2001 Reports

Where Does Smoothness Count the Most For Fredholm Equations of the Second Kind With Noisy Information?

Werschulz, Arthur G.

We study the complexity of Fredholm problems (I − Tk)u = f of the second kind on the I d = [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(I d ) with r > d/p and that k ∈ Ws,∞(I 2d ) 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 2(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.