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Solving PDEs with Incomplete Information

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arxiv 2301.05540 v2 pith:GTYPNYE4 submitted 2023-01-13 math.NA cs.NA

Peter Binev , Andrea Bonito , Albert Cohen , Wolfgang Dahmen , Ronald DeVore , Guergana Petrova This is my paper
classification math.NAcs.NA
keywords problemsolutionboundaryinformationmainwhenalgorithmsapproximating
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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.

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Cited by 1 Pith paper

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  1. Nonlinear Optimal Recovery in Hilbert Spaces

    math.NA 2025-05 conditional novelty 7.0 of 10

    The paper proves convergence and finite-dimensional representability for minimum-norm recovery applied to nonlinear problems in Hilbert spaces with finite measurements.

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