A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

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• 作者：Ali Safdari-Vaighani ; Alfa Heryudono ; Elisabeth Larsson
• 关键词：Collocation method ; Meshfree ; Radial basis function ; Partition of unity ; RBF–PUM ; Convection–diffusion equation ; American option ; MSC 65M70 ; MSC 35K15
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：64
• 期：2
• 页码：341-367
• 全文大小：1,789 KB
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• 作者单位：Ali Safdari-Vaighani (1)
  Alfa Heryudono (2)
  Elisabeth Larsson (3)
  
  1. Department of Mathematics, Allameh Tabataba’i University, Tehran, Iran
  2. Department of Mathematics, University of Massachusetts Dartmouth, Dartmouth, MA, USA
  3. Department of Information Technology, Uppsala University, Uppsala, Sweden
  
• 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

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF–PUM) proposed here. The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type. The stability and accuracy of RBF–PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection–diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF–PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF–PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.