A Modified Hermite Radial Basis Function for Accurate Interpolation

TL;DR

The paper addresses ill-conditioning in interpolation with infinitely smooth kernels by proposing a Modified Hermite Radial Basis Function (MHRBF) that injects a monomial scaling term into the kernel and uses polynomial augmentation. This approach preserves first-derivative contributions while reducing reliance on higher-order derivatives, enabling accurate interpolation in double precision with stable conditioning across a range of shape parameters $\varepsilon$ and node densities. Key findings show that near $\varepsilon \approx 1$ and monomial degree $n \approx 4$, MHRBF delivers near-minimal errors for both trigonometric and polynomial test functions, with linear polynomial augmentation often sufficing for efficiency. The results demonstrate improved accuracy, robust convergence, and favorable computational cost for MHRBF relative to standard HRBF, making it attractive for high-performance computing applications in PDEMesh-free contexts.

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.

Figures (12)

• Figure 1: Effect of the monomial term on the GA kernel for different values of $n$ using $x_0=0$.
• Figure 2: Distribution of data and error evaluation nodes within a circle of radius 0.1.
• Figure 3: Contour plots of the $L_\infty$ error for the MHRBF method as a function of shape parameter $\varepsilon$ and monomial degree $n$. Separate plots are shown for function values and first derivatives ($f_x$, $f_y$), revealing optimal parameter combinations that minimize error.
• Figure 4: Comparison of MHRBF accuracy with polynomial augmentation of degrees 1 and 9: The errors (function, $f$, and its first derivatives, $f_x$ and $f_y$) computed in double precision using the GA kernel, as functions of shape parameter $\varepsilon$, with $n=4$.
• Figure 5: Comparison of Accuracy for HRBF and MHRBF: The errors (function, $f$, and its first derivatives, $f_x$ and $f_y$) computed in double precision for GA kernel, as functions of shape parameter $\varepsilon$, with polynomial degree of $1$ and $n=4$ for MHRBF. Results correspond to the trigonometric test function.