On optimal recovering high order partial derivatives of bivariate functions

TL;DR

This paper develops an optimal numerical differentiation algorithm for recovering high-order partial derivatives of bivariate functions with finite smoothness, balancing accuracy and information usage, supported by numerical demonstrations.

Contribution

It introduces a truncation-based method that is optimal in order for high-order derivative recovery, considering both accuracy and data efficiency.

Findings

The algorithm achieves optimal order accuracy.

Numerical experiments confirm successful implementation.

The method effectively balances accuracy and data usage.

Abstract

The problem of recovering partial derivatives of high orders of bivariate functions with finite smoothness is studied. Based on the truncation method, a numerical differentiation algorithm was constructed, which is optimal by the order, both in the sense of accuracy and in the sense of the amount of Galerkin information involved. Numerical demonstrations are provided to illustrate that the proposed method can be implemented successfully.