Mesh-Free RBF-FD Method with Polyharmonic Splines and Polynomials for High-Dimensional PDEs and Financial Option Pricing

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.