Optimization & Operations Research
Narges Hosseinzadeh; Elyas Shivanian; Saeid Abbasbandy
Abstract
This study employs the radial basis function-generated finite difference (RBF-FD) method to address high-dimensional elliptic differential equations under Dirichlet boundary conditions. The method utilizes polyharmonic spline functions (PHSs) combined with polynomials for approximation. ...
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This study employs the radial basis function-generated finite difference (RBF-FD) method to address high-dimensional elliptic differential equations under Dirichlet boundary conditions. The method utilizes polyharmonic spline functions (PHSs) combined with polynomials for approximation. A notable benefit of this approach is that PHSs do not require a shape parameter, simplifying implementation and enhancing numerical stability. The proposed method offers several advantages, including high accuracy, rapid computation, and adaptability to complex geometries and irregular node arrangements. It is particularly effective for high-dimensional problems, providing a mesh-free alternative that scales efficiently with increased complexity. Beyond scientific computing, the method is also applied to financial option pricing, where integro-differential equations are transformed into a series of second-order elliptic partial differential equations (PDEs). Numerical experiments demonstrate that the proposed algorithm significantly outperforms existing RBF-based approaches in both accuracy and efficiency. These strengths make it a robust tool for solving a wide range of PDEs in both regular and irregular domains.
