AHermitian C^(2) Differential Reproducing Kernel Interpolation Meshless Method for the 3D Microstructure-Dependent Static Flexural Analysis of Simply Supported and Functionally Graded Microplates

This work develops a Hermitian C^(2) differential reproducing kernel interpolation meshless(DRKIM)method within the consistent couple stress theory(CCST)framework to study the three-dimensional(3D)microstructuredependent static flexural behavior of a functionally graded(FG)microplate subjected to mechanical loads and placed under full simple supports.In the formulation,we select the transverse stress and displacement components and their first-and second-order derivatives as primary variables.Then,we set up the differential reproducing conditions(DRCs)to obtain the shape functions of the Hermitian C^(2) differential reproducing kernel(DRK)interpolant’s derivatives without using direct differentiation.The interpolant’s shape function is combined with a primitive function that possesses Kronecker delta properties and an enrichment function that constituents DRCs.As a result,the primary variables and their first-and second-order derivatives satisfy the nodal interpolation properties.Subsequently,incorporating ourHermitianC^(2)DRKinterpolant intothe strong formof the3DCCST,we develop a DRKIM method to analyze the FG microplate’s 3D microstructure-dependent static flexural behavior.The Hermitian C^(2) DRKIM method is confirmed to be accurate and fast in its convergence rate by comparing the solutions it produces with the relevant 3D solutions available in the literature.Finally,the impact of essential factors on the transverse stresses,in-plane stresses,displacements,and couple stresses that are induced in the loaded microplate is examined.These factors include the length-to-thickness ratio,the material length-scale parameter,and the inhomogeneity index,which appear to be significant.

Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.

An Interpretable Denoising Layer for Neural Networks Based on Reproducing Kernel Hilbert Space and its Application in Machine Fault Diagnosis

Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.

Stability of Fredholm Integral Equation of the First Kind in Reproducing Kernel Space

It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a, b] or L2 [a, b]. In this paper, the representation of the solution for Fredholm integral equation of the first kind is given if it has a unique solution. The stability of the solution is proved in the reproducing kernel space, namely, the measurement errors of the experimental data cannot result in unbounded errors of the true solution. The computation of approximate solution is also stable with respect to ｜｜· ｜｜c or ｜｜L2· A numerical experiment shows that the method given in this paper is stable in the reproducing kernel space.

A Geometrically Exact Triangular Shell Element Based on Reproducing Kernel DMS-Splines

Tomodel amultibody systemcomposed of shell components,a geometrically exact Kirchho-Love triangular shell element is proposed.The middle surface of the shell element is described by using the DMS-splines,which can exactly represent arbitrary topology piecewise polynomial triangular surfaces.The proposed shell element employs only nodal displacement and can automatically maintain C1 continuity properties at the element boundaries.A reproducing DMS-spline kernel skill is also introduced to improve computation stability and accuracy.The proposed triangular shell element based on reproducing kernel DMS-splines can achieve an almost optimal convergent rate.Finally,the proposed shell element is validated via three static problems of shells and the dynamic simulation of aexible multibody system undergoing both overall motions and large deformations.

RECURSIVE REPRODUCING KERNELS HILBERT SPACES USING THE THEORY OF POWER KERNELS

The main objective of this work is to decompose orthogonally the reproducing kernels Hilbert space using any conditionally positive definite kernels into smaller ones by introducing the theory of power kernels, and to show how to do this decomposition recur- sively. It may be used to split large interpolation problems into smaller ones with different kernels which are related to the original kernels. To reach this objective, we will reconstruct the reproducing kernels Hilbert space for the normalized and the extended kernels and give the recursive algorithm of this decomposition.

Convergence analysis for complementary-label learning with kernel ridge regression

Complementary-label learning(CLL)aims at finding a classifier via samples with complementary labels.Such data is considered to contain less information than ordinary-label samples.The transition matrix between the true label and the complementary label,and some loss functions have been developed to handle this problem.In this paper,we show that CLL can be transformed into ordinary classification under some mild conditions,which indicates that the complementary labels can supply enough information in most cases.As an example,an extensive misclassification error analysis was performed for the Kernel Ridge Regression(KRR)method applied to multiple complementary-label learning(MCLL),which demonstrates its superior performance compared to existing approaches.

Minimax designs for linear regression models with bias in a reproducing kernel Hilbert space in a discrete set

Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs. These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.

A NOVEL METHOD FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS IN BROKEN REPRODUCING KERNEL SPACE

In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.

ON APPROXIMATION BY SPHERICAL REPRODUCING KERNEL HILBERT SPACES

The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.

The Expansion of the Function with Two Unknowns on the Reproducing Kernel Space

In this paper we make use of a special procedure on the repro ducing kernel space to give an expansion theorem for the function with two unkno wns and a surface approximation formula. The error of the surface possesses mono tonically decreasing and uniformly convergent characteristics in the sense of t he norm on the space.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.

Laguerre reproducing kernel method in Hilbert spaces for unsteady stagnation point ow over a stretching/shrinking sheet

This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid.To solve this equation,a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method.Using the operational matrices of derivative,we reduced the problem to a set of algebraic equations.We also compare this work with some other numerical results and present a solution that proves to be highly accurate.

h-ADAPTIVITY ANALYSIS BASED ON MULTIPLE SCALE REPRODUCING KERNEL PARTICLE METHOD

An h-adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two node refinement strategies were constructed using searching-neighbor-nodes(SNN) and local-Delaunay-triangulation(LDT) tech-niques, which were suitable and effective for h-adaptivity analysis on 2-D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h-adaptivity analyses on 2-D linear elastostatics and bending plate problems demonstrate that the improper high-gradient indicator will reduce the convergence property of the h-adaptivity analysis, and that the efficiency of the LDT node refinement strategy is better than SNN, and that the presented h-adaptivity analysis scheme is provided with the validity, stability and good convergence property.

h-ADAPTIVITY ANALYSIS BASED ON MULTIPLE SCALE REPRODUCING KERNEL PARTICLE METHOD

An h-adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two node refinement strategies were constructed using searching-neighbor-nodes（SNN） and local-Delaunay-triangulation（LDT） techniques, which were suitable and effective for h-adaptivity analysis on 2-D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h- adaptivity analyses on 2-D linear elastostatics and bending plate problems demonstrate that the improper high-gradient indicator will reduce the convergence property of the h- adaptivity analysis, and that the efficiency of the LDT node refinement strategy is better than SNN, and that the presented h-adaptivity analysis scheme is provided with the validity, stability and good convergence property.

The Boundedness of Littlewood-Paley Operators with Rough Kernels on Weighted (L^q , L^p )~α (R^n) Spaces

In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^＊ on the weighted amalgam spaces （Lω^q,L^p）^α（R^n）as 1〈q≤α〈p≤∞.

The complex variable reproducing kernel particle method for two-dimensional elastodynamics

On the basis of the reproducing kernel particle method （RKPM）, a new meshless method, which is called the complex variable reproducing kernel particle method （CVRKPM）, for two-dimensional elastodynamics is presented in this paper. The advantages of the CVRKPM are that the correction function of a two-dimensional problem is formed with one-dimensional basis function when the shape function is obtained. The Galerkin weak form is employed to obtain the discretised system equations, and implicit time integration method, which is the Newmark method, is used for time history analysis. And the penalty method is employed to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional elastodynamics are obtained. Three numerical examples of two-dimensional elastodynamics are presented, and the CVRKPM results are compared with the ones of the RKPM and analytical solutions. It is evident that the numerical results of the CVRKPM are in excellent agreement with the analytical solution, and that the CVRKPM has greater precision than the RKPM.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

ANALYSIS OF THREE-DIMENSIONAL UPSETTING PROCESS BY THE RIGID-PLASTIC REPRODUCING KERNEL PARTICLE METHOD

A meshless approach, called the rigid-plastic reproducing kernel particle method （RKPM）, is presented for three-dimensional （3D） bulk metal forming simulation. The approach is a combination of RKPM with the flow theory of 3D rigid-plastic mechanics. For the treatments of essential boundary conditions and incompressibility constraint, the boundary singular kernel method and the modified penalty method are utilized, respectively. The arc-tangential friction model is employed to treat the contact conditions. The compression of rectangular blocks, a typical 3D upsetting operation, is analyzed for different friction conditions and the numerical results are compared with those obtained using commercial rigid-plastic FEM （finite element method） software Deform^3D. As results show, when handling 3D plastic deformations, the proposed approach eliminates the need of expensive meshing and remeshing procedures which are unavoidable in conventional FEM and can provide results that are in good agreement with finite element predictions.

FRACTIONAL INTEGRALS WITH VARIABLE KERNELS ON HARDY SPACES

The fractional integral operators with variable kernels are discussed.It is proved that if the kernel satisfies the Dini-condition,then the fractional integral operators with variable kernels are bounded from Hp(Rn) into Lq(Rn) when 0<p≤1 and 1/q=1/p-α/n.The results in this paper improve the results obtained by Ding,Chen and Fan in 2002.