Local approximation of operators

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces mathfrakX and mathfrakY. We study the problem of determining the degree of approximation of such operators on a compact subset K_mathfrakXsubset mathfrakX using a finite amount of information. If F: K_mathfrakX→ K_mathfrakY, a well established strategy to approximate F(F) for some F∈ K_mathfrakX is to encode F (respectively, F(F)) in terms of a finite number d (repectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space ℝd, equivalently, the unit sphere mathbbSd embedded in ℝd+1. The problem is challenging because d, m, as well as the complexity of the approximation on mathbbSd are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on mathbbSd being O(d1/6). We study different smoothness classes for the operators, and also propose a method for approximation of F(F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved.