Beyond the Bakushinkii veto: regularising linear inverse problems without knowing the noise distribution

This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is...

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Vydáno v:Numerische Mathematik Ročník 145; číslo 3; s. 581 - 603
Hlavní autoři: Harrach, Bastian, Jahn, Tim, Potthast, Roland
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020
Springer Nature B.V
Témata:
Error analysis
Hilbert space
Ill posed problems
Inverse problems
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Noise levels
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
65J22 (Numerical solution to inverse problems in abstract spaces)
ISSN:0029-599X, 0945-3245
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Shrnutí:This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.
Bibliografie:ObjectType-Article-1
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01122-2