[Paper Review] A shape preserving quasi-interpolation operator based on a new transcendental RBF

Mohammad Heidari, Maryam Mohammadi+1 more|arXiv (Cornell University)|Jun 10, 2021

Advanced Numerical Analysis Techniques4 citations

TL;DR

This paper introduces a novel shape-preserving quasi-interpolation operator using a new transcendental radial basis function (RBF) based on the hyperbolic tangent function, $\phi(r) = r \tanh(r/c)$. The method improves upon the classical Multiquadric (MQ) RBF by achieving $O(h^2)$ convergence, preserving convexity and monotonicity, and demonstrating superior accuracy and stability in numerical tests without requiring linear system solves.

ABSTRACT

It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as a smooth approximant to f(r)=r with higher accuracy and better convergence properties than the multiquadric. Then Wu-Schaback's quasi-interpolation formula is rewritten using the proposed RBF. It preserves convexity and monotonicity. We prove that the proposed scheme converges with a rate of O(h^2). So it has a higher degree of smoothness. Some numerical experiments are given in order to demonstrate the efficiency and accuracy of the method.

Motivation & Objective

• To develop a higher-order, shape-preserving quasi-interpolation operator for univariate scattered data that avoids solving linear systems.
• To replace the classical Multiquadric (MQ) RBF with a new transcendental RBF based on the hyperbolic tangent function for improved accuracy and convergence.
• To ensure the operator preserves monotonicity and convexity of the underlying data, critical for reliable function reconstruction.
• To achieve $O(h^2)$ convergence rate independent of the shape parameter $c$, enhancing robustness and smoothness.
• To mitigate classical numerical instabilities such as the Runge and Gibbs phenomena in function approximation.

Proposed method

• Propose a new RBF: $\phi(r) = r \tanh(r/c)$, which smoothly approximates $|x|$ and offers better convergence than the MQ RBF $\sqrt{r^2 + c^2}$.
• Reformulate the Wu–Schaback quasi-interpolation formula using the new RBF to construct the operator $\mathcal{L}_{RTH}$.
• Ensure the operator is monotonicity- and convexity-preserving by construction, based on the properties of the new RBF and its derivative behavior.
• Achieve $O(h^2)$ convergence rate by proving theoretical bounds independent of the shape parameter $c$.
• Use function values directly as coefficients without solving a linear system, maintaining computational efficiency.
• Validate the method through numerical experiments on smooth, discontinuous, and oscillatory test functions.

Experimental results

Research questions

• RQ1 Can a new transcendental RBF based on the hyperbolic tangent function outperform the classical MQ RBF in terms of convergence rate and accuracy for quasi-interpolation?
• RQ2 Does the proposed quasi-interpolation operator preserve monotonicity and convexity of the input data?
• RQ3 Can the new RBF-based operator mitigate the Runge and Gibbs phenomena in function approximation?
• RQ4 What is the convergence rate of the proposed operator, and how does it depend on the shape parameter $c$ and step size $h$?
• RQ5 Is the new operator more robust and accurate than the standard MQ quasi-interpolation for scattered data sets?

Key findings

• The proposed RBF $\phi(r) = r \tanh(r/c)$ provides higher accuracy and better convergence than the MQ RBF, especially for small $c$.
• The quasi-interpolation operator $\mathcal{L}_{RTH}$ achieves $O(h^2)$ convergence rate, confirmed numerically with convergence rates up to 2.0943 and 3.4210 in different test problems.
• Numerical experiments show that $\mathcal{L}_{RTH}$ reduces the $L^\infty$ error to as low as $7.5 \times 10^{-14}$ for smooth functions with $h = 0.0125$ and $c = 0.01$.
• The operator successfully mitigates the Runge phenomenon for the Runge function, with relative errors decreasing significantly as $h$ is reduced.
• The Gibbs phenomenon is substantially attenuated for discontinuous functions, with oscillations decreasing as $c$ is reduced to $0.001$.
• The method outperforms the standard MQ quasi-interpolation in accuracy and stability across all test problems, including piecewise analytic and discontinuous functions.