Where does smoothness count the most for Fredholm equations of the second kind with noisy information?

• Authors: Arthur G. Werschulz
• Venue: Journal of Complexity
• Year: 2003

We study the complexity of Fredholm problems (I - Tk)u = f of the second kind on Id = [0,1]d, where Tk is an integral operator with kernel k. Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that f ∈ Wr,p(Id) with r d/p and that k ∈ Ws, ∞(I2d) with s 0. In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Θ(n-µ + δ), where µ = min{r/d, s/(2d)} and δ is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the ε-complexity for this problem. These bounds depend on the cost c(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation is proportional to δ-t, then the ε-complexity is roughly (1/ε)t+1/µ.