On optimal random nets.
- Mathé, Peter
2010 Mathematics Subject Classification
- 65C05 41A65 65J05
- ε-entropy, Sobolev embeddings, random nets
The possibility to approximate bounded linear mappings between Banach spaces depends on the degree of compactness. One way to measure this degree of compactness is the scale of entropy numbers, cf. [CS90]. In the usual (worst-case) setting of numerical analysis this scale has been studied for a long time. Recent reserach is concerned with the study of the so-called average-case and randomized (Monte Carlo) settings. We propose the respective counterparts of the entropy numbers in these settings and obtain their behavior for Sobolev embeddings. It turns out that, at least in this situation, randomly chosen nets may not improve the approximability of operators in the Monte Carlo setting. However, we can use the results to improve previous estimates for average Kolmogorov numbers, as obtained in [Mat91].