Hilbert Norm Research Articles

On the stability of nonlocal difference schemes in subspaces

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions. The original problem, as well as the approximating finite-difference schemes, is not stable under perturbations of the initial data. For such problems, we introduce the definition of stability of a finite-difference scheme in a subspace and derive estimates in terms of Hilbert norms expressing the stability with respect to the initial data in subspaces spanned by eigenvectors of the main difference operator.

Precise small deviations in L 2 of some Gaussian processes appearing in the regression context

Abstract We find precise small deviation asymptotics with respect to the Hilbert norm for some special Gaussian processes connected to two regression schemes studied by MacNeill and his coauthors. In addition, we also obtain precise small deviation asymptotics for the detrended Brownian motion and detrended Slepian process.

Study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge

A study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge is presented in the case of the gauge group SU(2) and for different Euclidean space–time dimensions. Explicit examples of classes of normalizable zero modes and corresponding gauge field configurations are constructed by taking into account two boundary conditions, namely: (i) the finite Euclidean Yang–Mills action, (ii) the finite Hilbert norm.

New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Models of Parabolic PDEs

The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data. Furthermore, the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces $ H $ and $ V $, with $ V \subset H $, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches of Kunisch and Volkwein [SIAM J. Numer. Anal., 40 (2002), pp. 492--515] and Chapelle, Gariah, an...

Exact Small Deviation Asymptotics for Some Brownian Functionals

We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

Hilbert space of symmetric functions on ℓ1

We consider a completion of a space of symmetric polynomials on l1 with respect to a certain Hilbert norm. It is shown that this completion consists of analytic functions on a unit ball in l1.

APPLICATIONS OF HILBERT SPACE DISSIPATIVE NORM

The concept of Hilbert space dissipative norm was introduced in [8] to obtain necessary and sufficient conditions for exponential stability of contraction semigroups. In the present paper we show that the same concept can also be used to derive further properties of contraction semigroups, as well as to characterize strongly stable semigroups that are not exponentially stable.

An operator-difference scheme for abstract Cauchy problems

An abstract Cauchy problem for second-order hyperbolic differential equations containing the unbounded self-adjoint positive linear operator A ( t ) with domain in an arbitrary Hilbert space is considered. A new second-order difference scheme, generated by integer powers of A ( t ) , is developed. The stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are established in Hilbert norms with respect to space variable. To support the theoretical statements for the solution of this difference scheme, the numerical results for the solution of one-dimensional wave equation with variable coefficients are presented.

Diffusion phenomenon in Hilbert spaces and applications

We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.

Locally Parallel Texture Modeling

This article presents a new adaptive framework for locally parallel texture modeling. Oscillating patterns are modeled with functionals that constrain the local Fourier decomposition of the texture. We first introduce a texture functional which is a weighted Hilbert norm. The weights on the local Fourier atoms are optimized to match the local orientation and frequency of the texture. This adaptive model is used to solve image processing inverse problems, such as image decomposition and inpainting. The local orientation and frequency of the texture component are adaptively estimated during the minimization process. To improve inpainting performances over large missing regions, we introduce a highly nonconvex generalization of our texture model. This new model constrains the amplitude of the texture and allows one to impose an arbitrary oscillation profile. Numerical results illustrate the effectiveness of the method.

Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm

We obtain results on small deviations of Bogoliubov’s Gaussian measure occurring in the theory of the statistical equilibrium of quantum systems. For some random processes related to Bogoliubov processes, we find the exact asymptotic probability of their small deviations with respect to a Hilbert norm.

Semi-reproducing kernel Hilbert spaces, splines and increment kriging

A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions on an index set E, say, for which function evaluation is continuous in the Hilbert norm. One of the classic applications of such spaces is to show that the solution of a certain penalised least squares problem is given by a smoothing spline. However, the formulation of this problem in an RKHS setting involves several arbitrary choices. In this paper, we propose the use of a semi-RKHS (SRKHS), which provides a more natural setting for the smoothing spline solution. In addition, a systematic study is made of the properties of an SRKHS. It is well known that there is a one-to-one correspondence between an RKHS and positive semi-definite (psd) function on E, which in turn can be viewed as the covariance function of a stochastic process on E. In this paper, we extend this result to show that there is a one-to-one correspondence between an SRKHS and conditionally positive semi-definite (cpsd) function on E, which in turn defines the covariance behaviour of certain increments on E. Further, it is shown how optimal smoothing in the functional setting corresponds to optimal prediction in the stochastic process setting.

A review on consistency and robustness properties of support vector machines for heavy-tailed distributions

Support vector machines (SVMs) belong to the class of modern statistical machine learning techniques and can be described as M-estimators with a Hilbert norm regularization term for functions. SVMs are consistent and robust for classification and regression purposes if based on a Lipschitz continuous loss and a bounded continuous kernel with a dense reproducing kernel Hilbert space. For regression, one of the conditions used is that the output variable Y has a finite first absolute moment. This assumption, however, excludes heavy-tailed distributions. Recently, the applicability of SVMs was enlarged to these distributions by considering shifted loss functions. In this review paper, we briefly describe the approach of SVMs based on shifted loss functions and list some properties of such SVMs. Then, we prove that SVMs based on a bounded continuous kernel and on a convex and Lipschitz continuous, but not necessarily differentiable, shifted loss function have a bounded Bouligand influence function for all distributions, even for heavy-tailed distributions including extreme value distributions and Cauchy distributions. SVMs are thus robust in this sense. Our result covers the important loss functions \({\epsilon}\) -insensitive for regression and pinball for quantile regression, which were not covered by earlier results on the influence function. We demonstrate the usefulness of SVMs even for heavy-tailed distributions by applying SVMs to a simulated data set with Cauchy errors and to a data set of large fire insurance claims of Copenhagen Re.

A Stochastic Poisson Structure

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

Heat flow on Finsler manifolds

AbstractThis paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → ℝ+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L2(M, m) for the energy as gradient flow on the reverse L2‐Wasserstein space 𝒫2(M) of probability measures on M for the relative entropy Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 𝒞1, α regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be 𝒞2. Moreover, we derive pointwise comparison results à la Cheeger‐Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc.

Small deviations for Matérn random fields and processes in the Hilbert norm

Small deviations for Matérn random fields and processes in the Hilbert norm

A general multipurpose interpolation procedure: the magic points

Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-defined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.

Consistency of the Group Lasso and Multiple Kernel Learning

We consider the least-square regression problem with regularization by a block l1-norm, that is, a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the l1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic group selection consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model mis specification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covar iance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.

ACCELERATED FAILURE TIME MODELS WITH NONLINEAR COVARIATES EFFECTS

SummaryAs a flexible alternative to the Cox model, the accelerated failure time (AFT) model assumes that the event time of interest depends on the covariates through a regression function. The AFT model with non‐parametric covariate effects is investigated, when variable selection is desired along with estimation. Formulated in the framework of the smoothing spline analysis of variance model, the proposed method based on the Stute estimate (Stute, 1993[Consistent estimation under random censorship when covariables are present, J. Multivariate Anal.45, 89–103]) can achieve a sparse representation of the functional decomposition, by utilizing a reproducing kernel Hilbert norm penalty. Computational algorithms and theoretical properties of the proposed method are investigated. The finite sample size performance of the proposed approach is assessed via simulation studies. The primary biliary cirrhosis data is analyzed for demonstration.

Exact small deviation asymptotics for the Slepian and Watson processes in the Hilbert norm

We find the exact asymptotic behavior of small ball probabilities in the Hilbert norm for the simplest form of the Slepian process and for the Watson process appearing in nonparametric statistics. Bibliography: 23 titles.