Where does smoothness count the most for Fredholm equations of the second kind with noisy information?
✍ Scribed by Arthur G. Werschulz
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 352 KB
- Volume
- 19
- Category
- Article
- ISSN
- 0885-064X
No coin nor oath required. For personal study only.
✦ Synopsis
We study the complexity of Fredholm problems ðI À T k Þu ¼ f of the second kind on I d ¼ ½0; 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 f AW r;p ðI d Þ with r4d=p and that kAW s;N ðI 2d Þ with s40: In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Yðn Àm þ dÞ; where m ¼ minfr=d; s=ð2dÞg and d 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 e-complexity for this problem. These bounds depend on the cost cðdÞ of calculating a d-noisy information value. As an example, if the cost of a d-noisy evaluation is proportional to d Àt ; then the e-complexity is roughly ð1=eÞ tþ1=m :