An optimal method for high order mixed derivatives of bivariate functions

TL;DR

This work develops an optimal truncation-based algorithm for recovering high-order mixed derivatives $f^{(r,r)}$ of non-periodic bivariate functions from perturbed Fourier-Legendre data. By employing a hyperbolic-cross index set, the method achieves order-optimal accuracy while using minimal Galerkin information, with rigorous $L_2$ and $C$ error bounds and perturbation stability. The authors establish lower and upper bounds for the minimal information radius, demonstrating the efficiency of the truncation scheme compared to standard dense-data approaches. Computational experiments in MATLAB on synthetic examples verify the predicted convergence behavior and robustness to noise, highlighting potential for practical numerical differentiation in multidimensional settings.

Abstract

The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.

Figures (3)

• Figure 1: Recovery of the derivative $F_1^{(2,2)}$ with random noise in the input data . The exact derivative $F_1^{(2,2)}$ (Fig. a ); approximation to $F_1^{(2,2)}$ for $\delta= 10^{-6}$ (Fig. b); for $\delta= 10^{-7}$ (Fig. c) and $\delta= 10^{-8}$ (Fig. d),
• Figure 2: Recovery of the derivative $F_1^{(2,2)}$ with noise in the input data arising from the quadrature formula. The exact derivative $F_1^{(2,2)}$ (Fig. a); approximation to $F_1^{(2,2)}$ for $\delta= 10^{-6}$ (Fig. b); for $\delta= 10^{-7}$ (Fig. c) and $\delta= 10^{-8}$ (Fig. d).
• Figure 3: Recovery of the derivative $F_2^{(2,2)}$ with noise in the input data arising from the quadrature formula. The exact derivative $F_2^{(2,2)}$ (Fig. a); approximation to $F_2^{(2,2)}$ for $\delta= 10^{-6}$ (Fig. b); for $\delta= 10^{-7}$ (Fig. c) and $\delta= 10^{-8}$ (Fig. d).

Theorems & Definitions (18)

• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Corollary 2.6
• Remark 2.7
• Lemma 3.1
• Remark 3.2
• Lemma 3.3