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Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem
Mo Y.; Qian T.
2014-10-15
Source PublicationApplied Mathematics and Computation
ISSN00963003
Volume245Pages:509-519
Abstract

Numerical solutions of partial differential equations are traditional topics that have been studied by many researchers. During the last decade, support vector machine (SVM) has been widely used for approximation problems. The contribution of this paper is two folds. One is to combine the reproducing kernel-SVM method with the Tikhonov regularization method, called the SVM-Tik methods, in which the kernels K and Kλσ (see below) are newly developed. In the paper they are respectively phrased as the SVM-Tik-K and SVM-Tik-Kλσ methods. The second contribution is to use the two models, SVM-Tik-K and SVM-Tik-Kλσ, to solve the Dirichlet problem. The methods are meshless. They produce sparse representations in the linear combination form of specific functions (the K and Kλσ kernels). The generalization bound result in learning theory is used to give an estimation of the approximation errors. With the illustrative examples the sparseness and robustness properties, as well as the effectiveness of the methods are presented. The proposed methods are compared with currently the most commonly used finite difference method (FDM) showing promising results. © 2014 Elsevier Inc. All rights reserved.

KeywordDirichlet Problem Gaussian Rkhs Reproducing Kernel Sobolev Space Support Vector Machine Tikhonov Regularization
DOIhttps://doi.org/10.1016/j.amc.2014.07.089
URLView the original
Indexed BySCI
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000343613900044
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Cited Times [WOS]:8   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorQian T.
AffiliationDepartment of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau, China
First Author AffilicationFaculty of Science and Technology
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/T 7714
Mo Y.,Qian T.. Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem[J]. Applied Mathematics and Computation,2014,245:509-519.
APA Mo Y.,&Qian T..(2014).Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem.Applied Mathematics and Computation,245,509-519.
MLA Mo Y.,et al."Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem".Applied Mathematics and Computation 245(2014):509-519.
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