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Constructing Embedded Lattice-Based Algorithms for multivariate function approximation with a Composite Number of Points

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CONSTRUCTIVE approximation 2025年第1期61卷 81-113页

作者： Kuo, Frances Y. Mo, Weiwen Nuyens, Dirk UNSW Sydney Sch Math & Stat Sydney NSW 2052 Australia Dept Comp Sci KU Leuven Celestijnenlaan 200A B-3001 Leuven Belgium

We approximate d-variate periodic functions in weighted Korobov spaces with general weight parameters using n function values at lattice points. We do not limit n to be a prime number, as in currently available literature, but allow any number of points, including powers of 2, thus providing the fundamental theory for construction of embedded lattice sequences. Our results are constructive in that we provide a component-by-component algorithm which constructs a suitable generating vector for a given number of points or even a range of numbers of points. It does so without needing to construct the index set on which the functions will be represented. The resulting generating vector can then be used to approximate functions in the underlying weighted Korobov space. We analyse the approximation error in the worst-case setting under both the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} and L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty }$$\end{document} norms. Our component-by-component construction under the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} norm achieves the best possible rate of convergence for lattice-based algorithms, and the theory can be applied to lattice-based kernel methods and splines. Depending on the value of the smoothness parameter alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackag

关键词： Lattice rules Lattice algorithms Embedded lattice sequences multivariate function approximation Component-by-component construction Composite number of points Non-prime number of points

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A RANDOMIZED ALGORITHM FOR multivariate function approximation

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第3期39卷 A983-A1002页

作者： Shin, Yeonjong Xiu, Dongbin Ohio State Univ Dept Math 231 W 18th Ave Columbus OH 43210 USA

The randomized Kaczmarz (RK) method is a randomized iterative algorithm for solving (overdetermined) linear systems of equations. In this paper, we extend the RK method to function approximation in a bounded domain. We demonstrate that by conducting the approximation randomly one sample at a time the method converges. Convergence analysis is conducted in terms of expectation, where we establish sharp upper and lower bounds for both the convergence rate of the algorithm and the error of the resulting approximation. The analysis also establishes the optimal sampling probability measure to achieve the optimal rate of convergence. Various numerical examples are provided to validate the theoretical results.

关键词： multivariate function approximation Kaczmarz algorithm

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Liberating the dimension for function approximation

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JOURNAL OF COMPLEXITY 2011年第1期27卷 86-110页

作者： Wasilkowski, G. W. Wozniakowski, H. Columbia Univ Dept Comp Sci New York NY 10027 USA Univ Warsaw Inst Appl Math Warsaw Poland

We study approximation of functions that may depend on infinitely many variables. We assume that such functions belong to a separable weighted Hilbert space that is the direct sum of tensor product Hilbert spaces of functions with finitely many variables. The weight sequence gamma = {gamma(u)} is used to moderate the influence of terms depending on finitely many variables from u. We consider algorithms that use finitely many arbitrary linear functionals. Each linear functional is an inner product whose cost depends on the number of active variables of the inner product generator. That is, if the generator has d active variables then the cost is $(d) for a given non-decreasing function $. The error of an algorithm is defined in the worst case setting in a norm given by weighted L-2 norms for terms depending on finitely many variables. The epsilon-complexity is understood as the minimal cost among all algorithms with errors at most epsilon. We are especially interested in polynomial tractability, where the epsilon-complexity is bounded by a polynomial in epsilon(-1), and weak tractability, where the epsilon-complexity is sub-exponential in epsilon(-1). The results are as follows. An algorithm whose cost is equal to the epsilon-complexity. It turns out the algorithm does not depend on the cost function $. Necessary and sufficient conditions on polynomial tractability. It turns out that we may have polynomial tractability even when $(d) is exponential in d. This holds since the minimal number of active variables that must be used to compute an epsilon-approximation may be surprisingly small, e.g., o(In(epsilon(-1))) or even smaller. Necessary and sufficient conditions on weak tractability. It turns out that we have two quite different cases depending on whether the largest eigenvalue for the univariate case is simple or not. We may have weak tractability even when $(d) is doubly exponential in d. Specializing tractability conditions for product and finite-order weights.

关键词： multivariate function approximation Path integration Optimal algorithms Complexity Tractability

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approximation of infinitely differentiable multivariate functions is not strongly tractable

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JOURNAL OF COMPLEXITY 2007年第1期23卷 73-81页

作者： Huang, F. L. Zhang, S. Anhui Univ Sch Math & Computat Sci Hefei 230039 Peoples R China

This article studies tractability and strong tractability for multivariate approximation of infinitely differentiable functions, using either standard information or continuous linear information. We prove that this a... 详细信息

关键词： information-based complexity multivariate function approximation tractability and strong tractability

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CFD-based optimization of hovering rotors using radial basis functions for shape parameterization and mesh deformation

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OPTIMIZATION AND ENGINEERING 2013年第1期14卷 97-118页

作者： Allen, Christian B. Rendall, Thomas C. S. Univ Bristol Dept Aerosp Engn Bristol BS8 1TR Avon England

Aerodynamic shape optimization of a helicopter rotor in hover is presented, using compressible CFD as the aerodynamic model. An efficient domain element shape parameterization method is used as the surface control and deformation method, and is linked to a radial basis function global interpolation, to provide direct transfer of domain element movements into deformations of the design surface and the CFD volume mesh, and so both the geometry control and volume mesh deformation problems are solved simultaneously. This method is independent of mesh type (structured or unstructured) or size, and optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced parallel gradient-based algorithm by finite-difference, resulting in a flexible method of 'wrap-around' optimization. This paper presents results of the method applied to hovering rotors using local and global design parameters, allowing a large geometric design space. Results are presented for two transonic tip Mach numbers, with minimum torque as the objective, and strict constraints applied on thrust, internal volume and root moments. This is believed to be the first free form design optimization of a rotor blade using compressible CFD as the aerodynamic model, and large geometric deformations are demonstrated, resulting in significant torque reductions, with off-design performance also improved.

关键词： multivariate function approximation Radial basis functions Numerical simulation Numerical optimization Volume mesh deformation Shape parameterization Aerodynamics Rotors CFD

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Comparison of Two Search Criteria for Lattice-Based Kernel approximation 15th

Comparison of Two Search Criteria for Lattice-Based Kernel A...

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15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC)

作者： Kuo, Frances Y. Mo, Weiwen Nuyens, Dirk Sloan, Ian H. Srikumar, Abirami UNSW Sydney Sch Math & Stat Sydney NSW 2052 Australia Katholieke Univ Leuven Dept Comp Sci B-3001 Leuven Belgium

ISBN: (纸本)9783031597640;9783031597626;9783031597619

The kernel interpolant in a reproducing kernel Hilbert space is optimal in the worst-case sense among all approximations of a function using the same set of function values. In this paper, we compare two search criteria to construct lattice point sets for use in lattice-based kernel approximation. The first candidate, P-n*, is based on the power function that appears in machine learning literature. The second, S-n*, is a search criterion used for generating lattices for approximation using truncated Fourier series. We find that the empirical difference in error between the lattices constructed using P-n* and S-n* is marginal. The criterion S-n* is preferred as it is computationally more efficient and has a bound with a superior convergence rate.

关键词： Lattice rules Component-by-component construction Kernel interpolation multivariate function approximation

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Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2022年第5期48卷 66-66页

作者： Saibaba, Arvind K. Minster, Rachel Kilmer, Misha E. North Carolina State Univ Dept Math Raleigh NC 27695 USA Wake Forest Univ Dept Comp Sci Winston Salem NC 27101 USA Tufts Univ Dept Math Medford MA 02155 USA

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points;the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We present an adaptive version of the function and kernel approximation that determines an approximation that satisfies a user-specified relative error over a set of random points. We extend our approach to the case where the kernel requires repeated evaluations for many values of (hyper)parameters that govern the kernel. We give detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes. We observe speedups up to 18X over standard matrix-based approaches.

关键词： multivariate function approximation Tucker format Randomized algorithms Low-rank approximations Kernel methods

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Automatic smoothing spline projection pursuit

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Journal of Computational and Graphical Statistics 1994年第3期3卷 235-248页

作者： Roosen, Charles B. Hastie, Trevor J. Department of Statistics Stanford University Stanford CA 94305 United States Statistics and Data Analysis Research Department AT and T Bell Laboratories NJ 07974 Murray Hill United States

A highly flexible nonparametric regression model for predicting a response y given covariates (xk)kd=1 is the projection pursuit regression (PPR) model Ŷ = h(x) =β+ 12j βj βj(αjdX)’ where the fj, are general smoo... 详细信息

A highly flexible nonparametric regression model for predicting a response y given covariates (xk)_k^d=1 is the projection pursuit regression (PPR) model Ŷ = h(x) =β+ 12_j β_j β_j(α_j^dX)’ where the f_j, are general smooth functions with mean 0 and norm1, and Yt= l Qfej = 1- The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions fj using the supersmoother nonparametric scatterplot smoother. Friedman’s algorithm constructs a model with Mmax linear combinations, then prunes back to a simpler model of size M ≤ M_max, where M and M_max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients ay, the amount of smoothing in each direction, and the number of terms M and M_max are determined to optimize a single generalized cross-validation measure. © 1994 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.

关键词： GCV multivariate function approximation Neural networks Nonparametric regression

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