Local Adaption for Approximation and Minimization of Univariate Functions

TL;DR

This paper introduces new locally adaptive algorithms for univariate function approximation and minimization that guarantee user-specified accuracy for a broad class of functions, with optimal computational complexity.

Contribution

The paper presents the first theoretically guaranteed locally adaptive algorithms for univariate approximation and minimization with optimal complexity.

Findings

Algorithms achieve $\mathcal{O}(\sqrt{\|f''\|_{1/2}/\varepsilon})$ complexity for approximation.

Algorithms automatically adapt sampling density based on the second derivative.

Numerical experiments show superior performance over existing methods.

Abstract

Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, $\mathcal{C}$, of non-spiky input functions in the Sobolev space $W^{2,\infty}[a,b]$. Our algorithms automatically determine where to sample the function---sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function $f$ on a bounded interval with $L^{\infty}$-error no greater than $\varepsilon$ is $\mathcal{O}\Bigl(\sqrt{{\left\|f"\right\|}_{\frac12}/\varepsilon}\Bigr)$ as $\varepsilon \to 0$. This is the same order as that of the best function approximation algorithm for functions in $\mathcal{C}$. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if $f$ significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.