The Experts below are selected from a list of 19023 Experts worldwide ranked by ideXlab platform
Jiun-shyan Chen - One of the best experts on this subject based on the ideXlab platform.
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A study on convergence and complexity of Reproducing Kernel collocation method
Interaction and multiscale mechanics, 2009Co-Authors: Chiu-kai Lai, Jiun-shyan ChenAbstract:In this work, we discuss a Reproducing Kernel collocation method (RKCM) for solving order PDE based on strong formulation, where the Reproducing Kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of equations. We investigate the convergence and the computational complexity for this proposed method. An important result obtained from the analysis is that the degree of basis in the Reproducing Kernel approximation has to be greater than one for the method to converge. Some numerical experiments are provided to validate the error analysis. The complexity of RKCM is also analyzed, and the complexity comparison with the weak formulation using Reproducing Kernel approximation is presented.
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Filters, Reproducing Kernel, and adaptive meshfree method
Computational Mechanics, 2003Co-Authors: Y. You, Jiun-shyan ChenAbstract:Reproducing Kernel, with its intrinsic feature of moving averaging, can be utilized as a low-pass filter with scale decomposition capability. The discrete convolution of two nth order Reproducing Kernels with arbitrary support size in each Kernel results in a filtered Reproducing Kernel function that has the same Reproducing order. This property is utilized to separate the numerical solution into an unfiltered lower order portion and a filtered higher order portion. As such, the corresponding high-pass filter of this Reproducing Kernel filter can be used to identify the locations of high gradient, and consequently serves as an operator for error indication in meshfree analysis. In conjunction with the naturally conforming property of the Reproducing Kernel approximation, a meshfree adaptivity method is also proposed.
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Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched Reproducing Kernel particle method
Finite Elements in Analysis and Design, 2003Co-Authors: Dongdong Wang, Jiun-shyan Chen, Lizhi SunAbstract:A formulation is proposed for homogenization of magnetostrictive particle-filled elastomers (MPFE) based on an interface-enriched Reproducing Kernel particle method. A variational equation for obtaining the local fluctuating deformation of MPFE is introduced. The magnetostrictive effect in the metal inclusion is modeled as an eigen-deformation. An interface-enriched Reproducing Kernel approximation with embedded derivative discontinuities on the material interface is presented. This approach does not require additional degrees of freedom in the approximation of displacement field for the interface conditions compared to the conventional Reproducing Kernel approximation. Microscopic solution and homogenized constitutive behavior of uniaxial tension and simple shear deformation of MPFE are presented.
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a lagrangian Reproducing Kernel particle method for metal forming analysis
Computational Mechanics, 1998Co-Authors: Jiun-shyan Chen, C Pan, Cristina Maria Oliveira Lima Roque, Huiping WangAbstract:A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the Reproducing Kernel approximation that satisfies consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current configuration. A Lagrangian Kernel function, and its corresponding Reproducing Kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian Reproducing Kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.
Jun Zhang - One of the best experts on this subject based on the ideXlab platform.
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Reproducing Kernel banach spaces for machine learning
Journal of Machine Learning Research, 2009Co-Authors: Haizhang Zhang, Jun ZhangAbstract:We introduce the notion of Reproducing Kernel Banach spaces (RKBS) and study special semi-inner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its Reproducing Kernel are investigated. As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and Kernel principal component analysis. In particular, existence, uniqueness and representer theorems are established.
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Reproducing Kernel Banach spaces for machine learning
2009 International Joint Conference on Neural Networks, 2009Co-Authors: Haizhang Zhang, Yuesheng Xu, Jun ZhangAbstract:Reproducing Kernel Hilbert space (RKHS) methods have become powerful tools in machine learning. However, their Kernels, which measure similarity of inputs, are required to be symmetric, constraining certain applications in practice. Furthermore, the celebrated representer theorem only applies to regularizers induced by the norm of an RKHS. To remove these limitations, we introduce the notion of Reproducing Kernel Banach spaces (RKBS) for pairs of reflexive Banach spaces of functions by making use of semi-inner-products and the duality mapping. As applications, we develop the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, and support vector machines. In particular, existence, uniqueness and representer theorems are established.
Kam Liu - One of the best experts on this subject based on the ideXlab platform.
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Enrichment of the Finite Element Method With the Reproducing Kernel Particle Method
Journal of Applied Mechanics, 1997Co-Authors: Kam Liu, R.a. Uras, Y. ChenAbstract:Based on the Reproducing Kernel particle method on enrichment procedure is introduced to enhance the effectiveness of the finite element method. The basic concepts for the Reproducing Kernel particle method are briefly reviewed. By adopting the well-known completeness requirements, a generalized form of the Reproducing Kernel particle method is developed. Through a combination of these two methods their unique advantages can be utilized. An alternative approach, the multiple field method is also introduced.
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Multiresolution Reproducing Kernel particle methods
Computational Mechanics, 1997Co-Authors: Kam Liu, Sukky Jun, Wei Hao, Yuli Chen, J. GoszAbstract:Reproducing Kernel Particle Methods (RKPM) with a built-in feature of multiresolution analysis are addressed. Some fundamental concepts such as Reproducing conditions, and correction function are constructed to systematize the framework of RKPM. In particular, Fourier analysis, as a tool, is exploited to further elaborate RKPM in the frequency domain. Furthermore, we address error estimation and convergence properties. We present several applications which confirm the widespread applicability of multiresolution RKPM.
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Generalized multiple scale Reproducing Kernel particle methods
Computer Methods in Applied Mechanics and Engineering, 1996Co-Authors: Kam Liu, Y. Chen, R.a. Uras, Chin Tang ChangAbstract:Abstract An approach to unify Reproducing Kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed. The study is divided into three major topics. The groundwork is set by revisiting the Fourier analysis of discrete systems. The multiresolution concept and its significance in devising the Reproducing Kernel methods and its discrete counterpart, Reproducing Kernel particle methods, are explained. An edge detection technique based on multiresolution analysis is developed. This wavelet approach, together with particle methods, gives rise to a straightforward h -adaptivity algorithm. By using this framework, a Hermite Reproducing Kernel method is also proposed, and its relation to wavelet methods is presented. It is also shown that the new approach generalizes existing Kernel methods, and it can easily be degenerated into other widely used methods such as partition of unity, moving least-square interpolants, smooth particle hydrodynamics, scaling functions and wavelets, and multiple scale analysis. Furthermore, the Hermite Reproducing Kernel particle method, a particle based discrete version of the Hermite Reproducing Kernel method is developed. Finally, multiple-scale methods based on frequency and wave number shifting techniques are presented. A stability analysis is also presented for Newmark time-integration schemes for the low frequency equation. Numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods.
Haizhang Zhang - One of the best experts on this subject based on the ideXlab platform.
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Reproducing Kernel banach spaces for machine learning
Journal of Machine Learning Research, 2009Co-Authors: Haizhang Zhang, Jun ZhangAbstract:We introduce the notion of Reproducing Kernel Banach spaces (RKBS) and study special semi-inner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its Reproducing Kernel are investigated. As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and Kernel principal component analysis. In particular, existence, uniqueness and representer theorems are established.
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Reproducing Kernel Banach spaces for machine learning
2009 International Joint Conference on Neural Networks, 2009Co-Authors: Haizhang Zhang, Yuesheng Xu, Jun ZhangAbstract:Reproducing Kernel Hilbert space (RKHS) methods have become powerful tools in machine learning. However, their Kernels, which measure similarity of inputs, are required to be symmetric, constraining certain applications in practice. Furthermore, the celebrated representer theorem only applies to regularizers induced by the norm of an RKHS. To remove these limitations, we introduce the notion of Reproducing Kernel Banach spaces (RKBS) for pairs of reflexive Banach spaces of functions by making use of semi-inner-products and the duality mapping. As applications, we develop the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, and support vector machines. In particular, existence, uniqueness and representer theorems are established.
Ali Akgul - One of the best experts on this subject based on the ideXlab platform.
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Reproducing Kernel functions and homogenizing transforms
Thermal Science, 2021Co-Authors: Elif Nuray Yildirim, Ali AkgulAbstract:A lot of problems of the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution and its effectiveness are as important as the modelling. In this paper, on the basis of Reproducing Kernel theory, the Reproducing Kernel functions have been obtained for solving some non-linear higher order differential equations. Additionally, for each problem the homogenizing transforms have been obtained.
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New Reproducing Kernel functions in the Reproducing Kernel Sobolev spaces
AIMS Mathematics, 2020Co-Authors: Ali Akgul, Esra Karatas Akgül, Sahin KorhanAbstract:In this paper we construct some new Reproducing Kernel functions in the Reproducing Kernel Sobolev space. These functions are new in the literature. We can solve many problems by these functions in the Reproducing Kernel Sobolev spaces.
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Reproducing Kernel Hilbert space method based on Reproducing Kernel functions for investigating boundary layer flow of a Powell–Eyring non-Newtonian fluid
Journal of Taibah University for Science, 2019Co-Authors: Ali AkgulAbstract:In this work, the boundary layer flow of a Powell–Eyring non-Newtonian fluid over a stretching sheet has been investigated by a Reproducing Kernel method. Reproducing Kernel functions are used to o...
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Reproducing Kernel method for fractional derivative with non local and non singular Kernel
2019Co-Authors: Ali AkgulAbstract:Atangana and Baleanu introduced a derivative with fractional order to answer some outstanding questions that were posed by many investigators within the field of fractional calculus. Their derivative has a non-singular and nonlocal Kernel. Therefore, we apply the Reproducing Kernel method to fractional differential equations with non-local and non-singular Kernel. In this work, a new method has been developed for the newly established fractional differentiation. Examples are given to illustrate the numerical effectiveness of the Reproducing Kernel method when properly applied in the Reproducing Kernel space. The comparison of approximate and exact solutions leaves no doubt believing that the Reproducing Kernel method is very efficient and converges toward exact solution very rapidly.
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Reproducing Kernel functions for the generalized Kuramoto-Sivashinsky equation
ITM Web of Conferences, 2018Co-Authors: Ali Akgul, Esra Karatas Akgül, Sahin KorhanAbstract:Reproducing Kernel functions are obtained for the solution of generalized Kuramoto–Sivashinsky (GKS) equation in this paper. These Reproducing Kernel functions are valuable in the Reproducing Kernel Hilbert space method. They will be useful for interested researchers.
