Fast algorithms for interpolation with clamped $L$-splines of order four

TL;DR

This paper develops fast, stable algorithms for interpolation with clamped L-splines of order four, extending previous methods for natural L-splines and enabling applications in PDE solving.

Contribution

It explicitly constructs and proves the invertibility of the linear system for clamped L-splines, ensuring numerical stability of the fast algorithm.

Findings

The linear system's matrix is strictly row diagonally dominant.

The method is implemented in MATLAB.

Clamped L-splines can be used for PDEs as an alternative to PINNs.

Abstract

Interpolation and smoothing using cubic and generalized splines are fundamental tools in data analysis and statistical modeling. Recently, fast computational algorithms were developed for natural $L$-splines of order four, which arise as piecewise solutions to the differential operator $L_{\xi}^2 = (\frac{d^2}{dt^2} - \xi^2)^2$. In this paper, we extend this mathematical framework to the important case of clamped (or complete) boundary conditions, where the first derivatives at the interval endpoints are prescribed. We explicitly construct the governing linear system for the interpolation problem and mathematically prove that the resulting tridiagonal matrix is strictly row diagonally dominant, thereby guaranteeing its invertibility and the numerical stability of the fast algorithm. The proposed method is implemented in MATLAB. Furthermore, the developed clamped $L$-splines provide a foundation for constructing multivariate clamped polysplines, which serve as a promising alternative to Physics-Informed Neural Networks (PINNs) for solving partial differential equations in Mathematical Physics.