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 litera...
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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
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 functionapproximation in a bounded domain. W...
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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 functionapproximation 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.
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 f...
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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.
This article studies tractability and strong tractability for multivariateapproximation of infinitely differentiable functions, using either standard information or continuous linear information. We prove that this a...
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This article studies tractability and strong tractability for multivariateapproximation of infinitely differentiable functions, using either standard information or continuous linear information. We prove that this approximation problem is not strongly tractable. (c) 2006 Elsevier Inc. All rights reserved.
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...
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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.
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 criter...
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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.
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 randomize...
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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 functionapproximation 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.
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...
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