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

We study the complexity of Fredholm problems (I−T_k)u=f of the second kind on I^d=0,1]^d, where T_k 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 fW^r,p(I^d) with r>d/p and that kW^s,∞(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 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 -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 -complexity is roughly (1/)^t+1/μ.