Least squares solvers for ill-posed PDEs that are conditionally stable

This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to es...

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Bibliographic Details
Published in:ESAIM. Mathematical modelling and numerical analysis 2023-07, Vol.57 (4), p.2227
Main Authors: Dahmen, Wolfgang, Monsuur, Harald, Stevenson, Rob
Format: Article
Language:English
Subjects:
Hilbert space
Ill posed problems
Least squares
Mathematics
Norms
Projectors
Regularization
Solvers
Stability
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Summary:This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.
ISSN:1290-3841
DOI:10.1051/m2an/2023050