Constructive Approximation of Nonlinear Operators Based on Piecewise Interpolation Technique

Suppose KY and KX are the image and the preimage of a nonlinear operator F:KY→KX. It is supposed that the cardinality of each KY and KX is N and N is large. We provide an approximation to the map F that requires prior information only on a few elements p from KY, where p≪N, but still effectively represents F(KY). It is achieved under Lipschitz continuity assumptions. The device behind the proposed method is based on a special extension of the piecewise linear interpolation technique to the case of sets of stochastic elements. The proposed technique provides a single operator that transforms any element from the arbitrarily large set KY. The operator is determined in terms of pseudo-inverse matrices so that it always exists.