An Adaptive Algorithm Employing Continuous Linear Functionals
This work provides a theoretically grounded adaptive method for linear problems in Hilbert spaces, offering a practical solution for users needing automatic error control.
The paper presents an automatic adaptive algorithm for solving general linear problems on Hilbert spaces using Fourier coefficients, which guarantees a user-specified error tolerance without prior knowledge of the solution's norm or decay rate, and achieves near-optimal computational cost.
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.