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On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

JOURNAL OF COMPLEXITY 2022年 68卷

作者： Ugulava, Duglas Zarnadze, David Georgian Tech Univ Niko Muskhelishvili Inst Computat Math Atlanta GA USA

The equation Au = f with a linear symmetric positive definite operator A : D(A) subset of H -> H having a discrete spectrum and dense image in a complex Hilbert space H is considered. This equation is transferred into the Hilbert space of finite orbits D(A(n)) as well as into the Frechet space of all orbits D(A(infinity)), that is, the projective limit of the sequence of spaces {D(A(n))}. For an approximate solution of the inverse of A, linear spline central algorithms in these spaces are constructed. The convergence of the sequence of approximate solutions to the exact solution is proved. The obtained results are applied to the quantum harmonic oscillator operator Au (t) = -u ''(t) + t(2)u(t), t is an element of R, in the Hilbert space of finite orbits D(A(n)), and in the Frechet space of all orbits D(A(infinity)) that in this case coincides with the Schwartz space of rapidly decreasing functions. Some quantum mechanical interpretations of obtained results are also given. (C) 2021 Published by Elsevier Inc.

关键词： Spline algorithm central algorithm Discrete spectrum Space of all orbits Quantum harmonic oscillator Hamiltonian

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Convergence analysis of central and minimax algorithms in scalar regressor models

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MATHEMATICS OF CONTROL SIGNALS AND SYSTEMS 2006年第1期18卷 66-99页

作者： Akçay, H At, N Anadolu Univ Dept Elect & Elect Engn TR-26470 Eskisehir Turkey

In this paper, the estimation of a scalar parameter is considered with given lower and upper bounds of the scalar regressor. We derive non-asymptotic, lower and upper bounds on the convergence rates of the parameter estimate variances of the central and the minimax algorithms for noise probability density functions characterized by a thin tail distribution. This presents an extension of the previous work for constant scalar regressors to arbitrary scalar regressors with magnitude constraints. We expect our results to stimulate further research interests in the statistical analysis of these set-based estimators when the unknown parameter is multi-dimensional and the probability distribution function of the noise is more general than the present setup.

关键词： system identification membership-set central algorithm minimax algorithm Chebyshev center parameter estimate variance convergence analysis

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Interpolation of Lipschitz functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2006年第1期196卷 20-44页

作者： Beliakov, Gleb Deakin Univ Sch Informat Technol Burwood NSW 3125 Australia

This paper describes a new computational approach to multivariate scattered data interpolation. It is assumed that the data is generated by a Lipschitz continuous function f. The proposed approach uses the central interpolation scheme, which produces an optimal interpolant in the worst case scenario. It provides best uniform error bounds on f, and thus translates into reliable learning off. This paper develops a computationally efficient algorithm for evaluating the interpolant in the multivariate case. We compare the proposed method with the radial basis functions and natural neighbor interpolation, provide the details of the algorithm and illustrate it on numerical experiments. The efficiency of this method surpasses alternative interpolation methods for scattered data. (c) 2005 Elsevier B.V. All rights reserved.

关键词： scattered data interpolation Lipschitz approximation optimal interpolation central algorithm multivariate approximation

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The Iyengar inequality, revisited

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BIT NUMERICAL MATHEMATICS 2007年第4期47卷 839-852页

作者： Wang, X. H. Xie, C. C. Yang, S. J. Zhejiang Univ Dept Math Hangzhou 310028 Peoples R China Hangzhou Normal Coll Dept Math Hangzhou 310036 Peoples R China

Detailed analysis shows that the famous Iyengar inequality actually says that the Trapezoidal formula is a central algorithm for approximating integrals over an appropriate interval for the class of functions whose derivatives are bounded by a positive number K in L infinity-sense. The inherent nonlinearity from central algorithms reflects the importance of the Iyengar inequality and thus makes familiar linear methods malfunction when one tries to generalize it. It is shown that the generalization depends on a nonlinear system of equations satisfied by a set of free nodes of a perfect spline. Explicit constructions are obtained in the spirit of the Iyengar inequality for the class of functions whose rth (r <= 4) derivatives are bounded by a positive number K in L infinity-sense because a closed solution to the nonlinear system is only available for r <= 4. Connections with computational mathematics, especially with best interpolation and best quadrature, are discussed. Numerical experiments are also included.

关键词： Iyengar inequality central algorithm perfect spline best quadrature best interpolation

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Monotonicity preserving approximation of multivariate scattered data

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BIT NUMERICAL MATHEMATICS 2005年第4期45卷 653-677页

作者： Beliakov, G Deakin Univ Sch Informat Technol Burwood 3125 Australia

This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.

关键词： monotone approximation isotone approximation scattered data central algorithm optimal approximation

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CONVERGENCE ANALYSIS OF central AND MINIMAX algorithmS IN SCALAR REGRESSOR MODELS

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IFAC Proceedings Volumes 2006年第1期39卷 594-599页

作者： Hüseyin Akçay Nuray At Department of Electrical and Electronics Engineering Anadolu University 26470 Eskisehir Turkey

In this paper, estimation of a scalar parameter is considered with given lower and upper bounds of the scalar regressor. We derive non-asymptotic, lower and upper bounds on the convergence rates of the parameter estimate variances for noise probability density functions charecterized by a thin tail distribution. This presents an extension of the previous work for constant scalar regressors to arbitrary scalar regressors with magnitude constraints. We expect our results to stimulate further research interests in the statistical analysis of these set-based estimators when the unknown parameter is multi-dimensional and the probability distribution function of the noise is more general than the present setup.

关键词： System identification Membership set central algorithm Minimax algorithm Chebyshev center Parameter estimate variance Convergence analysis

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