Stochastic Convergence of A Nonconforming Finite Element Method for the Thin Plate Spline Smoother for Observational Data
Provides theoretical guarantees for a finite element approach to thin plate spline smoothing, relevant for statistical and numerical analysts working with noisy scattered data.
The paper proves stochastic convergence of a nonconforming Morley finite element method for the thin plate spline smoother, characterizing the tail property of the finite element error distribution. A self-consistent iterative algorithm for smoothing parameter selection is proposed and validated with numerical examples.
The thin plate spline smoother is a classical model for fnding a smooth function from the knowledge of its observation at scattered locations which may have random noises. We consider a nonconforming Morley finite element method to approximate the model. We prove the stochastic convergence of the finite element method which characterizes the tail property of the probability distribution function of the finite element error. We also propose a self-consistent iterative algorithm to determine the smoothing parameter based on our theoretical analysis. Numerical examples are included to confirm the theoretical analysis and to show the competitive performance of the self- consistent algorithm for finding the smoothing parameter.