Fast algorithms for interpolation with clamped $L$-splines of order four
For researchers in data analysis and numerical methods, this provides a numerically stable and efficient algorithm for clamped L-splines, enabling their use in solving PDEs via polysplines.
This paper extends fast algorithms for natural L-splines of order four to clamped boundary conditions, proving the resulting tridiagonal system is strictly diagonally dominant, ensuring invertibility and stability. The method is implemented in MATLAB and provides a foundation for multivariate clamped polysplines as an alternative to PINNs.
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 $L_ξ^2 = (\frac{d^2}{dt^2} - ξ^2)^2$. 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.