Multiscale Support Vector Approach for Solving Ill-Posed Problems

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• 作者：Min Zhong ; Yiu Chung Hon ; Shuai Lu
• 关键词：Multiscale support vector approach ; Compactly supported radial basis functions ; Ill ; posed problems ; Regularization methods ; 45Q05 ; 47A52 ; 65J20
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：64
• 期：2
• 页码：317-340
• 全文大小：767 KB
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• 作者单位：Min Zhong (1)
  Yiu Chung Hon (2)
  Shuai Lu (3)
  
  1. Department of Mathematics, Southeast University, Nanjing, 210096, China
  2. Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China
  3. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach to a multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik’s \(\epsilon \)-intensive function is adopted to replace the standard \(l^2\) loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik’s cut-off parameter and the regularization parameter. For noisy data case, we demonstrate that a corresponding choice for the Vapnik’s cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.