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A Note on the Direct Approximation of Derivatives in Rational Radial Basis Functions Partition of Unity Method

Vahid Mohammadi, Stefano De Marchi

TL;DR

The paper addresses the challenge of accurately differentiating functions with steep gradients or discontinuities using RRBF-PU interpolants. It introduces Direct RRBF-PU (DRRBF-PU), which computes derivatives by differentiating only the local rational RRBF pieces and weighting them with PU functions, thereby avoiding the costly differentiation of PU weights. A simple, robust error bound shows the derivative error is bounded by the worst local derivative error, and the method remains mesh-free and compatible with discontinuous PU weights. Numerical tests in two dimensions with a Matérn kernel demonstrate high-accuracy approximations for both functions and their derivatives, illustrating the approach's practicality for challenging interpolation tasks. The work offers a computationally efficient alternative for derivative estimation in PU-based RBF frameworks, with potential extensions to error analysis in terms of fill distance.

Abstract

This paper proposes a Direct Rational Radial Basis Functions Partition of Unity (D-RRBF-PU) approach to compute derivatives of functions with steep gradients or discontinuities. The novelty of the method concerns how derivatives are approximated. More precisely, all derivatives of the partition of unity weight functions are eliminated while we compute the derivatives of the local rational approximants in each patch. As a result, approximate derivatives are obtained more easily and quickly than those obtained in the standard formulation. The corresponding error bounds are briefly discussed. Some numerical results are presented to show the technique's potential.

A Note on the Direct Approximation of Derivatives in Rational Radial Basis Functions Partition of Unity Method

TL;DR

The paper addresses the challenge of accurately differentiating functions with steep gradients or discontinuities using RRBF-PU interpolants. It introduces Direct RRBF-PU (DRRBF-PU), which computes derivatives by differentiating only the local rational RRBF pieces and weighting them with PU functions, thereby avoiding the costly differentiation of PU weights. A simple, robust error bound shows the derivative error is bounded by the worst local derivative error, and the method remains mesh-free and compatible with discontinuous PU weights. Numerical tests in two dimensions with a Matérn kernel demonstrate high-accuracy approximations for both functions and their derivatives, illustrating the approach's practicality for challenging interpolation tasks. The work offers a computationally efficient alternative for derivative estimation in PU-based RBF frameworks, with potential extensions to error analysis in terms of fill distance.

Abstract

This paper proposes a Direct Rational Radial Basis Functions Partition of Unity (D-RRBF-PU) approach to compute derivatives of functions with steep gradients or discontinuities. The novelty of the method concerns how derivatives are approximated. More precisely, all derivatives of the partition of unity weight functions are eliminated while we compute the derivatives of the local rational approximants in each patch. As a result, approximate derivatives are obtained more easily and quickly than those obtained in the standard formulation. The corresponding error bounds are briefly discussed. Some numerical results are presented to show the technique's potential.
Paper Structure (5 sections, 16 equations, 2 figures, 2 tables)

This paper contains 5 sections, 16 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The RRBF-PU method
  3. The DRRBF-PU method
  4. Tests
  5. Conclusion

Figures (2)

  • Figure 1: The approximate (left) and exact (right) solutions to function \ref{['Test-1']} (first row), and the approximate (left) and exact derivatives (second row).
  • Figure 2: The approximate (left) and exact (right) solutions to function \ref{['Test-2']} (first row), and the approximate (left) and exact derivatives (second row).