A Modified Hermite Radial Basis Function for Accurate Interpolation (2503.05752v3)

Abstract: Accurate interpolation of functions and derivatives is crucial in solving partial differential equations (PDEs). The Radial Basis Function (RBF) method has become an extremely popular and robust approach for interpolation on scattered data. Hermite Radial Basis Function (HRBF) methods are an extension of the RBF and improve the overall accuracy by incorporating both function and derivative information. Infinitely smooth kernels, such as the Gaussian, use a shape-parameter to describe the width of support and are widely used due to their excellent approximation accuracy and ability to capture fine-scale details. Unfortunately, the use of infinitely smooth kernels suffers from ill-conditioning at low to moderate shape parameters, which affects the accuracy. This work proposes a Modified HRBF (MHRBF) method that introduces an additional polynomial term to balance kernel behavior, improving accuracy while maintaining or lowering computational cost. Using standard double-precision mathematics, the results indicate that compared to the HRBF method, the MHRBF method achieves lower error for all values of the shape parameter and domain size. The MHRBF is also able to achieve low errors at a lower computational cost as compared to the standard HRBF method.