Academic Commons


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.



More About This Work

Academic Units
Computer Science
Department of Computer Science, Columbia University
Columbia University Computer Science Technical Reports, CUCS-017-01
Published Here
April 22, 2011
Academic Commons provides global access to research and scholarship produced at Columbia University, Barnard College, Teachers College, Union Theological Seminary and Jewish Theological Seminary. Academic Commons is managed by the Columbia University Libraries.