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

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 imple...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 64; no. 2; pp. 341 - 367
Main Authors: Safdari-Vaighani, Ali, Heryudono, Alfa, Larsson, Elisabeth
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2015
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Approximation
Collocation methods
Computational efficiency
Computational Mathematics and Numerical Analysis
Computing costs
Computing time
Convection-diffusion equation
Convergence
Linear systems
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical functions
Mathematics
Mathematics and Statistics
Methods
Parabolic differential equations
Partial differential equations
Radial basis function
Spectral methods
Theoretical
Valuation
Radial basis function
MSC 65M70
MSC 35K15
American option
RBF–PUM
Convection–diffusion equation
Partition of unity
Collocation method
Meshfree
ISSN:0885-7474, 1573-7691, 1573-7691
Online Access:Get full text
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Summary: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.
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ISSN:0885-7474
1573-7691
1573-7691
DOI:10.1007/s10915-014-9935-9