Hyperbolic cross designs for approximation of random fields

✍ Scribed by Thomas Müller-Gronbach

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 916 KB
• Volume: 66
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(97)00088-8

✦ Synopsis

We consider a random field on the p-dimensional cube with a covariance function of tensor product type. The quality of an approximation which is based on finitely many observations of the field is measured by the integrated mean-squared error and the maximum mean-squared error. We use the optimal affine linear approximation and analyze the asymptotic performance of hyperbolic cross designs which are constructed from regular sequences. These designs are weakly asymptotically optimal in the sense that the corresponding approximation errors are of the same order as the minimal errors. For the integrated mean-squared error we explicitly determine the corresponding asymptotic constants. For the maximum mean-squared error we determine the constants up to the factor (-~)P. Moreover, we provide a simple alternative approximation which neither depends on the mean function nor on the covariance function of the field and which performs asymptotically as well as the optimal approximation.