Approximation In Hilbert Space Research Articles

On the linear convergence of circumcentered isometry methods

The circumcentered Douglas–Rachford method (C–DRM), introduced by Behling, Bello Cruz and Santos, iterates by taking the circumcenter of associated successive reflections. It is an acceleration of the well-known Douglas-Rachford method (DRM) for finding the best approximation onto the intersection of finitely many affine subspaces. Inspired by the C–DRM, we introduced the more flexible circumcentered reflection method (CRM) and circumcentered isometry method (CIM). The CIM essentially chooses the closest point to the solution among all of the points in an associated affine hull as its iterate and is a generalization of the CRM. The circumcentered–reflection method introduced by Behling, Bello Cruz and Santos to generalize the C–DRM is a special class of our CRM. We consider the CIM induced by a set of finitely many isometries for finding the best approximation onto the intersection of fixed point sets of the isometries which turns out to be an intersection of finitely many affine subspaces. We extend our previous linear convergence results on CRMs in finite-dimensional spaces from reflections to isometries. In order to better accelerate the symmetric method of alternating projections (MAP), the accelerated symmetric MAP first applies another operator to the initial point. (Similarly, to accelerate the DRM, the C–DRM first applies another operator to the initial point as well.) Motivated by these facts, we show results on the linear convergence of CIMs in Hilbert spaces with first applying another operator to the initial point. In particular, under some restrictions, our results imply that some CRMs attain the known linear convergence rate of the accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs converging to the best approximation in Hilbert spaces with a convergence rate no worse than the sharp convergence rate of MAP. The fact that some CRMs attain the linear convergence rate of MAP or accelerated symmetric MAP is entirely new.

A Characterization of Polynomial Density on Curves via Matrix Algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

Rescaled Pure Greedy Algorithm for convex optimization

We suggest a new greedy strategy for convex optimization in Banach spaces and prove its convergence rates under a suitable behavior of the modulus of uniform smoothness of the objective function. We show that this algorithm is a generalization of the recently discovered Rescaled Pure Greedy Algorithm for approximation in Hilbert spaces.

Breaking the curse for uniform approximation in Hilbert spaces via Monte Carlo methods

We study the L∞-approximation of d-variate functions from Hilbert spaces via linear functionals as information. It is a common phenomenon in tractability studies that unweighted problems (with each dimension being equally important) suffer from the curse of dimensionality in the deterministic setting, that is, the number n(ϵ,d) of information needed in order to solve a problem within a given accuracy ϵ>0 grows exponentially in d. We show that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness r>1∕2, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability, namely n(ϵ,d)⪯ϵ−2d(1+logd). Similar benefits from Monte Carlo methods in terms of tractability have only been known for integration problems so far.

Optimal Monte Carlo Methods for $$L^2$$ L 2 -Approximation

We construct Monte Carlo methods for the $$L^2$$ -approximation in Hilbert spaces of multivariate functions sampling not more than n function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling n pieces of arbitrary linear information, including function values.

On the Approximation of Periodic Functions in L2 and the Values of the Widths of Certain Classes of Functions

The sharp Jackson–Stechkin inequalities are received, in which a special module of continuity Ωe m (f;t) determined by Steklov’s function is used instead the usual modulus of continuity of mth order ω m (f;t). Such generalized modulus of continuity of mth order were introduced by V.A. Abilov and F.V. Abilova. The introduced modulus of continuity found their application in the theory of polynomial approximation in Hilbert space in the works by M.Sh. Shabozov and G.A. Yusupov, S.B. Vakarchuk and V.I. Zabutnaya and others. While continuing and developing these direction for some classes of functions defined by modulus of continuity, the new values of n-widths in the Hilbert space L₂were found.

A comparison between Markov approximations and other methods for large spatial data sets

The Matérn covariance function is a popular choice for modeling dependence in spatial environmental data. Standard Matérn covariance models are, however, often computationally infeasible for large data sets. Recent results for Markov approximations of Gaussian Matérn fields based on Hilbert space approximations are extended using wavelet basis functions. Using a simulation-based study, these Markov approximations are compared with two of the most popular methods for computationally efficient model approximations, covariance tapering and the process convolution method. The methods are compared with respect to their computational properties when used for spatial prediction (kriging), and the results show that, for a given computational cost, the Markov methods have a substantial gain in accuracy compared with the other methods.

The stochastic Hopf bifurcation analysis in Brusselator system with random parameter

Hopf bifurcation in Brusselator system with random parameter is explored in this paper. Firstly according to the orthogonal polynomial approximation in Hilbert space, the Brusselator system with random parameter can be reduced into the deterministic equivalent system. Then the Hopf bifurcation in deterministic equivalent system is discussed by the mathematical analysis and the first Lyapunov coefficient method. The studies discovered that different from the deterministic system, the critical value of stochastic Hopf bifurcation is determined not only by deterministic parameters in stochastic system, but also by the intensity of random parameter. As the intensity of random parameter is increased, the critical value of stochastic Hopf bifurcation is decreased. At last theoretical results are verified by numerical simulations.

Estimates for the rate of strong approximation in Hilbert space

We obtain infinite-dimensional corollaries of our recent results. We show that the finite-dimensional results imply meaningful estimates for the accuracy of strong Gaussian approximation of sums of independent identically distributed Hilbert space-valued random vectors with finite power moments. We establish that the accuracy of approximation depends substantially on the decay rate of the sequence of eigenvalues of the covariance operator of the summands.

Application of stochastic approximation in the Hilbert space to solving the problem of detecting the instant of arrival of an object in a sequence of noisy images

Application of stochastic approximation in an infinite-dimensional Hilbert space to the problem of object detection on one observed image distorted by noise and to the problem of fast detection of object arrival in a sequence of noisy images is considered. No a priori information is assumed to be available. The observer is supposed to have two stochastically independent sequences of independent “exact” noisy images, one containing and one not containing the object that has to be (repeatedly) detected by the observer in the nearest time.

Hopf bifurcation in a kind of stochastic van der Pol system

The Hopf bifurcation of van der Pol system with random parameter is studied. Firstly according to the orthogonal polynomial approximation in Hilbert space, the van der Pol system with random parameter can be reduced into the equivalent deterministic system. Then the Hopf bifurcation can be explored by the traditional methods in deterministic bifurcation theory. After the critical point of Hopf bifurcation in stochastic van der Pol system is obtained, the influence of the random parameter on Hopf bifurcation in stochastic van der Pol system is analyzed. At last we verified these results by numerical simulations.

Non-traditional stochastic models for ocean waves

We present two flexible stochastic models for 2D and 3D ocean waves with potential to reproduce severe and non-homogeneous sea conditions. The first family consists of generalized Lagrange models for the movements of individual water particles. These models can generate crest-trough and front-back statistically asymmetric waves, with the same degree of asymmetry as measured ocean waves. They are still in the Gaussian family and it is possible to calculate different slope distributions exactly from a wave energy spectrum. The second model is a random field model that is generated by a system of nested stochastic partial differential equations. This model can be adapted to spatially non-homogeneous sea conditions and it can approximate standard wave spectra. One advantage with this model is that Hilbert space approximations can be used to obtain computationally efficient representations with Markov-type properties that facilitate the use of sparse matrix techniques in simulation and estimation.

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener–Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener–Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.

Stochastic approximation in Hilbert space and the problem of detecting an object in a sequence of noisy images

Under study is the problem of fast detection of the appearance of an object in a sequence of noisy images (using a minimal number of images once the object appears). Let the probabilistic characteristics of the images be unknown to the observer; however, two sequences of independent images are available, statistically independent of each other, containing and not containing the object of interest. In order to detect the appearance of the object it seems reasonable to use the method of cumulative summation of the values of a suitable functional of the observed images. A construction of a functional of this type is proposed by using the method of stochastic approximation in infinite dimensional Hilbert space.

Development of additive isotropic site potential for exchange-repulsion energy, based on intermolecular perturbation theory

We have developed an additive spherical site potential for exchange-repulsion energy by applying the local density approximation in Hilbert space, the local-site approximation, and the s-type auxiliary basis set to the equation derived from intermolecular perturbation theory. The method efficiently addresses the decomposition of molecular interactions derived from quantum chemistry into additive spherical site potentials, required as force field input in a statistical-mechanical, reference interaction site model (RISM and 3D-RISM), molecular theory of solvation. The present method reproduces the exchange-repulsion energy between simple molecules obtained from quantum chemical calculations.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

This paper presents a finite dimensional approach to stochastic approximation in infinite dimensional Hilbert space. The problem was motivated by applications in the field of stochastic programming wherein we minimize a convex function defined on a Hilbert space. We define a finite dimensional approximation to the Hilbert space minimizer. A justification is provided for this finite dimensional approximation. Estimates of the dimensionality needed are also provided. The algorithm presented is a two time-scale Newton-based stochastic approximation scheme that lives in this finite dimensional space. Since the finite dimensional problem can be prohibitively large dimensional, we operate our Newton scheme in a projected, randomly chosen smaller dimensional subspace.

ON SMOOTHING PROBLEMS WITH ONE ADDITIONAL EQUALITY CONDITION

Two problems of approximation in Hilbert spaces are considered with one additional equality condition: the smoothing problem with a weight and the smoothing problem with an obstacle. This condition is a generalization of the equality, which appears in the problem of approximation of a histogram in a natural way. We characterize the solutions of these smoothing problems and investigate the connection between them.

Analytic approximation with real constraints, with applications to inverse diffusion problems

The methods of constrained approximation in Hilbert spaces of analytic functions are applied to the solution of the inverse problems of detecting cracks or sources in a two-dimensional material by means of boundary measurements. Issues of well-posedness are discussed, and results on continuity and robustness with respect to the given data are established. Constructive and efficient methods for resolution of the above approximation problems are presented. The techniques are illustrated by numerical examples incorporating a further rational approximation step.

A modified model and linearization method for solid state power amplifier

In this paper we present a modified model of solid state power amplifiers. Also using the mathematical concept of best approximation in Hilbert spaces, a predistorter structure is suggested to compensate both AM/AM and AM/PM conversion characteristics of a solid state power amplifier. To verify the effects of this linearization we generally consider it in a QAM signal transmission in an additive white Gaussian noise channel and the results are compared with the ideal linear case. It is shown that the presented method is able to compensate the nonlinearity up to a good extent. The performance of proposed method has been analyzed in frequency domain through spectrum simulations. To prove the validity of presented method we have made similar considerations on continuous spectrum.

Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space—A further study

This paper discusses optimal approximation of autocorrelation functions of teletraffic series by introducing a generalization of autocorrelation function form of fractional Gaussian noise (FGN). The demonstrations with real-traffic series are given.