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High order approximation to non-smooth multivariate functions

COMPUTER AIDED GEOMETRIC DESIGN 2018年 63卷 31-65页

作者： Amir, Anat Levin, David School of Mathematical Sciences Tel Aviv University Ramat Aviv Tel Aviv 6997801 Israel

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r(+) where g, r is an element of CM+1 (R-n) and the function r(+) is defined by r(+)(Y) = {r(y), r(y) >= 0 , for all y is an element of R-n. 0, r(y) < 0 Given scattered (or uniform) data points X subset of R-n, we investigate approximation by quasi- interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Quasi-interpolation multivariate functions Non-smooth functions

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Constancy maximization based weight optimization in high dimensional model representation for multivariate functions

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JOURNAL OF MATHEMATICAL CHEMISTRY 2011年第9期49卷 1996-2012页

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

High Dimensional Model Representation (HDMR) method is a technique that represents a multivariate function in terms of less-variate functions. Even though the method has a finite expansion, to determine the components of this expansion is very expensive due to integration based natures of the components. Hence, the HDMR expansion is generally truncated at certain multivariance level and such approximations are produced to represent the given multivariate function approximately. The weight function selection becomes an important issue for the HDMR based applications when it is desired to give different importances to function values at different points. An appropriately chosen weight function may increase the quality of the approximation incredibly. This work aims at a multivariate weight function optimization to obtain high quality approximations through the HDMR method to represent multivariate functions. The proposed optimization considers constancy measurer maximization which produces a quadratic vector equation to be solved. Another contribution of this work is to use a recently developed method, fluctuation free integration, with HDMR, to solve this equation easily. This work is an extension of a previous work about weight optimization in HDMR for univariate functions.

关键词： High Dimensional Model Representation multivariate functions Approximation Optimization Fluctuation expansion

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AUTOMATED ESTABLISHMENT AND PLOTTING OF multivariate functions

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ANALYTICA CHIMICA ACTA 1990年第2期234卷 493-497页

作者： YAN, ZQ TANG, XH Department of Chemical Engineering Beijing Institute of Chemical Technology Beijing China

The earlier package for univariate functions is extended to multivariate functions in the QQN series software, which contains millions of models. The models are characterized by numbered basic and compound functions which are selected by the user. Plots for a selected model are displayed in rotatablethree-dimensional coordinates. The package can be run on an IBM-PC or compatible computer.

关键词： multivariate functions Computerized establishment of multivariate functions

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On the points of regularity of multivariate functions of bounded variation.

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Real Analysis Exchange 2003年第2期29卷 647-656页

作者： Burkhard Lenze

In the one-dimensional case it is well-known that functions of bounded variation on ℝ possess at most a countable number of non-regular points. In this paper we will show that multivariate functions f:ℝn→ℝ of bounded variation satisfying the condition lim|x|→∞f(X) are non-regular at most on a subset of ℝn of Lebesgue measure zero. Moreover, we will point out that this result is best possible.

关键词： 26B05 26B35 Bounded variation multivariate functions points of regularity

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multivariate RATIONAL APPROXIMATION OF functions WITH CURVES OF SINGULARITIES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第6期46卷 A3401-A3426页

作者： Boulle, Nicolas Herremans, Astrid Huybrechs, Daan Imperial Coll London Dept Math London SW7 2AZ England Katholieke Univ Leuven Dept Comp Sci B-3001 Leuven Belgium

functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types of grading strategies. Rational functions are an exception to this rule: for univariate functions with point singularities, such as branch points, rational approximations exist with root-exponential convergence in the rational degree. This is typically enabled by the clustering of poles near the singularity. Both the theory and computational practice of rational functions for function approximation have focused on the univariate case, with extensions to two dimensions via identification with the complex plane. multivariate rational functions, i.e., quotients of polynomials of several variables, are relatively unexplored in comparison. Yet, apart from a steep increase in theoretical complexity, they also offer a wealth of opportunities. A first observation is that singularities of multivariate rational functions may be continuous curves of poles, rather than isolated ones. By generalizing the clustering of poles from points to curves, we explore constructions of multivariate rational approximations to functions with curves of singularities.

关键词： rational approximation multivariate functions singularity least-squares

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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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Empirical Interpolation Decomposition

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ACTA APPLICANDAE MATHEMATICAE 2019年第1期164卷 49-64页

作者： Toumi, Asma De Vuyst, Florian Ecole Normale Super Paris Saclay CMLA Paris France CNRS Paris France Univ Technol Compiegne Sorbonne Univ LMAC Compiegne France

Many physical problems need a multidimensional description and involve high dimensional spaces. Standard discretization techniques often lead to an excessive computation time. To solve this problem, we develop in this paper an empirical interpolation decomposition (EID) for multivariate functions. This method provides an approximate representation of a given function in separate form. Error estimates of the developed EID are derived and some properties are given.

关键词： Empirical interpolation decomposition multivariate functions

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A direct construction of complete complementary code with zero correlation zone property for prime-power length

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2024年第2期16卷 403-426页

作者： Kumar, Nishant Majhi, Sudhan Upadhyay, Ashish K. IIT Patna Dept Math Patna 801106 Bihar India IISc Bangalore Dept Elect Commun Engn CV Raman Rd Bengaluru 560012 Karnataka India BHU Dept Math Varanasi 221005 Uttar Pradesh India

In this paper, we propose a direct construction of a novel type of code set which has combined properties of complete complementary code (CCC) and zero correlation zone (ZCZ) sequence set and whichwe call complete complementary-ZCZ (CC-ZCZ) code set. The code set is constructed using multivariate functions. The proposed construction also provides Golay-ZCZ codes of prime-power lengths. The proposed Golay-ZCZ codes are optimal and asymptotically optimal for binary and non-binary cases, respectively, by Tang-Fan-Matsufuzi bound. Furthermore, the proposed direct construction provides novel ZCZ sequences of length pk, where p is a prime number and k is an integer >= 2. We establish a relationship between the proposed CC-ZCZ code set and the first-order generalized Reed-Muller (GRM) code, and prove that both have the same Hamming distance. We also count the number of CC-ZCZ code sets in first-order GRM codes. The column sequence peak-to-mean envelope power ratio (PMEPR) of the proposed CC-ZCZ code set is derived and compared with existing works. From the proposed construction, the Golay-ZCZ code and ZCZ sequences are also derived and compared with the existing works. The proposed construction generalizes many of the existing works.

关键词： Golay complementary sets (GCSs) Complete complementary code (CCC) multivariate functions Zero correlation zone (ZCZ) sequence set Golay-ZCZ sequence set Complete complementary-zero correlation zone (CC-ZCZ) code set Prime-power length Peak-to-mean envelope power ratio (PMEPR) Orthogonal frequency division multiplexing (OFDM) Multi-carrier code division multiple access (MC-CDMA) Generalized Reed-Muller (GRM) codes

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A novel approximation method for multivariate data partitioning Fluctuation free integration based HDMR

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ENGINEERING COMPUTATIONS 2012年第7-8期29卷 743-765页

作者： Tunga, Burcu Demiralp, Metin Istanbul Tech Univ Inst Informat TR-80626 Istanbul Turkey

Purpose - The plain High Dimensional Model Representation (HDMR) method needs Dirac delta type weights to partition the given multivariate data set for modelling an interpolation problem. Dirac delta type weight imposes a different importance level to each node of this set during the partitioning procedure which directly effects the performance of HDMR. The purpose of this paper is to develop a new method by using fluctuation free integration and HDMR methods to obtain optimized weight factors needed for identifying these importance levels for the multivariate data partitioning and modelling procedure. Design/methodology/approach - A common problem in multivariate interpolation problems where the sought function values are given at the nodes of a rectangular prismatic grid is to determine an analytical structure for the function under consideration. As the multivariance of an interpolation problem increases, incompletenesses appear in standard numerical methods and memory limitations in computer-based applications. To overcome the multivariance problems, it is better to deal with less-variate structures. HDMR methods which are based on divide-and-conquer philosophy can be used for this purpose. This corresponds to multivariate data partitioning in which at most univariate components of the Plain HDMR are taken into consideration. To obtain these components there exist a number of integrals to be evaluated and the Fluctuation Free Integration method is used to obtain the results of these integrals. This new form of HDMR integrated with Fluctuation Free Integration also allows the Dirac delta type weight usage in multivariate data partitioning to be discarded and to optimize the weight factors corresponding to the importance level of each node of the given set. Findings - The method developed in this study is applied to the six numerical examples in which there exist different structures and very encouraging results were obtained. In addition, the new method is compar

关键词： Interpolation Modelling High dimensional model representation multivariate functions Approximation Fluctuation expansion

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Bound analysis in univariately truncated Generalized High Dimensional Model Representation for random-data partitioning: Interval GHDMR

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APPLIED NUMERICAL MATHEMATICS 2009年第6期59卷 1431-1448页

作者： Tunga, M. Alper Demiralp, Metin Bahcesehir Univ Fac Engn Dept Comp Engn TR-34100 Istanbul Turkey Istanbul Tech Univ Inst Informat Computat Sci & Engn Program Grp Sci & Methods Comp TR-34469 Istanbul Turkey

Generalized High Dimensional Model Representation (GHDMR) method is based on a divide-and-conquer philosophy and it approximately reduces the N-dimensional interpolation to N number of one-dimensional interpolations. It is an exact representation which uses Dirac delta functions as the weight functions. GHDMR's components with multivariances higher than univariance are quite complicated and we prefer to use only its constant and univariate terms. In this work, we try to construct a new high dimensional model representation based method, Interval GHDMR, to be used when a random discrete multivariate data set which has construction errors coming from incapabilities or limited capabilities of the measurement devices or tools is given and an analytical structure for a multivariate function that passes through the nodes of this data set is to be determined. The errors coming from the construction of the data set that describes the sought multivariate function result in a band structure for the sought function instead of an analytical structure. The main purpose, here, is to estimate the total error formed through the given method in the given multivariate interpolation problem and, to this end, to partition the given multivariate data into at most univariate data and to determine an approximate band structure (or interval) for the sought function. By this way, the computational complexity and CPU time needed for computer based applications of the problem decrease. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： High Dimensional Model Representation Generalized High Dimensional Model Representation multivariate functions Interpolation Multidimensional problems Bound analysis Approximation

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