# Fast numerical derivatives based on multi-interval Fourier extension

## TL;DR

This paper introduces a fast, stable numerical differentiation algorithm that combines multi-interval Fourier extension with adaptive domain partitioning, effectively handling noisy data and complex functions.

## Contribution

The paper proposes a novel method that integrates multi-interval Fourier extension with adaptive partitioning for efficient and stable numerical differentiation from noisy data.

## Key findings

- Significant accuracy improvements over existing methods.
- Robustness to noise and function oscillations.
- Efficient computation through precomputed matrices and SVDs.

## Abstract

We present a computationally efficient algorithm for stable numerical differentiation from noisy, uniformly-sampled data on a bounded interval. The method combines multi-interval Fourier extension approximations with an adaptive domain partitioning strategy: a global precomputation of local Fourier sampling matrices and their thin SVDs is reused throughout a recursive bisection procedure that selects locally-resolved Fourier fits. Each accepted subinterval stores a compact set of Fourier coefficients that are subsequently used to reconstruct the derivative via a precomputed differentiation operator. The stopping criterion balances fitting error and an explicit noise-level bound, and the algorithm automatically refines the partition where the function exhibits rapid oscillations or boundary activity. Numerical experiments demonstrate significant improvements over existing methods, achieving accurate derivative reconstruction for challenging functions. The approach provides a robust framework for ill-posed differentiation problems while maintaining computational efficiency.