AN ERROR ANALYSIS FOR THE NUMERICAL CALCULATION OF CERTAIN RANDOM INTEGRALS: PART 1
成果类型：
Article
署名作者：
Pitt, Loren D.; Robeva, Raina; Wang, Da Yi
署名单位：
University of Virginia; National University of Singapore
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/aoap/1177004835
发表日期：
1995
页码：
171-197
关键词：
摘要：
For a wide sense stationary random field Phi = {phi(x): x is an element of R-2}, we investigate the asymptotic errors made in the numerical integration of line intergrals of the form integral(Gamma) f(x)phi(x)d sigma(x). It is shown, for example, that if f and Gamma are smooth, and if the spectral density rho(lambda) satisfies rho(lambda) approximate to k vertical bar lambda vertical bar(-4) as lambda -> infinity, then there is a constant c' with (NE)-E-3 vertical bar integral(Gamma) f(x)phi(x)d sigma(x)-Sigma beta(J)phi(x(j))vertical bar(2) >= c' N-3 for all finite sets {x(j): 1 <= j <= N} and all choices of coefficients {beta(J)}. And, if any fixed parameterization x(t) of Gamma is given and the integral integral(1)(0) f(x(t))phi(x(t))vertical bar x'(t)vertical bar dt is numerically integrated using the midpoint method, the exact asymptotics of the mean squared error is derived. This leads to asymptotically optimal designs, and generalizes to other power laws and to nonstationary and nonisotropic fields.