcal o h d degrees of freedom required by the latter

boundary finding with parametrically deformable models segmentation using boundary finding is enhanced both by considering the boundary as a whole and by using model based global shape information the authors apply flexible constraints in the form of a probabilistic deformable model to the problem of segmenting natural 2 d objects whose diversity and irregularity of shape make them poorly represented in terms of fixed features or form the parametric model is based on the elliptic fourier decomposition of the boundary probability distributions on the parameters of the representation bias the model to a particular overall shape while allowing for deformations boundary finding is formulated as an optimization problem using a maximum a posteriori objective function results of the method applied to real and synthetic images are presented including an evaluation of the dependence of the method on prior information and image quality

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

convergence of the combination technique for second order elliptic differential equations the combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization the advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points in this paper the combination technique is used with a bilinear finite element discretization depending on the smoothness of the solution and the coefficients it is proved for general second order elliptic differential equations on the unit square that the combined solution converges with order o h or o h log h 1 in the energy norm and with order o h2 log h 1 or o h3 2 in the l2 norm respectively this holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite the proof does not use an asymptotic error expansion but sobolev space techniques

wavelet modeling of contour deformations in sobolev spaces for fitting and tracking applications abstract this paper proposes a new model for contour deformations using wavelets this model uses sobolev spaces to control the smoothness of the contour deformation this formulation defines a probabilistic model that induces a prior distribution for contour deformation based on this distribution the fitting problem is solved in bayesian terms the deformation model is also used to generate a prior dynamic model for contour evolution in time this probabilistic model is then applied to solve the tracking problem computational results for several real image problems are given for both the kalman and condensation filters

multi index monte carlo when sparsity meets sampling we propose and analyze a novel multi index monte carlo mimc method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients the mimc method is both a stochastic version of the combination technique introduced by zenger griebel and collaborators and an extension of the multilevel monte carlo mlmc method first described by heinrich and giles inspired by giles s seminal work we use in mimc high order mixed differences instead of using first order differences as in mlmc to reduce the variance of the hierarchical differences dramatically this in turn yields new and improved complexity results which are natural generalizations of giles s mlmc analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence mathcal o mathrm tol 2 o tol 2 moreover in mimc the rate of increase of required memory with respect to mathrm tol tol is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using mlmc we motivate the setting of mimc by first focusing on a simple full tensor index set we then propose a systematic construction of optimal sets of indices for mimc based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance under standard assumptions on the convergence rates of the weak error variance and work per sample the optimal index set turns out to be the total degree type in some cases using optimal index sets mimc achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets we also show the asymptotic normality of the statistical error in the resulting mimc estimator and justify in this way our error estimate which allows both the required accuracy and the confidence level in our computational results to be prescribed finally we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of mimc

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed