Solving PDEs with Incomplete Information

TL;DR

This paper develops a theoretical and numerical framework for approximating PDE solutions when boundary data is incomplete, using optimal recovery and Riesz representers to handle limited information.

Contribution

It introduces a novel approach to PDE approximation under incomplete boundary data by formulating it as an optimal recovery problem with new algorithms.

Findings

Effective numerical algorithms for PDEs with incomplete boundary data.

Theoretical foundation linking optimal recovery to PDE solution approximation.

Demonstrated accuracy of the proposed methods in example scenarios.

Abstract

We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.