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Optimized refinable enclosures of multivariate polynomial pieces

COMPUTER AIDED GEOMETRIC DESIGN 2001年第9期18卷 851-863页

作者： Lutterkort, D Peters, J Univ Florida Dept Comp Sci & Informat Engn CISE Gainesville FL 32611 USA

An enclosure is a two-sided approximation of a uni- or multivariate function b is an element of B by a pair of typically simpler functions b(+), b(-) is an element of H not equal B such that b(-) less than or equal to b less than or equal to b(+) over the domain U of interest. Enclosures are optimized by minimizing the width max(U) b(+) - b(-) and refined by enlarging the space R. This paper develops a framework for efficiently computing enclosures for multivariate polynomials and, in particular, derives piecewise bilinear enclosures for bivariate polynomials in tensor-product Bezier form. Runtime computation of enclosures consists of looking up s < dim B pre-optimized enclosures and linearly combining them with the second differences of b. The width of these enclosures scales by a factor 1/4 under midpoint subdivision. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： refinable enclosures multivariate functions efficiently computable polynomials control net Bezier B-spline

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Hybrid high dimensional model representation (HHDMR) on the partitioned data

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2006年第1期185卷 107-132页

作者： Tunga, MA Demiralp, M Isik Univ Fac Engn Dept Comp Sci & Engn TR-34398 Istanbul Turkey Istanbul Tech Univ Inst Informat Computat Sci & Engn Program TR-34469 Istanbul Turkey

A multivariate interpolation problem is generally constructed for appropriate determination of a multivariate function whose values are given at a finite number of nodes of a multivariate grid. One way to construct the solution of this problem is to partition the given multivariate data into low-variate data. High dimensional model representation (HDMR) and generalized high dimensional model representation (GHDMR) methods are used to make this partitioning. Using the components of the HDMR or the GHDMR expansions the multivariate data can be partitioned. When a cartesian product set in the space of the independent variables is given, the HDMR expansion is used. On the other band, if the nodes are the elements of a random discrete data the GHDMR expansion is used instead of HDMR. These two expansions work well for the multivariate data that have the additive nature. If the data have multiplicative nature then factorized high dimensional model representation (FHDMR) is used. But in most cases the nature of the given multivariate data and the sought multivariate function have neither additive nor multiplicative nature. They have a hybrid nature. So, a new method is developed to obtain better results and it is called hybrid high dimensional model representation (HHDMR). This new expansion includes both the HDMR (or GHDMR) and the FHDMR expansions through a hybridity parameter. In this work, the general structure of this hybrid expansion is given. It has tried to obtain the best value for the hybridity parameter. According to this value the analytical structure of the sought multivariate function can be determined via HHDMR. (c) 2005 Elsevier B.V. All rights reserved.

关键词： high dimensional model representation factorized high dimensional model representation multivariate functions interpolation multidimensional problems approximation optimization

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An Efficient and Accurate Method for the Identification of the Most Influential Random Parameters Appearing in the Input Data for PDEs

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SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION 2014年第1期2卷 82-105页

作者： Labovsky, A. Gunzburger, Max Florida State Univ Dept Sci Comp Tallahassee FL 32309 USA

Cut-HDMR expansions (also referred to as anchored-ANOVA expansions) have often been used to represent multivariate functions in high dimensions because they can be used to identify unimportant variables. Past efforts in this direction have examined only the separate influence of each variable. However, simple examples show that variables that have small separate influences can have large interactions, and thus those variables should not be ignored. In this paper, a methodology is developed for determining the importance of variables by examining both their separate and pairwise effects. As a result, not only are all unimportant terms omitted from the cut-HDMR expansion, but all important terms are retained. This is in contrast to existing methods that can omit important pairwise interaction terms. The application and effectiveness of the new methodology are demonstrated for a nonlinear system of partial differential equations having random inputs;specifically, we consider a magnetohydrodynamics setting which also serves to illustrate that a realistic problem can indeed involve random input parameters that have small individual influences but large pairwise influences. In such settings, the new method is not only computationally attractive, but it is the only method that can correctly identify all important individual effects and pairwise interactions. Also discussed is a possible further application of the new methodology, namely reducing the cost of sparse-grid approximations of quantities of interest that depend on the solution of a partial differential equation.

关键词： cut-HDMR expansions important interactions analysis of variance important inputs partial differential equations multivariate functions magnetohydrodynamics

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multivariate RATIONAL INTERPOLATION

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COMPUTING 1985年第1期34卷 41-61页

作者： CUYT, AAM VERDONK, BM Department of Mathematics and Computer Science Universiteit Antwerpen (UIA) Wilrijk Belgium

Many papers have already been published on the subject of multivariate polynomial interpolation and also on the subject of multivariate Padé approximation. But the problem of multivariate rational interpolation has only very recently been considered; we refer among others to [8] and [3]. The computation of a univariate rational interpolant can be done in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, or start a recursive algorithm, or calculate the convergent of a continued fraction. In this paper we will generalize each of those methods from the univariate to the multivariate case. Although the generalization is simple, the equivalence of the computational methods is completely lost in the multivariate case. This was to be expected since various authors have already remarked [2,7] that there is no link between multivariate Padé approximants calculated by matching the Taylor series and those obtained as convergents of a continued fraction.

关键词： AMS Subject Classifications: 41A05, 41A20, 41A63 branched continued fractions multivariate functions multivariate inverse differences multivariate Padé approximants rational interpolation recursive calculation of interpolants

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A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid

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APPLIED MATHEMATICS AND COMPUTATION 2005年第3期164卷 865-883页

作者： Tunga, MA Demiralp, M Isik Univ Fac Engn Dept Comp Sci & Engn TR-34398 Istanbul Turkey Tech Univ Istanbul Inst Informat Comp Sci & Engn Program TR-34469 Istanbul Turkey

When the values of a multivariate function f(x(1),...,x(N)), having N independent variables like x(1),...,x(N) are given at the nodes of a cartesian, product set in the space of the independent variables and ail interpolation problem is defined to find out the analytical structure of this function some difficulties arise in the standard methods due to the multidimensionality of the problem. Here, the main purpose is to partition this multivariate data into low-variate data and to obtain the analytical structure of the multivariate function by using this partitioned data. High dimensional model representation (HDMR) is used for these types of problems. However, if HDMR requires all components, which means 2(N) number of components, to get a desired accuracy then factorized high dimensional model representation (FHDMR) can be used. This method uses the components of HDMR. This representation is needed when the sought multivariate function has a multiplicative nature. In this work we introduce how to utilize FHDMR for these problems and present illustrative examples. (c) 2004 Elsevier Inc. All rights reserved.

关键词： high dimensional model representation multivariate functions interpolation multidimensional problems approximation

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Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation

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JOURNAL OF MATHEMATICAL CHEMISTRY 2012年第8期50卷 2223-2238页

作者： Tunga, Burcu Demiralp, Metin Istanbul Tech Univ Fac Comp & Informat TR-34469 Istanbul Turkey Istanbul Tech Univ Computat Sci & Engn Program Inst Informat TR-34469 Istanbul Turkey

High Dimensional Model Representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on Hybrid HDMR which is composed of Plain HDMR and Logarithmic HDMR. The Plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the Logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the Hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of Plain and Logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness Approximation Theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.

关键词： High dimensional model representation multivariate functions Approximation by polynomials Fluctuation expansion Optimization

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Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation

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JOURNAL OF MATHEMATICAL CHEMISTRY 2011年第4期49卷 894-909页

作者： Tunga, Burcu Demiralp, Metin Istanbul Tech Univ Inst Informat Comp Sci & Engn Program Grp Sci & Methods Computat Maslak TR-34469 Istanbul Turkey

This paper focuses on the Logarithmic High Dimensional Model Representation (Logarithmic HDMR) method which is a divide-and-conquer algorithm developed for multivariate function representation in terms of less-variate functions to reduce both the mathematical and the computational complexities. The main purpose of this work is to bypass the evaluation of N-tuple integrations appearing in Logarithmic HDMR by using the features of a new theorem named as Fluctuationlessness Approximation Theorem. This theorem can be used to evaluate the complicated integral structures of any scientific problem whose values can not be easily obtained analytically and it brings an approximation to the values of these integrals with the help of the matrix representation of functions. The Fluctuation Free multivariate Integration Based Logarithmic HDMR method gives us the ability of reducing the complexity of the scientific problems of chemistry, physics, mathematics and engineering. A number of numerical implementations are also given at the end of the paper to show the performance of this new method.

关键词： High dimensional model representation multivariate functions Approximation

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Constructive Representation of functions in Low-Rank Tensor Formats

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CONSTRUCTIVE APPROXIMATION 2013年第1期37卷 1-18页

作者： Oseledets, I. V. Russian Acad Sci Inst Numer Math Moscow Russia

In this paper, we obtain explicit representations of several multivariate functions in the Tensor Train (TT) format and explicit TT-representations of tensors that stem from the tensorization of univariate functions on grids. Previous results contained only estimates on the number of parameters (tensor ranks), and this paper fills this gap by providing explicit low-parametric representations for these functions and tensors.

关键词： TT-format QTT-format Tensor decompositions multivariate functions Explicit representations

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The multivariate De Pril transform

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INSURANCE MATHEMATICS & ECONOMICS 2000年第1期27卷 123-136页

作者： Sundt, B Univ Bergen Bergen Norway Univ Melbourne Melbourne Vic Australia

Ln the present paper we extend the definition of the De Pril transform to a class of multivariate functions and discuss various properties of this multivariate De Pril transform. In particular, we show that, like in the univariate case, it is additive for convolutions, and discuss De Pril transforms of compound functions and higher order functions. Finally, we introduce a multivariate Dhaene-De Pril transform, which we compare with the De Pril transform. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： multivariate functions recursions De Pril transforms convolutions compound distributions

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Approximation in the extended functional tensor train format

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2024年第3期50卷 1-28页

作者： Strossner, Christoph Sun, Bonan Kressner, Daniel Ecole Polytech Fed Lausanne EPFL Inst Math CH-1015 Lausanne Switzerland

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96 % \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$96\%$$\end{document} compared to the algorithm from Gorodetsky et al. (Comput. Methods Appl. Mech. Eng. 347, 59-84 2019).

关键词： multivariate functions Polynomial approximation Tensors Low-rank approximation

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