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- W1976537103 abstract "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." @default.
- W1976537103 created "2016-06-24" @default.
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- W1976537103 date "2003-12-01" @default.
- W1976537103 modified "2023-09-23" @default.
- W1976537103 title "Where does smoothness count the most for Fredholm equations of the second kind with noisy information?" @default.
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- W1976537103 doi "https://doi.org/10.1016/s0885-064x(03)00030-x" @default.
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