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A formally exact Bethe-Salpeter-like equation for the linear-response function is introduced with a kernel which depends only on the one frequency of the applied field. This is in contrast with the standard Bethe-Salpeter equation (BSE) which involves multiple-frequency integrals over the kernel and response functions. From the one-frequency kernel, known approximations are straightforwardly recovered. However, the present formalism lends itself to more powerful approximations. This is demonstrated with the exact analytical solution of the Hubbard molecule. Similarities and differences of the GW + BSE approach with the self-consistent random-phase approximation are also discussed. [ABSTRACT FROM AUTHOR]

We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient. [ABSTRACT FROM AUTHOR]

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines. [ABSTRACT FROM AUTHOR]

We report an implementation of the spin-flip (SF) variant of time-dependent density functional theory (TD-DFT) within the Tamm-Dancoff approximation and non-collinear (NC) formalism for local, generalized gradient approximation, hybrid, and range-separated functionals. The performance of different functionals is evaluated by extensive benchmark calculations of energy gaps in a variety of diradicals and open-shell atoms. The benchmark set consists of 41 energy gaps. A consistently good performance is observed for the Perdew-Burke-Ernzerhof (PBE) family, in particular PBE0 and PBE50, which yield mean average deviations of 0.126 and 0.090 eV, respectively. In most cases, the performance of original (collinear) SF-TDDFT with 50-50 functional is also satisfactory (as compared to non-collinear variants), except for the same-center diradicals where both collinear and non-collinear SF variants that use LYP or B97 exhibit large errors. The accuracy of NC-SF-TDDFT and collinear SF-TDDFT with 50-50 and BHHLYP is very similar. Using PBE50 within collinear formalism does not improve the accuracy. [ABSTRACT FROM AUTHOR]

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations. [ABSTRACT FROM AUTHOR]

Time-dependent current-density-functional theory (TDCDFT) provides an, in principle, exact scheme to calculate efficiently response functions for a very broad range of applications. However, the lack of approximations valid for a range of parameters met in experimental conditions has so far delayed its extensive use in inhomogeneous systems. On the other side, in many-body perturbation theory accurate approximations are available, but at a price of a higher computational cost. In the present work, the possibility of combining the advantages of both approaches is exploited. In this way, an exact equation for the exchange-correlation kernel of TDCDFT is obtained, which opens the way for a systematic improvement of the approximations adopted in practical applications. Finally, an approximate kernel for an efficient calculation of spectra of solids and molecular conductances is suggested and its validity is discussed. [ABSTRACT FROM AUTHOR]

The excluded volume (EV) effect between nonreactive like-particles of diffusion-influenced pseudo-first-order reaction A + B → C is investigated by the hierarchical Smoluchowski approach of Kuzovkov and Kotomin [Rep. Prog. Phys. 51, 1479 (1988)] and the many-particle kernel formalism of Lee et al. [J. Chem. Phys. 113, 8686 (2000)] in one dimension. Contrary to the three-dimensional analysis, the latter theory can be formulated without additional approximations in one dimension so that more accurate results are obtained. Although formulations and resulting expressions are different, these two theories show almost identical results numerically. The EV effect becomes significant at higher concentrations of B molecules as in three dimensions. However, we found that the EV effect in one dimension is more pronounced than in three dimensions. A similar trend appears as the size of the B molecule increases. Theoretical results are compared with Monte Carlo simulations. The simulation results reveal much larger EV effect than that predicted by both theories. This behavior may be attributed to the ''cage'' effect which is not considered in both theories. [ABSTRACT FROM AUTHOR]

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast‐slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast‐slow systems parameters, which can influence the detection of potential early‐warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change‐point location induced by the critical transition. We cross‐validate our results by numerical simulations for a van der Pol‐type model. [ABSTRACT FROM AUTHOR]

Kernel-based methods exhibit well-documented performance in various nonlinear learning tasks. Most of them rely on a preselected kernel, whose prudent choice presumes task-specific prior information. Especially when the latter is not available, multi-kernel learning has gained popularity thanks to its exibility in choosing kernels from a prescribed kernel dictionary. Leveraging the random feature approximation and its recent orthogonality-promoting variant, the present contribution develops a scalable multi-kernel learning scheme (termed Raker) to obtain the sought nonlinear learning function 'on the y,' first for static environments. To further boost performance in dynamic environments, an adaptive multi-kernel learning scheme (termed AdaRaker) is developed. AdaRaker accounts not only for data-driven learning of kernel combination, but also for the unknown dynamics. Performance is analyzed in terms of both static and dynamic regrets. AdaRaker is uniquely capable of tracking nonlinear learning functions in environments with unknown dynamics, and with with analytic performance guarantees. Tests with synthetic and real datasets are carried out to showcase the effectiveness of the novel algorithms. [ABSTRACT FROM AUTHOR]

Kernel k-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear k-means clustering algorithm. However, kernel k- means clustering is computationally expensive when the non-linear feature map is highdimensional and there are many input points. Kernel approximation, e.g., the Nyström method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel k-means clustering, and shows that applying the linear k-means clustering algorithm to k/∊ (1 + o(1)) features constructed using a so-called rank-restricted Nyström approximation results in cluster assignments that satisfy a 1+∊ approximation ratio in terms of the kernel k-means cost function, relative to the guarantee provided by the same algorithm without the use of the Nyström method. As part of the analysis, this work establishes a novel 1+ relative-error trace norm guarantee for low-rank approximation using the rank-restricted Nyström approximation. Empirical evaluations on the 8:1 million instance MNIST8M dataset demonstrate the scalability and usefulness of kernel k-means clustering with Nyström approximation. This work argues that spectral clustering using Nyström approximation|a popular and computationally effcient, but theoretically unsound approach to non-linear clustering| should be replaced with the effcient and theoretically sound combination of kernel k-means clustering with Nyström approximation. The superior performance of the latter approach is empirically veriöed. [ABSTRACT FROM AUTHOR]

This paper discusses Lp approximation error estimates for hybrid interpolation on the unit sphere. This interpolation scheme is integrated by spherical polynomials and radial basis functions. The smooth radial basis functions generated by a strictly positive definite zonal kernel are embedded in a larger native space generated by a less smooth kernel, and the error estimates for hybrid interpolation to a target function from the larger native space are given. In a sense, the results of this paper show that the hybrid interpolation associated with the smooth kernel enjoys the same order of error estimate as hybrid interpolation associated with the less smooth kernel for a target function from the rough native space. [ABSTRACT FROM AUTHOR]

The shaking of a camera can easily cause blurs in an image. Thus, deblurring is a problem that is worth solving and has always been an active research interest. The color information in an image is an important feature and contains clues for image deblurring that have not been widely exploited. In this paper, we present an efficient and stable blurring kernel estimation method by solving an energy function constructed by a weighted color approximation regularization term. The term is derived from a two-color model, and we use a defined weight to alleviate the color change through the blurring process. Then we select salient edges in an effective way to apply the proposed method on the patches centered at these edges. Experiments on synthetic and real-world images show the efficiency and stability of our proposed method. [ABSTRACT FROM AUTHOR]

Voice conversion (VC) is a technique that aims to transform the individuality of a source speech so as to mimic that of a target speech while keeping the message unaltered. In our previous work, Gaussian process (GP) was introduced into the literature of VC for the first time, for the sake of overcoming the "over-fitting" problem inherent in the state-of-the-art VC methods, which gives very promising results. However, standard GP usually acts as somewhat a smoothing device more than a universal approximator. In this paper, we further attempt to improve the flexibility of GP-based VC by resorting to the expressive kernels that are derived to model the spectral density with Gaussian mixture model (GMM). Our new method benefits from the expressiveness of the new kernel while the inference of GP remains simple and analytic as usual. Experiments demonstrate both objectively and subjectively that the individualities of the converted speech are much more closer to those of the target while speech quality obtained is comparable to the standard GP-based method. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical solution of certain matrix-valued PDEs. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyze a new meshfree discretization scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth-order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems. [ABSTRACT FROM AUTHOR]

Highlights • A new CKN model is proposed that is more efficient regarding accuracy and time. • It is structurally more similar to convolutional networks than ordinary CKNs. • A fast approximation method based on RFF is employed for training the proposed model. • A novel data-dependent method is proposed for approximating shift-invariant kernels. Abstract Convolutional Kernel Networks (CKNs) are efficient multilayer kernel machines, which are constructed by approximating a convolution kernel with a mapping based on Gaussian functions. In this paper, we introduce a new approximation of the same convolution kernel based on a convex combination of cosine kernels. CKNs are structurally similar to Convolutional Neural Networks (CNNs), but the convolution operation in CKNs is based on the Euclidean distance, which is not common in convolutional networks. We show that the CKN model obtained by the proposed approximation leads to the ordinary convolution operation, which is based on the inner product. From this point of view, the proposed model is a step forward towards bridging the gap between kernel methods and deep learning. In this paper, we use two methods for learning filters of the proposed CKN: Random Fourier Features, which is a randomized data-independent method for approximating shift-invariant kernels, and a novel method based on the minimization of the sum of squared errors of approximating shift-invariant kernels. Although the RFF method is much faster than ordinary CKN, it requires a high number of random features in order to obtain an acceptable accuracy. To overcome this problem, we proposed the second method, in which the filters are learned in a data-dependent fashion. We evaluate the proposed model on visual recognition datasets MNIST, CIFAR-10, C-Cube, and FERET. Our experiments show that the proposed model surpasses ordinary CKNs in terms of accuracy. Specifically, on CIFAR-10, the accuracy of the proposed method is 1.7% higher than ordinary CKN. [ABSTRACT FROM AUTHOR]

Suggested by the max‐product sampling operators based on sinc‐Fejér kernels, in this paper, we introduce truncated max‐product Kantorovich operators based on generalized type kernels depending on two functions ϕ and ψ satisfying a set of suitable conditions. Pointwise convergence, quantitative uniform convergence in terms of the moduli of continuity, and quantitative Lp‐approximation results in terms of a K‐functional are obtained. Previous results in sampling and neural network approximation are recaptured, and new results for many concrete examples are obtained. [ABSTRACT FROM AUTHOR]

In the paper, we provide sufficient conditions assuring existence of viable solutions of differential inclusions with fractional derivative without singular kernel, namely, Caputo‐Fabrizio derivative of order α∈(0,1). A construction of an approximate solution is presented. A modified condition of tangency, according to specifity of the system with this new fractional derivative, is given. [ABSTRACT FROM AUTHOR]

We compute the values of the best approximations of the Cauchy-Szeg¨o kernel in the mean on the unit circle by quasipolynomials with respect to the Takenaka-Malmquist system. [ABSTRACT FROM AUTHOR]

We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions are satisfied for the analyzed systems. The reproducing kernel functions are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based on generating the orthonormal basis with an aim to formulate the solution via the evolution of the algorithm. The analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed components. In this connection, some numerical examples are presented to show the good performance and applicability of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution of fractional models encountered in various fields of sciences and engineering. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We motivate and describe a new parameterized approximation paradigm which studies the interaction between approximation ratio and running time for any parametrization of a given optimization problem. As a key tool, we introduce the concept of an α - shrinking transformation , for α ≥ 1 . Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving the approximation ratio of α (or α-fidelity ). Moving even beyond the approximation ratio, we call for a new type of approximative kernelization race . Our α -shrinking transformations can be used to obtain approximative kernels which are smaller than the best known for a given problem. The smaller “ α -fidelity” kernels allow us to obtain an exact solution for the reduced instance more efficiently, while obtaining an approximate solution for the original instance. We show that such fidelity preserving transformations exist for several fundamental problems, including Vertex Cover , d-Hitting Set , Connected Vertex Cover and Steiner Tree . [ABSTRACT FROM AUTHOR]

Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider discrete versions of the modified projection method and of the iterated modified projection method for solution of a Urysohn integral equation with a smooth kernel. For r ≥ 1 , a space of piecewise polynomials of degree ≤ r − 1 with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at r Gauss points. The orders of convergence which we obtain for these discrete versions indicate the choice of numerical quadrature which preserves the orders of convergence. Numerical results are given for a specific example. [ABSTRACT FROM AUTHOR]

The problem of density estimation arises naturally in many contexts. In this paper, we consider the approach using a piecewise constant function to approximate the underlying density. We present a new density estimation method via the random forest method based on the Bayesian Sequential Partition (BSP) (Lu, Jiang, and Wong 2013). Extensive simulations are carried out with comparison to the kernel density estimation method, BSP method, and four local kernel density estimation methods. The experiment results show that the new method is capable of providing accurate and reliable density estimation, even at the boundary, especially for i.i.d. data. In addition, the likelihood of the out-of-bag density estimation, which is a byproduct of the training process, is an effective hyperparameter selection criterion. [ABSTRACT FROM AUTHOR]

When high-order numerical approximation method is applied to model the propagation of shock wave or discontinuity, it usually creates unstable unreal numerical oscillations around the discontinuous regions. In this research, we propose a non-oscillation meshfree scheme based on reproducing kernel particle method (RKPM) which can maintain the accuracy and minimize the oscillation in the modeling of shock wave propagation. In the proposed method, the original influence domain of high-order RK approximation is divided into several subdomains. Then we apply low-order RK approximation within each subdomain. Instead of directly using the discrete particles to build the numerical approximation, we consider that the high-order approximation is constructed by the summation of those low-order approximations multiplied by a local weight function. By adjusting these local weights with the "smoothness indicator", we can determine the "effect" of the corresponding subdomain and the discrete particles inside this subdomain. Therefore, the subdomain containing discontinuity would not participate in the high-order approximation, and the numerical oscillation is automatically suppressed. The proposed method does not need artificial viscosity or numerical damping to stabilize the solution. Several benchmark problems with shock wave propagation are tested. The results show that the proposed method can maintain high-order accuracy without numerical oscillation. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the Berry-Esseen bounds of weighted kernel estimator for a nonparametric regression model based on linear process errors under a LNQD random variable sequence. The rate of the normal approximation is shown as $O(n^{-1/6})$ under some appropriate conditions. The results obtained in the article generalize or improve the corresponding ones for mixing dependent sequences in some sense. [ABSTRACT FROM AUTHOR]

We apply reproducing kernel method to the density-dependent diffusion Nagumo equation. Powerful method has been applied by reproducing kernel functions. The approximations to the exact solution are obtained. In particular, series solutions are obtained. These solutions demonstrate the certainty of the method. The results acquired in this work conceive many attracted behaviors that assure further work on the Nagumo equation. [ABSTRACT FROM AUTHOR]

In this paper, the multiple kernel learning (MKL) is formulated as a supervised classification problem. We dealt with binary classification data and hence the data modelling problem involves the computation of two decision boundaries of which one related with that of kernel learning and the other with that of input data. In our approach, they are found with the aid of a single cost function by constructing a global reproducing kernel Hilbert space (RKHS) as the direct sum of the RKHSs corresponding to the decision boundaries of kernel learning and input data and searching that function from the global RKHS, which can be represented as the direct sum of the decision boundaries under consideration. In our experimental analysis, the proposed model had shown superior performance in comparison with that of existing two stage function approximation formulation of MKL, where the decision functions of kernel learning and input data are found separately using two different cost functions. This is due to the fact that single stage representation helps the knowledge transfer between the computation procedures for finding the decision boundaries of kernel learning and input data, which inturn boosts the generalisation capacity of the model. [ABSTRACT FROM AUTHOR]

We introduce the reproducing kernel method to approximate solutions of difference equations. Reproducing kernel functions for difference equations are obtained. Examples that illustrate the accuracy and power of the method are given. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Gradient averaging-type a posteriori error estimators applied to the finite element method enjoy great popularity in the engineering community. This is mainly because they are easy in their construction and computer implementation and usually provide constant-free and sharp error estimates on the expense of losing the bounding property. However, it proves difficult to transfer this error estimation procedure to Galerkin meshfree methods. One reason is that the meshfree gradient approximation is generally rather accurate per se and difficult to improve. Moreover, in many Galerkin meshfree methods the shape functions are no interpolants, as required in the original idea of constructing the recovered and thus smooth gradient field. In this paper, a novel error estimation procedure is presented that is based on the enhanced assumed strain (EAS) method and applied to the reproducing kernel particle method (RKPM) as a representative of Galerkin meshfree methods. The error estimator is naturally tailored to Galerkin meshfree methods if stabilized conforming nodal integration (SCNI) is employed, which provides two gradient fields: a globally smooth one (the compatible strain) and a cellwise constant one (the enhanced assumed strain). It is shown that the difference (the enhanced strain) can be used to derive an error estimator that follows the notion of gradient averaging-type error estimators and resolves its issues when applied to Galerkin meshfree methods. In addition, the enhanced-strain error estimator is even easier to implement and computationally less expensive than gradient averaging-type error estimators. Numerical examples of engineering interest illustrate the performance of the enhanced-strain error estimator presented in this paper. [ABSTRACT FROM AUTHOR]

We developed a new self-adjoint, consistent, and stable coupling strategy for nonlocal diffusion models, inspired by the quasi-nonlocal atomistic-to-continuum method for crystalline solids. The proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the L² norm, and it satisfies the maximum principle. A finite difference approximation is used to discretize the coupled system, which inherits the property from the continuous formulation. Furthermore, we design a numerical example that shows the discrepancy between the fully nonlocal and fully local diffusions, whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we aim at analyzing the approximation abilities of shallow networks in reproducing kernel Hilbert spaces (RKHSs). We prove that there is a probability measure such that the achievable lower bound for approximating by shallow nets can be realized for all functions in balls of reproducing kernel Hilbert space with high probability, which is different with the classical minimax approximation error estimates. This result together with the existing approximation results for deep nets shows the limitations for shallow nets and provides a theoretical explanation on why deep nets perform better than shallow nets. [ABSTRACT FROM AUTHOR]

In this article we prove a central limit theorem for theLpdistancewhere μ is a weight function andfnis the kernel density estimator proposed by Jones (1991) for length-biased data. The approach is based on the invariance principle for the empirical processes proved by Horváth (1985). We study the differenceIn(p) with its approximation in terms of its rates of convergence to zero. We subsequently present a central limit theorem for approximation ofIn(p). [ABSTRACT FROM AUTHOR]

In this paper, an effective numerical method to solve the Cauchy type singular Fredholm integral equations (CSFIEs) of the first kind is proposed. The collocation technique based on Bernstein polynomials is used for approximation the solution of various cases of CSFIEs. By transforming the problem into systems of linear algebraic equations, we see that this approach is computationally simple and attractive. Then the approximate solution of the problem in truncated series form is obtained by using the matrix form of this method. Convergence and error analyses of the presented method are mentioned. Finally, numerical experiments show the validity, accuracy, and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In the Vector Connectivity problem we are given an undirected graph $$G=(V,E)$$ , a demand function $$\lambda :V\rightarrow \{0,\ldots ,d\}$$ , and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex $$v\in V{\setminus } S$$ has at least $$\lambda (v)$$ vertex-disjoint paths to S; this abstractly captures questions about placing servers or warehouses relative to demands. The problem is $$\mathsf {NP}$$ -hard already for instances with $$d=4$$ (Cicalese et al., Theoretical Computer Science '15), admits a log-factor approximation (Boros et al., Networks '14), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished '14). We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d- Connectivity where the upper bound d on demands is a fixed constant. For Vector d- Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, that is, an efficient reduction to an equivalent instance with $$f(d)k=O(k)$$ vertices. For Vector Connectivity we have a factor $$\mathsf {opt} $$ -approximation and we can show that it has no kernelization to size polynomial in k or even $$k+d$$ unless $$\mathsf {NP} \subseteq \mathsf {coNP}/\mathsf {poly}$$ , which shows that $$f(d){\text {poly}}(k)$$ is optimal for Vector d- Connectivity. Finally, we give a simple randomized fixed-parameter algorithm for Vector Connectivity with respect to k based on matroid intersection. [ABSTRACT FROM AUTHOR]

In this article, we proposed a new numerical method to obtain the approximation solution for the time variable fractional order mobile–immobile advection–dispersion model based on reproducing kernel theory and collocation method. The equation is obtained from the standard advection–dispersion equation (ADE) by adding the Coimbra's variable fractional derivative in time of order γ ( x , t ) ∈ [ 0 , 1 ] . In order to solve this kind of equation, we discuss and derive the ε -approximate solution in the form of series with easily computable terms in the bivariate spline space. At the same time, the stability and convergence of the approximation are investigated. Finally, numerical examples are provided to show the accuracy and effectiveness. [ABSTRACT FROM AUTHOR]

In this work, we apply the reproducing kernel method for investigating Blasius equations with two different boundary conditions in semi-infinite domains. Convergence analysis of the reproducing kernel method is given. The numerical approximations are presented and compared with some other techniques, Howarth's numerical solution and Runge-Kutta Fehlberg method. [ABSTRACT FROM AUTHOR]

In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous functions (signals) are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Finally, several examples of (duration-limited) kernels which satisfy the assumptions of the present theory have been provided, and also the problem of the linear prediction by sampling values from the past is analyzed. [ABSTRACT FROM AUTHOR]

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker–Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well-known diffusion approximation under convenience scaling and on the basic properties of this diffusion. Eventually, we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel. [ABSTRACT FROM AUTHOR]

Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szegő–Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian, the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szegő kernel. We do not know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question. [ABSTRACT FROM AUTHOR]

In this paper, efficient and computationally attractive methods based on the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations for the numerical solution of a system of Volterra integral equations with weakly singular kernels are presented. Simplicity for performing even in the presence of singularities is one of the advantages of Sinc methods. Convergence analysis of the proposed methods is given and an exponential convergence is achieved as well. Numerical results are presented which demonstrate the efficiency and high accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper we derive optimal subspaces for Kolmogorov n -widths in the L 2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases. [ABSTRACT FROM AUTHOR]

We prove two new universality results for polynomial reproducing kernels of compactly supported measures. The first applies to measures on the unit circle with a jump and a singularity in the weight at 1 and the second applies to area-type measures on a certain disconnected polynomial lemniscate. In both cases, we apply methods developed by Lubinsky to obtain our results. [ABSTRACT FROM AUTHOR]

Scaling kernel machines to massive data sets is a major challenge due to storage and computation issues in handling large kernel matrices, that are usually dense. Recently, many papers have suggested tackling this problem by using a low-rank approximation of the kernel matrix. In this paper, we first make the observation that the structure of shift-invariant kernels changes from low-rank to block-diagonal (without any low-rank structure) when varying the scale parameter. Based on this observation, we propose a new kernel approximation framework { Memory Efficient Kernel Approximation (MEKA), which considers both low-rank and clustering structure of the kernel matrix. We show that the resulting algorithm outperforms state-of-the-art low-rank kernel approximation methods in terms of speed, approximation error, and memory usage. As an example, on the covtype dataset with half a million samples, MEKA takes around 70 seconds and uses less than 80 MB memory on a single machine to achieve 10% relative approximation error, while standard Nyström approximation is about 6 times slower and uses more than 400MB memory to achieve similar approximation. We also present extensive experiments on applying MEKA to speed up kernel ridge regression. [ABSTRACT FROM AUTHOR]

One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are ‘flat’ leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Padé method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson's equation in a 3-D spherical shell. [ABSTRACT FROM AUTHOR]

There are many methods to construct kernel with given data attribute information. Gaussian radial basis function (RBF) kernel is one of the most popular ways to construct a kernel. The key observation is that in real-world data, besides the data attribute information, data label information also exists, which indicates the data class. In order to make use of both data attribute information and data label information, in this work, we propose a supervised kernel construction method. Supervised information from training data is integrated into standard kernel construction process to improve the discriminative property of resulting kernel. A supervised Laplacian embedding cellular automaton model is another key application developed for two-lane heterogeneous traffic flow with the safe distance and large-scale truck. Based on the properties of traffic flow in China, we re-calibrate the cell length, velocity, random slowing mechanism and lane-change conditions and use simulation tests to study the relationships among the speed, density and flux. The numerical results show that the large-scale trucks will have great effects on the traffic flow, which are relevant to the proportion of the large-scale trucks, random slowing rate and the times of the lane space change. [ABSTRACT FROM AUTHOR]