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In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision. [ABSTRACT FROM AUTHOR]

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman-Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin-Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side. [ABSTRACT FROM AUTHOR]

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral-differential equations and comparing with other methods, we find that the two methods are useful. [ABSTRACT FROM AUTHOR]

We revisit in this paper the problem of inferring a diffusion network from information cascades. In our study, we make no assumptions on the underlying diffusion model, in this way obtaining a generic method with broader practical applicability. Our approach exploits the pairwise adoption-time intervals from cascades. Starting from the observation that different kinds of information spread differently, these time intervals are interpreted as samples drawn from unknown (conditional) distributions. In order to statistically distinguish them, we propose a novel method using Reproducing Kernel Hilbert Space embeddings. Experiments on both synthetic and real-world data from Twitter and Flixster show that our method significantly outperforms the state-of-the-art methods. We argue that our algorithm can be implemented by parallel batch processing, in this way meeting the needs in terms of efficiency and scalability of real-world applications. [ABSTRACT FROM AUTHOR]

Abstract The Swift–Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift–Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The solution is represented in the form of a series in the reproducing kernel space, and truncating this series representation we obtain the n -term approximate solution. In the first approach, we aim to explain how to construct a reproducing kernel method without using Gram-Schmidt orthogonalization, as orthogonalization is computationally expensive. This approach will therefore be most practical for obtaining numerical solutions. Gram-Schmidt orthogonalization is later applied in the second approach, despite the increased computational time, as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the Swift–Hohenberg equation. We demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal. [ABSTRACT FROM AUTHOR]

A new fracture criterion based on the crack opening displacement for peridynamic (PD) and dual-horizon peridynamics (DH-PD) is proposed. When the relative deformation of the PD bond between the particles reaches the critical crack tip opening displacement of the fracture mechanics, we assume that the bond force vanishes. A new damage rule similar to the local damage rule in conventional PD is introduced to simulate fracture. The new formulation is developed for a linear elastic solid though the extension to nonlinear materials is straightforward. The performance of the new fracture criterion is demonstrated by four examples, i.e. a bilateral crack problem, double parallel crack, monoclinic crack and the double inclined crack. The results are compared to experimental data and the results obtained by other computational methods. [ABSTRACT FROM AUTHOR]

We consider the weighted polyharmonic Bergman space , where is the space of all real-valued polyharmonic functions of degree m on and . has the reproducing kernel which is called by the weighted polyharmonic Bergman kernel. [ABSTRACT FROM AUTHOR]

In this article, we construct a novel iterative approach that depends on reproducing kernel method for Cahn-Allen equation with Caputo derivative. Representation of solution and convergence analysis are presented theoretically. Numerical results are given as tables and graphics with intent to show efficiency and power of method. The results demonstrate that approximate solution uniformly converges to exact solution for Cahn-Allen equation with fractional derivative. [ABSTRACT FROM AUTHOR]

An analysis was performed in this paper to study the problem of magnetohydrodynamic squeezing flow of nanofluid between parallel disks. This paper also proposed a reproducing kernel method for solving fourth-order four-point boundary value problems. In order to eliminate the singularity of the equation, a transform was used. Convergence analysis and error estimation for the present method in -space were also discussed. The exact solution was represented in the form of series in the reproducing kernel Hilbert space which proved higher accuracy in numerical computation. The algorithm derived from this approach can be easily implemented. The ideas and techniques presented in this paper will be useful for solving many other problems. [ABSTRACT FROM AUTHOR]

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value α is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test. [ABSTRACT FROM AUTHOR]

If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a reflexive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint. [ABSTRACT FROM AUTHOR]

The standard Sobolev space W 2 s (ℝ d) , with arbitrary positive integers s and d for which s > d / 2 , has the reproducing kernel K d , s (x , t) = ∫ ℝ d ∏ j = 1 d cos (2 π (x j − t j) u j) 1 + ∑ 0 < | α | 1 ≤ s ∏ j = 1 d (2 π u j) 2 α j d u for all x , t ∈ ℝ d , where x j , t j , u j , α j are components of d -variate x , t , u , α , and | α | 1 = ∑ j = 1 d α j with non-negative integers α j . We obtain a more explicit form for the reproducing kernel K 1 , s and find a closed form for the kernel K d , ∞ . Knowing the form of K d , s , we present applications on the best embedding constants between the Sobolev space W 2 s (ℝ d) and L ∞ (ℝ d) , and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in d , whereas worst case integration errors of algorithms using n function values are also exponentially small in d and decay at least like n − 1 / 2 . This yields strong polynomial tractability in the worst case setting for the absolute error criterion. [ABSTRACT FROM AUTHOR]

Kernel methods have been successfully applied to nonlinear problems in machine learning and signal processing. Various kernel-based algorithms have been proposed over the last two decades. In this paper, we investigate the kernel conjugate gradient (KCG) algorithms in both batch and online modes. By expressing the solution vector of CG algorithm as a linear combination of the input vectors and using the kernel trick, we developed the KCG algorithm for batch mode. Because the CG algorithm is iterative in nature, it can greatly reduce the computations by the technique of reduced-rank processing. Moreover, the reduced-rank processing can provide the robustness against the problem of overlearning. The online KCG algorithm is also derived, which converges as fast as the kernel recursive least squares (KRLS) algorithm, but the computational cost is only a quarter of that of the KRLS algorithm. Another attractive feature of the online KCG algorithm compared with other kernel adaptive algorithms is that it does not require the user-defined parameters. To control the growth of data size in online applications, a simple sparsification criterion based on the angles among elements in reproducing kernel Hilbert space is proposed. The angle criterion is equivalent to the coherence criterion but does not require the kernel to be unit norm. Finally, numerical experiments are provided to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A new binary relation associated with the core--EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the (dual) core partial order between core parts of operators and the minus partial order between quasinilpotent parts of operators, new pre-orders and partial orders are also introduced and characterized. [ABSTRACT FROM AUTHOR]

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

Reproducing kernel method for approximating solutions of linear boundary value problems is valid in Hilbert spaces composed of continuous functions, but its convergence is not satisfactory without additional smoothness assumptions. We prove 2nd order uniform convergence for regular problems with coefficient piecewise of Sobolev class H 2 . If the coefficients are globally of class H 2 , more refined phantom boundary NSC-RKHS method is derived, and the order of convergence rises to 3 or 4, according to whether the problem is piecewise of class H 3 or H 4 . The algorithms can be successfully applied to various non-local linear boundary conditions, e.g. of simple integral form. The paper contains also a new explicit formula for general spline reproducing kernels in H m [ a, b ], if the inner product 〈 f , g 〉 m , ξ = ∑ i < m f ( i ) ( ξ ) g ( i ) ( ξ ) + ∫ f ( m ) g ( m ) depends on any fixed reference point ξ  ∈ [ a, b ]. The piecewise NSC–RKHS methods are then applied to two example regular LBV problems in H 3 and H 5 . Exactness of the resulting numerical solutions, the degree of convergence, and their dependency of the reference point ξ  ∈ [ a, b ] are presented in attached figures. [ABSTRACT FROM AUTHOR]

Canonical correlation analysis (CCA) is a powerful statistical tool for detecting the linear relationship between two sets of multivariate variables. Kernel generalization of it, namely, kernel CCA is proposed to describe nonlinear relationship between two variables. Although kernel CCA can achieve dimensionality reduction results for high-dimensional data feature selection problem, it also yields the so called over-fitting phenomenon. In this paper, we consider a new kernel CCA algorithm via randomized Kaczmarz method. The main contributions of the paper are: (1) A new kernel CCA algorithm is developed, (2) theoretical convergence of the proposed algorithm is addressed by means of scaled condition number, (3) a lower bound which addresses the minimum number of iterations is presented. We test on both synthetic dataset and several real-world datasets in cross-language document retrieval and content-based image retrieval to demonstrate the effectiveness of the proposed algorithm. Numerical results imply the performance and efficiency of the new algorithm, which is competitive with several state-of-the-art kernel CCA methods. [ABSTRACT FROM AUTHOR]

One of the most important subject in financial mathematics is the option pricing. The most famous result in this area is Black–Scholes formula for pricing European options. This paper is concerned with a method for solving a generalized Black–Scholes equation in a reproducing kernel Hilbert space. Subsequently, the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method. Furthermore, the error estimates for obtained approximation in reproducing kernel Hilbert space are presented. Finally, a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the optimal harvesting problem for age distribution and weighted size competitive species with diffusion has been studied. The existence and uniqueness of solution for the system are proven using the Banach fixed-point theorem. The existence of optimal control is demonstrated and necessary optimality conditions are obtained via tangent-normal cone technique in nonlinear functional analysis. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is proposed to solve the nonlinear Bitsadze–Samarskii boundary value problems with multi-point boundary conditions. The algorithm is based on the reproducing kernel Hilbert space method. We use an iterative scheme to overcome the nonlinearity of the problem. The convergence and error estimate of the iterative scheme are established. The reproducing kernel Hilbert space method is used to generate an approximation of the linearized problem. In fact, the reproducing kernel Hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived. In order to show the efficiency and versatility of the proposed method, some numerical results are reported. The comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

We consider singularly perturbed differential-difference equation, which contains negative shift in the differentiated term with boundary layer behavior. RKHSM (Reproducing Kernel Hilbert Space Method) without Gram–Schmidt orthogonalization process, is considered in the present paper. We decompose the domain of the problem into two subintervals. One of them has not the boundary layer and the other one has. The side of the interval in which the boundary layer exists is important. If the boundary layer of this problem exists on the left side of interval, the RKHSM will provide a proper approximation of solution, otherwise for the implement of RKHSM, we need to change the variable of the singularly perturbed problem to shift the boundary layer region. [ABSTRACT FROM AUTHOR]

A number of graph kernel-based methods have been developed with great success in many fields, but very little research has been published that is concerned with a graph kernel in Reproducing Kernel Hilbert Space (RKHS). In this paper, we firstly start with a derived expression for two forms of information entropy of an undirected graph. They are approximated von Neumann entropy and Shannon entropy, and depend on vertex degree statistics. Secondly, we show the basic solution of a generalized differential operator. This solution is a specific reproducing kernel called the H 1 -reproducing kernel in H 1 -space, and then it is proven to satisfy the condition of Mercer kernel. Thirdly, based on the two aforementioned forms of information entropy and H 1 -reproducing kernel, we define two reproducing graph kernels: one is approximated von Neumann entropy reproducing graph kernel (AVNERGK), the other is Shannon entropy reproducing graph kernel (SERGK). And then we prove that they satisfy the condition of Mercer kernel. Finally, to obtain better classification results, we further propose a hybrid reproducing graph kernel (HRGK) based on the two reproducing graph kernels. We use the HRGK as a means to establish the similarity between a pair of graphs. Experimental results reveal that our method gives better classification performance on graphs extracted from several graph datasets. [ABSTRACT FROM AUTHOR]

We give a criterion for basicity of a sequence of complex exponentials in terms of the invertibility properties of a certain naturally associated Toeplitz operator. The criterion is similar to the well-known criterion of Hruschev, Nikolskii and Pavlov, the main difference being that we don't require preliminary translation of the frequency sequence to the upper half-plane. [ABSTRACT FROM AUTHOR]

Improvements in plant density tolerance have played an essential role in grain corn yield gains for ≈80 years; however, plant density effects on sweet corn biomass allocation to the ear (the reproductive 'sink') is poorly quantified. Moreover, optimal plant densities for modern white-kernel shrunken-2 (sh2) hybrids are unknown. The objectives of the study were to 1) quantify the effect of plant density and hybrid on the reproductive sink of sweet corn and 2) determine optimal plant densities for white-kernel sh2 sweet corn. Field experiments were conducted across 2 years on 10 white-kernel sh2 hybrids grown at plant densities ranging from 4.3 to 8.6 plants/m². Increasing plant density negatively influenced reproductive sink characteristics of individual sweet corn plants, including linear decreases in ear shoots/plant, marketable ears/plant, ear length, filled ear length, ear mass/plant, and kernel mass/plant. Reproductive traits varied widely among hybrids, including ear mass (15.6-20.6 Mt·ha-1) and recovery (32.3% to 42.4%), which is the contribution of fresh kernel mass to total ear mass. Hybrids had a common response to plant density, whereby ear yield was optimized at 5.5 plants/m2 and gross profit margin was optimized at 6.1 plants/m2. Plant density data from 586 growers' fields suggest current seeding rates have optimized the reproductive sink size for today's white-kernel sh2 hybrids. However, room exists for improving plant density tolerance, yield, and profitability. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a new iterative method to solve integral equations of the second kind in reproducing kernel Hilbert spaces (RKHS). At first, we make appropriate splitting in second kind integral equations and according to this splitting the iterative method will be constructed; then, bases of RKHS and reproducing kernel spaces properties are used to convert this problem to linear system of equations. We move between reproducing kernel spaces by changing bases in order to achieve more accurate approximate solutions. Classically, in iterative RKHS method, the number of iterations should be the same as the number of points; here, we present a type of iterative RKHS method without this limitation. Convergence of the proposed method is investigated, and the efficiency of the method is demonstrated through various examples. [ABSTRACT FROM AUTHOR]

As we known, reproducing kernel method (RKM) has been presented for solving differential equations for initial and boundary value problems. However, the direct application of the RKM presented in the previous works cannot produce good numerical results for Burgers’ equation. To solve this problem, this paper give a modified reproducing kernel method by piecewise technique. The exact solution is given by reproducing kernel functions in a series expansion form, the approximation solution is expressed by n-term summation of reproducing kernel functions. The three numerical experiments results show that the piecewise method is more easily implemented and effective. Some numerical results are also compared with the results obtained by other methods. [ABSTRACT FROM AUTHOR]

A number of recent papers have established connections between reproducing kernel Hilbert spaces H of entire functions, de Branges spaces, sampling formulas and a class of symmetric operators with deficiency indices ( 1 , 1 ) . In this paper analogous connections between reproducing kernel Hilbert spaces of entire vector valued functions, de Branges spaces of entire vector valued functions, sampling formulas and symmetric operators with deficiency indices ( p , p ) are obtained. Enroute, an analog of L. de Branges' characterization of the reproducing kernel Hilbert spaces of entire functions that are now called de Branges spaces is obtained for the p × 1 vector valued case. A special class of these de Branges spaces of p × 1 vector valued entire functions is identified as a functional model for M. G. Krein's class of entire operators with deficiency indices ( p , p ) . [ABSTRACT FROM AUTHOR]

This paper answers Bell's question: What does quantum information refer to? It is about quantum properties represented by subspaces of the quantum Hilbert space, or their projectors, to which standard (Kolmogorov) probabilities can be assigned by using a projective decomposition of the identity (PDI or framework) as a quantum sample space. The single framework rule of consistent histories prevents paradoxes or contradictions. When only one framework is employed, classical (Shannon) information theory can be imported unchanged into the quantum domain. A particular case is the macroscopic world of classical physics whose quantum description needs only a single quasiclassical framework. Nontrivial issues unique to quantum information, those with no classical analog, arise when aspects of two or more incompatible frameworks are compared. [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]

In this paper we present upper and lower estimates for the covering numbers of the unit ball of a reproducing kernel Hilbert space associated to a continuous isotropic kernel on a compact two-point homogeneous space (CTPHS). These estimates are obtained from estimates on the decay of the Fourier-Jacobi coefficients of the kernel via applications of the Funk-Hecke formula and the Schoenberg series representation of an isotropic kernel on CTPHS and also by the use of cubature formulas on these spaces. [ABSTRACT FROM AUTHOR]

In the product development process, structural optimization plays vital role because it deals with size, shape and topology of the structures. However, structural performance greatly depends on its geometric shape and hence structural shape optimization has remained one of the most active research areas since early 1970s. Conventional parametric shape optimization technique employs grid-based numerical tools like FEM and BEM for structural analysis, which experiences some innate limitations like mesh distortion and frequent remeshing, element locking and poor approximation while dealing with large shape changes during the optimization process. Meshless Methods (MMs) can alleviate these issues when used as a structural analysis tool in shape optimization. In last two decades, MMs have been explored for structural shape optimization along with various deterministic and stochastic optimization algorithms. The objective of present work is twofold, first is to review advanced parametric shape optimization techniques which are based on MMs like Element Free Galerkin (EFG) method and Reproducing Kernel Particle Method (RKPM) for linear elastic, thermoelastic, hyperelastic, frictional contact and structure dynamics optimization problems and second is to emphasize benefits of meshless techniques in shape optimization. Based on the review, the article presents some critical observations including Design Sensitivity Analysis (DSA) in meshless environment, numerical integration techniques in MMs and benefits of coupled FEM-MM approach in shape optimization. At the end, promising future research directions in shape optimization field based on MMs are presented along with concluding remarks. [ABSTRACT FROM AUTHOR]

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators. [ABSTRACT FROM AUTHOR]

The existence of unconditional bases of reproducing kernels in the Fock-type spaces F with radial weights φ is studied. It is shown that there exist functions φ( r) of arbitrarily slow growth for which ln r = o( φ( r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F . Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function. [ABSTRACT FROM AUTHOR]

We concentrate our attention on developing a meshfree flat-shell formulation for evaluating linear buckling loads and mode shapes (modes) of structural plates employing an eigen value analysis. A Galerkin-based shear deformable flat-shell formulation for that purpose is proposed. The in-plane and out-of-plane deformations are interpolated using the reproducing kernel particle method (RKPM), while the two membrane deformations, and the three deflection and rotational components are, respectively, approximated through a plane stress condition and Mindlin-Reissner plate theory. The meshfree discretization by which, as a consequence, constructs five degrees of freedom per node. A generalized eigenvalue problem for the solution of buckling loads and modes of the structural plates is then described. The stiffness matrices of the linear buckling analysis are numerically integrated based on the stabilized conforming nodal integration (SCNI) and sub-domain stabilized conforming integration (SSCI). The RKPM and SCNI/SSCI based on Galerkin meshfree formulation, i.e., stabilized meshfree Galerkin method, can overcome the shear locking problem by imposing the Kirchhoff mode reproducing condition. In addition, a singular kernel (SK) function is included in the meshfree interpolation functions to accurately impose the essential boundary conditions. The merits of the developed formulation are demonstrated through numerical buckling experiments of several examples of plates, by which the accuracy and performance of the proposed method are investigated and discussed in detail. It indicates from our numerical results of buckling loads and modes that the proposed meshfree formulation is accurate and useful in the simulation of buckling problems of structural stiffened plates. [ABSTRACT FROM AUTHOR]

This paper focuses on the high-dimensional additive quantile model, allowing for both dimension and sparsity to increase with sample size. We propose a new sparsity-smoothness penalty over a reproducing kernel Hilbert space (RKHS), which includes linear function and spline-based nonlinear function as special cases. The combination of sparsity and smoothness is crucial for the asymptotic theory as well as the computational efficiency. Oracle inequalities on excess risk of the proposed method are established under weaker conditions than most existing results. Furthermore, we develop a majorize-minimization forward splitting iterative algorithm (MMFIA) for efficient computation and investigate its numerical convergence properties. Numerical experiments are conducted on the simulated and real data examples, which support the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A novel approximate solution is obtained for viscoelastic fluid model by reproducing kernel method (RKM). The resulting equation for viscoelastic fluid with magneto-hydrodynamic flow is transformed to the nonlinear system by introducing the dimensionless variables. Results are presented graphically to study the efficiency and accuracy of the reproducing kernel method. Results show that this method namely RKM is an efficient method for solving nonlinear system in any engineering field. [ABSTRACT FROM AUTHOR]

This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible. [ABSTRACT FROM AUTHOR]

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method. [ABSTRACT FROM AUTHOR]

General natural image deblurring methods do not work well for document images. We exploit a two-tone prior to steer the intermediate latent image towards a piece-wise constant image with only two distinct gray levels. This prior is helpful for the process of kernel estimation to overcome undesirable local minima, and it is not too restrictive to deblur text images with complex backgrounds. Our kernel estimation method comprises two stages, where we first employ contrast-enhancing two-tone prior and then use intermediate-value inhibition regularizer. The resulting optimization formulation is solved by half-quadratic splitting and alternating minimization techniques. The experimental results show that the proposed method is capable of achieving accurate results and compares well with the state-of-the-art. [ABSTRACT FROM AUTHOR]

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind. First, we find the solution of integral equation in terms of reproducing kernel functions in series, then by truncating the series an approximate solution obtained. In addition, the calculation of Fourier coefficients solution of the integral equation in terms of reproducing kernel functions is notable. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

A non-parametric extension of control variates is presented. These leverage gradient information on the sampling density to achieve substantial variance reduction. It is not required that the sampling density be normalized. The novel contribution of this work is based on two important insights: a trade-off between random sampling and deterministic approximation and a new gradient-based function space derived from Stein's identity. Unlike classical control variates, our estimators improve rates of convergence, often requiring orders of magnitude fewer simulations to achieve a fixed level of precision. Theoretical and empirical results are presented, the latter focusing on integration problems arising in hierarchical models and models based on non-linear ordinary differential equations. [ABSTRACT FROM AUTHOR]

Various kernel-based methods have been developed with great success in many fields, but very little research has been published that is concerned with a multiple attribute kernel in reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel elastic kernel called a multiple attribute convolution kernel with reproducing property (MACKRP) and present improved classification results over conventional approaches in the RKHS rather than the more commonly used Hilbert space. The MACKRP consists of two major steps. First, we find the basic solution of a generalized differential operator by the delta function, and then we design a convolution function using this solution. This convolution function is proven to be a specific reproducing kernel called a convolution reproducing kernel (CRK) in H -space. Second, we prove that the CRK satisfies the condition of Mercer kernel. And the CRK is composed of three attributes ( L -norm, L -norm and Laplace kernel), and each attribute can capture a different feature, with all attributes generating a novel kernel which we call an MACKRP. The experimental results demonstrate that the MACKRP possesses approximation and regularization performance and that classification results are consistently comparable or superior to a number of other state-of-the-art kernel functions. [ABSTRACT FROM AUTHOR]

The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]

This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n -term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective. [ABSTRACT FROM AUTHOR]

We study two geometric properties of reproducing kernels in model spaces K θ where θ is an inner function: overcompleteness and existence of uniformly minimal systems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern–Clark point. It is shown that “uniformly minimal non-Riesz” sequences of reproducing kernels exist near each Ahern–Clark point which is not an analyticity point for θ , while overcompleteness may occur only near the Ahern–Clark points of infinite order and is equivalent to a “zero localization property”. In this context the notion of quasi-analyticity appears naturally, and as a by-product of our results we give conditions in the spirit of Ahern–Clark for the restriction of a model space to a radius to be a class of quasi-analyticity. [ABSTRACT FROM AUTHOR]

This paper applies the reproducing kernel Hilbert space method to the solutions of three types of Fornberg–Whitham equations: original, modified and time fractional. Comparison with Adomian decomposition method, homotopy analysis method and the variational iteration method shows the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

We present a formulation of smoothed particle hydrodynamics (SPH) that utilizes a first-order consistent reproducing kernel, a smoothing function that exactly interpolates linear fields with particle tracers. Previous formulations using reproducing kernel (RK) interpolation have had difficulties maintaining conservation of momentum due to the fact the RK kernels are not, in general, spatially symmetric. Here, we utilize a reformulation of the fluid equations such that mass, linear momentum, and energy are all rigorously conserved without any assumption about kernel symmetries, while additionally maintaining approximate angular momentum conservation. Our approach starts from a rigorously consistent interpolation theory, where we derive the evolution equations to enforce the appropriate conservation properties, at the sacrifice of full consistency in the momentum equation. Additionally, by exploiting the increased accuracy of the RK method's gradient, we formulate a simple limiter for the artificial viscosity that reduces the excess diffusion normally incurred by the ordinary SPH artificial viscosity. Collectively, we call our suite of modifications to the traditional SPH scheme Conservative Reproducing Kernel SPH, or CRKSPH. CRKSPH retains many benefits of traditional SPH methods (such as preserving Galilean invariance and manifest conservation of mass, momentum, and energy) while improving on many of the shortcomings of SPH, particularly the overly aggressive artificial viscosity and zeroth-order inaccuracy. We compare CRKSPH to two different modern SPH formulations (pressure based SPH and compatibly differenced SPH), demonstrating the advantages of our new formulation when modeling fluid mixing, strong shock, and adiabatic phenomena. [ABSTRACT FROM AUTHOR]