Statistical approach to some ill-posed problems for linear partial differential equations
成果类型：
Article
署名作者：
Chow, PL; Ibragimov, IA; Khasminskii, RZ
署名单位：
Wayne State University; Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s004400050212
发表日期：
1999
页码：
421-441
关键词：
摘要：
For linear partial differential equations, some inverse source problems are treated statistically based on nonparametric estimation ideas. By observing the solution in a small Gaussian white noise, the kernel type of estimators is used to estimate the unknown source function and its partial derivatives.. It is proved that such estimators are consistent as the noise intensity tends to zero. Depending on the principal part of the differential operator, the optimal asymptotic rate of convergence is ascertained within a wide class of risk functions in a minimax sense.