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Approximation by quasi-interpolation operators and smolyak's algorithm

JOURNAL OF COMPLEXITY 2022年 69卷

作者： Kolomoitsev, Yurii Univ Lubeck Inst Math Ratzeburger Allee 160 D-23562 Lubeck Germany NAS Ukraine Inst Appl Math & Mech Gen Batyuk Str 19 UA-84116 Slovyansk Donetsk Region Ukraine

We study approximation of multivariate periodic functions from Besov and Triebel-Lizorkin spaces of dominating mixed smoothness by the smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding smolyak algorithm in the Lq-norm for functions from the Besov spaces Bsp,theta(Td) and the Triebel-Lizorkin spaces Fsp,theta(Td) for all s > 0 and admissible 1 < p, theta < infinity as well as provide analogues of the Littlewood- Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators. (c) 2021 Elsevier Inc. All rights reserved.

关键词： smolyak algorithm Quasi-interpolation operators Kantorovich operators Besov-Triebel-Lizorkin spaces of mixed smoothness Error estimates Littlewood-Paley-type characterizations

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On weak tractability of the smolyak algorithm for approximation problems

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JOURNAL OF APPROXIMATION THEORY 2015年 192卷 347-361页

作者： Xu, Guiqiao Tianjin Normal Univ Dept Math Tianjin 300387 Peoples R China

We consider the problems of L-P-approximation of d-variate analytic functions defined on the cube with directional derivatives of all orders bounded by 1. For 1 <= p < infinity, it is shown that the smolyak algorithm based on polynomial interpolation at the extrema of the Chebyshev polynomials leads to weak tractability of these problems. This gives an affirmative answer to one of the open problems raised recently by Hinrichs et al. (2014). Our proof uses the polynomial exactness of the algorithm and an explicit bound on the operator norm of the algorithm. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Weak tractability smolyak algorithm Infinitely differentiable function class Standard information

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Infinite-dimensional integration and L2-approximation on Hermite spaces

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JOURNAL OF APPROXIMATION THEORY 2024年 300卷

作者： Gnewuch, M. Hinrichs, A. Ritter, K. Ruessmann, R. Univ Osnabruck Inst Math Albrechtstr 28A D-49076 Osnabruck Germany Johannes Kepler Univ Linz Inst Anal Altenberger Str 69 A-4040 Linz Austria Rheinland Pfalz Tech Univ Kaiserslautern Landau Fachbereich Math Postfach 3049 D-67653 Kaiserslautern Germany

We study integration and L2-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space RN. We establish upper and lower bounds for the minimal worst case errors under general assumptions;these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods. (c) 2024 Elsevier Inc. All rights reserved.

关键词： Infinite-variate numerical problems Hermite kernels Embedding theorems Countable tensor products of Hilbert spaces smolyak algorithm Multivariate decomposition method

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Randomized sparse grid algorithms for multivariate integration on Haar wavelet spaces

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IMA JOURNAL OF NUMERICAL ANALYSIS 2023年第1期43卷 73-98页

作者： Wnuk, M. Gnewuch, M. Univ Osnabruck Dept Math Albrechtstr 28a D-49076 Osnabruck Germany

The deterministic sparse grid method, also known as smolyak's method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about randomized versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the D-dimensional unit cube [0, 1)(D). Let d, s is an element of N be such that D = d . s. The s-dimensional building blocks of the sparse grid quadratures are based on stratified sampling for s = 1 and on scrambled (0, m, s)-nets for s >= 2. The spaces of integrands and the error criterion we consider are Haar wavelet spaces with parameter alpha and the randomized error (i.e., the worst case root mean square error), respectively. We prove sharp (i.e., matching) upper and lower bounds for the convergence rates of the Nth minimal errors for all possible combinations of the parameters d and s. Our upper error bounds still hold if we consider as spaces of integrands Sobolev spaces of mixed dominated smoothness with smoothness parameters 1/2 < alpha < 1 instead of Haar wavelet spaces.

关键词： smolyak algorithm multivariate integration Haar function spaces quasi-Monte Carlo (0,m,s)-nets randomized algorithms

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Sparse Grid Approximation in Weighted Wiener Spaces

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2023年第2期29卷 19-19页

作者： Kolomoitsev, Yurii Lomako, Tetiana Tikhonov, Sergey Gottingen Univ Inst Numer & Appl Math Lotzestr 16-18 D-37083 Gottingen Germany Ctr Recerca Matemat Campus BellaterraEdifici C Barcelona 08193 Spain ICREA Pg Lluis Co 23 Barcelona 08010 Spain Univ Autonoma Barcelona Bellaterra Barcelona 08193 Spain

We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grid methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grid methods in weighted Wiener norms as well as analogues of the Littlewood-Paley-type characterizations in terms of families of quasi-interpolation operators.

关键词： Sparse grid Weighted Wiener spaces Quasi-interpolation operators Kantorovich operators smolyak algorithm Littlewood-Paley-type characterizations

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Weighted Integral of Infinitely Differentiable Multivariate Functions is Exponentially Convergent

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 98-114页

作者： Guiqiao Xu Yongping Liu Jie Zhang Department of Mathematics Tianjin Normal UniversityTianjin300387PRC Department of Mathematics Beijing Normal UniversityBeijing100875PRC

We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by *** consider the algorithms that use finitely many function values as information(called standard information).On the one hand,we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature *** the other hand,by using the smolyak al-gorithm with the above interpolatory quadratures,we proved that the weighted integral problem is of exponential convergence in the worst case setting.

关键词： smolyak algorithm infinitely differentiable function class standard information worst case setting

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Analysis and Design of Robust Control for Linear Parameter-Varying Systems

Analysis and Design of Robust Control for Linear Parameter-V...

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作者： Hsu, Shao-Chen Texas A&M University

学位级别：Ph.D., Doctor of Philosophy

Gain-scheduling approach is a powerful tool but it only guarantees the local stability and performance for a slow varying system. Linear parameter varying (LPV) systems hence were developed to overcome this drawback. The LPV system is a linear system with parameter-dependent system matrices, which can be formulated from a nonlinear system via either approximation or function substitution. Three major control design methods includes linear fractional transformation, polytopic system design and gridding approach. All methods results in a convex optimization with either parameter-dependent or parameter-independent linear matrix inequalities (LMIs) and some conservatism may be introduced. Gridding based approach is the main focus in this research because it has no further assumptions about the structure and hence admits less conservatism. However, the number of samples for gridding approach grows up exponentially as the dimension of the problem increases. This drawback hence inspires the approach developed in this research. Several stability and performance conditions are introduced in this research and all controller syntheses arrive at optimization problems with parameter-dependent LMIs. Hence the objective of this research is to solve these problems. We present two methodologies to handle with generic LPV control systems. The first approach is to consider the problem in a stochastic framework so that the stability and performance are guaranteed in the stochastic sense. Two algorithms, i.e. polynomial chaos expansion and stochastic collocation, are used to formulate the convex optimization problems. The other method is to directly interpolate the parameter-dependent LMIs by sparse grid with smolyak algorithm, which extremely reduces the amount of the sample points and successfully solve the infinite-dimensional optimization by the proposed algorithm. Two examples are shown to compare two proposed controllers with existing methods, where the benefits of the method we d

关键词： Linear parameter-varying systems LPV control Polynomial chaos theory smolyak algorithm Convex optimizations Tensegrity systems

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Collocation methods for fuzzy uncertainty propagation in heat conduction problem

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INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER 2017年 107卷 631-639页

作者： Wang, Chong Qiu, Zhiping Xu, Menghui Beihang Univ Inst Solid Mech Beijing 100191 Peoples R China Ningbo Univ Fac Mech Engn & Mech Ningbo 315211 Zhejiang Peoples R China

Based on the combination of collocation technology and fuzzy theory, this paper proposes a full grid fuzzy collocation method (FGFCM) and a sparse grid fuzzy collocation method (SGFCM) for fuzzy uncertainty propagation in heat conduction problem. Converting fuzzy parameters into interval variables by level cut strategy, the Legendre polynomial series provides a surrogate function for temperature response. To calculate the expansion coefficients, FGFCM evaluates the deterministic solutions directly on the full tensor product grids, whereas smolyak algorithm is introduced in SGFCM to reduce the number of collocation points. According to the smoothness property of surrogate function and fuzzy decomposition theorem, the interval bounds and membership functions of uncertain temperature response are derived, respectively. Comparing result with traditional Monte Carlo simulation and parameter perturbation method, two numerical examples evidence the remarkable accuracy and effectiveness of proposed methods for fuzzy temperature field prediction in engineering. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Fuzzy uncertainty propagation Heat conduction Legendre polynomial series Collocation technology smolyak algorithm

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Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood-Paley type characterizations

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ANALYSIS MATHEMATICA 2017年第2期43卷 133-191页

作者： Byrenheid, G. Ullrich, T. Hausdorff Ctr Math Inst Numer Simulat Endenicher Allee 62 D-53115 Bonn Germany

In this paper we consider the L (q) -approximation of multivariate periodic functions f with L (q) -bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of n sampling nodes. The general performance is measured in terms of (non-)linear sampling widths I +/- (n) . We conduct a systematic analysis of smolyak type interpolation algorithms in the framework of Besov-Lizorkin-Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood-Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case 2 <= p < q < a and r > 1/p the linear sampling widths I +/- (n) (lin) (S (p) (r) W(T (d) ), L (q) (T (d) )) and I +/- (n) (lin) (S (p,a) (r) B(T (d) ), L (q) (T (d) )) show the asymptotic behavior of the corresponding Gelfand n-widths, whereas in case 1 < p < q <= 2 and r > 1/p the linear sampling widths match the corresponding linear widths. In the mentioned cases linear smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.

关键词： sampling recovery sparse grid sampling representation Besov-Triebel-Lizorkin space of mixed smoothness smolyak algorithm Gelfand n-width linear width

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Weighted smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2011年第11期85卷 1365-1389页

作者： Agarwal, Nitin Aluru, N. R. Univ Illinois Dept Mech Sci & Engn Beckman Inst Adv Sci & Technol Urbana IL 61801 USA

This paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non-uniform probability measures. Recently, there has been a growing interest in the stochastic collocation approach, which seeks to approximate the unknown stochastic solution using polynomial interpolation in the multi-dimensional random domain. Existing approaches employ sparse grid interpolation based on the smolyak algorithm, which leads to orders of magnitude reduction in the number of support nodes as compared with usual tensor product. However, such sparse grid interpolation approaches based on piecewise linear interpolation employ uniformly sampled nodes from the random domain and do not take into account the probability measures during the construction of the sparse grids. Such a construction based on uniform sparse grids may not be ideal, especially for highly skewed or localized probability measures. To this end, this work proposes a weighted smolyak algorithm based on piecewise linear basis functions, which incorporates information regarding non-uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： smolyak algorithm sparse grid non-uniform distributions stochastic collocation method stochastic differential equations uncertainty quantification

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