An Adaptive Algorithm Employing Continuous Linear Functionals

TL;DR

This paper introduces an adaptive algorithm for solving linear problems in Hilbert spaces that uses Fourier coefficients, automatically achieving desired accuracy without prior knowledge of the function's norm.

Contribution

The paper presents a new adaptive algorithm that guarantees error tolerance without requiring prior decay rate knowledge of Fourier coefficients.

Findings

Algorithm achieves error tolerance automatically

Computational cost comparable to optimal algorithms

Numerical experiments validate effectiveness

Abstract

Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a general linear problem on Hilbert spaces. The algorithm employs continuous linear functionals of the input function, specifically Fourier coefficients. We assume that the Fourier coefficients of the solution decay sufficiently fast, but do not require the decay rate to be known a priori. We also assume that the Fourier coefficients decay steadily, although not necessarily monotonically. Under these assumptions, our adaptive algorithm is shown to produce an approximate solution satisfying the desired error tolerance, without prior knowledge of the norm of the function to be approximated. Moreover, the computational cost of our algorithm is shown to be essentially no worse than that of the optimal algorithm. We provide a numerical experiment to illustrate our algorithm.