Nonlinear Optimal Recovery in Hilbert Spaces

TL;DR

This paper develops a theoretical framework for solving nonlinear problems in Hilbert spaces using limited measurements, establishing convergence and conditions for finite-dimensional solutions.

Contribution

It introduces a nonlinear optimal recovery formulation in Hilbert spaces, analyzes its properties, and provides conditions for finite-dimensional solutions with convergence guarantees.

Findings

Established well-posedness of the nonlinear optimal recovery problem.

Proved convergence of the solution to the true solution as measurements increase.

Provided a sufficient condition for finite-dimensional solutions in specific cases.

Abstract

This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data.