Uniform Distribution,Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Learning errors of linear programming support vector regression

In this paper, we give several results of learning errors for linear programming support vector regression. The corresponding theorems are proved in the reproducing kernel Hilbert space. With the covering number, the approximation property and the capacity of the reproducing kernel Hilbert space are measured. The obtained result (Theorem 2.1) shows that the learning error can be controlled by the sample error and regularization error. The mentioned sample error is summarized by the errors of learning regression function and regularizing function in the reproducing kernel Hilbert space. After estimating the generalization error of learning regression function (Theorem 2.2), the upper bound (Theorem 2.3) of the regularized learning algorithm associated with linear programming support vector regression is estimated.     

Solutions of fractional gas dynamics equation by a new technique

In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.     

The convergence rates of Shannon sampling learning algorithms

In the present paper,we provide an error bound for the learning rates of the regularized Shannon sampling learning scheme when the hypothesis space is a reproducing kernel Hilbert space(RKHS) derived by a Mercer kernel and a determined net.We show that if the sample is taken according to the determined set,then,the sample error can be bounded by the Mercer matrix with respect to the samples and the determined net.The regularization error may be bounded by the approximation order of the reproducing kernel Hilbert space interpolation operator.The paper is an investigation on a remark provided by Smale and Zhou.     

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

Application of integral operator for regularized least-square regression

In this paper, we study the consistency of the regularized least-square regression in a general reproducing kernel Hilbert space. We characterize the compactness of the inclusion map from a reproducing kernel Hilbert space to the space of continuous functions and show that the capacity-based analysis by uniform covering numbers may fail in a very general setting. We prove the consistency and compute the learning rate by means of integral operator techniques. To this end, we study the properties of the integral operator. The analysis reveals that the essence of this approach is the isomorphism of the square root operator.     

再生核希尔伯特空间连续线性泛函的范数及其应用北大核心

考虑了再生核希尔伯特空间连续线性泛函范数的表示,得到了用其范数平方等于该线性泛函连续两次作于再生核的简明表示.对于常见的Sobolev-Hibert空间而言,其再生核则可用截幂函数来表示,从而得到Sobolev-Hibert空间上连续线性泛函范数的简洁表示,以新视角解释和简化了文献中的现有结果.     

Solving the Dym initial value problem in reproducing kernel space

We consider two numerical solution approaches for the Dym initial value problem using the reproducing kernel Hilbert space method. For each solution approach, the solution is represented in the form of a series contained in the reproducing kernel space, and a truncated approximate solution is obtained. This approximation converges to the exact solution of the Dym problem when a sufficient number of terms are included. In the first approach, we avoid to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. The latter approach also permits a more straightforward convergence analysis. Therefore, there are benefits to both approaches. After developing the reproducing kernel Hilbert space method for the numerical solution of the Dym equation, we present several numerical experiments in order to show that the method is efficient and can provide accurate approximations to the Dym initial value problem for sufficiently regular initial data after relatively few iterations. We present the absolute error of the results when exact solutions are known and residual errors for other cases. The results suggest that numerically solving the Dym initial value problem in reproducing kernel space is a useful approach for obtaining accurate solutions in an efficient manner.     

Quasi-Monte Carlo integration in unanchored Sobolev spaces

We find a concrete sequence of N points, for which the squared worst-case quasi-Monte Carlo error in the Hilbert space of continuous functions defined on [0, 1] with square integrable second derivative is smaller than for the centered regular lattice point set.     

Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces

We study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tractability and tractability in terms of the weights of the spaces. For the three Sobolev spaces periodicity has no significant effect on strong tractability and tractability. In contrast, for general reproducing kernel Hilbert spaces anything can happen: we may have strong tractability or tractability for the non-periodic case and intractability for the periodic one, or vice versa.     

三阶微分方程边值问题的高效数值算法

基于再生核空间法提出了一个高效的数值算法来解决三阶微分方程的边值问题.利用再生性以及正交基的构造,得到了模型精确解的级数表示形式,并通过截断级数获得了其近似解.通过数值算例说明了此方法的有效性.     

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions,Optimal Deterministic Algorithms,and Higher-Order Convergence

We study the numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing-dimension algorithms. More precisely, the spaces of integrands we consider are weighted, reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst-case error. We study two cost models for function evaluations that depend on the number of active variables of the chosen sample points, and we study two classes of weights, namely product and order-dependent weights and the newly introduced finite projective dimension weights. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing-dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter \(\alpha \) and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing-dimension algorithms based on higher-order polynomial lattice rules.     

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

This article is concerned with a method for solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel Hilbert space. Convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method and some error estimates for the numerical approximation in reproducing kernel Hilbert space are presented. Finally, two numerical examples are considered to illustrate the computation efficiency and accuracy of the proposed method. © 2016 The Authors Numerical Methods for Partial Differential Equations Published by Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 174–198, 2017     

Greedy sparse linear approximations of functionals from nodal data

In many numerical algorithms, integrals or derivatives of functions have to be approximated by linear combinations of function values at nodes. This ranges from numerical integration to meshless methods for solving partial differential equations. The approximations should use as few nodal values as possible and at the same time have a smallest possible error. For each fixed set of nodes and each fixed Hilbert space of functions with continuous point evaluation, e.g. a fixed Sobolev space, there is an error–optimal method available using the reproducing kernel of the space. But the choice of the nodes is usually left open. This paper shows how to select good nodes adaptively by a computationally cheap greedy method, keeping the error optimal in the above sense for each incremental step of the node selection. This is applied to interpolation, numerical integration, and numerical differentiation. The latter case is particularly important for the design of meshless methods with sparse generalized stiffness matrices. The greedy algorithm is described in detail, and numerical examples are provided. In contrast to the usual practice, the greedy method does not always use nearest neighbors for local approximations of function values and derivatives. Furthermore, it avoids multiple points from clusters and it is better conditioned than choosing nearest neighbors.     

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Recently, error estimates have been made available for divergence-free radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the ``native space' of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates when the target function is less smooth than functions in the native space.

     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Numerical method for solving the nonlinear four-point boundary value problems

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.     

Learning Rates of Least-Square Regularized Regression

This paper considers the regularized learning algorithm associated with the least-square loss and reproducing kernel Hilbert spaces. The target is the error analysis for the regression problem in learning theory. A novel regularization approach is presented, which yields satisfactory learning rates. The rates depend on the approximation property and on the capacity of the reproducing kernel Hilbert space measured by covering numbers. When the kernel is C^∞ and the regression function lies in the corresponding reproducing kernel Hilbert space, the rate is m^ζ with ζ arbitrarily close to 1, regardless of the variance of the bounded probability distribution.     

On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.