Approximation of Functions of Few Variables in High Dimensions

作者：DeVore Ronald^*; Petrova Guergana; Wojtaszczyk Przemyslaw

来源：Constructive Approximation, 2011, 33(1): 125-143.

DOI：10.1007/s00365-010-9105-8

• Approximation of functions
• High dimension
• Significant variables
• Sensitivity analysis

Let f be a continuous function defined on Omega:=[0,1] (N) which depends on only a"" coordinate variables, . We assume that we are given m and are allowed to ask for the values of f at m points in Omega. If g is in Lip1 and the coordinates i (1),aEuro broken vertical bar,i (a"") are known to us, then by asking for the values of f at m=L (a"") uniformly spaced points, we could recover f to the accuracy |g|(Lip1) L (-1) in the norm of C(Omega). This paper studies whether we can obtain similar results when the coordinates i (1),aEuro broken vertical bar,i (a"") are not known to us. A prototypical result of this paper is that by asking for C(a"")L (a"") (log (2) N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|(Lip1) L (-1) when gaLip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance epsilon (which is not known) by functions of a"" variables.

• 出版日期2011-2