March 3, 2026Open Access

Constructive Approximation of Nonlinear Operators Based on Piecewise Interpolation Technique

Key Points

The proposed method effectively approximates nonlinear operators using minimal prior information from the set KY.
With only a few known elements, this technique maintains accuracy and adheres to Lipschitz continuity requirements.
Analysis employs a specially extended piecewise linear interpolation technique to handle stochastic elements efficiently.
The framework guarantees the existence of a transformation operator through the use of pseudo-inverse matrices.

Abstract

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.

Constructive Approximation of Nonlinear Operators Based on Piecewise Interpolation Technique

Key Points

Abstract

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