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...

Full description

Saved in:
Bibliographic Details
Published in:Numerische Mathematik Vol. 145; no. 3; pp. 581 - 603
Main Authors: Harrach, Bastian, Jahn, Tim, Potthast, Roland
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020
Springer Nature B.V
Subjects:
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
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01122-2