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A threshold for the use of Tikhonov regularization in inverse force determination

A threshold for the use of Tikhonov regularization in inverse force determination
A threshold for the use of Tikhonov regularization in inverse force determination
In the analysis of structure-borne sound from installed machinery, it is important to be able to estimate the operational forces. Assuming that their location is known, indirect approaches based on matrix inversion can be used to reconstruct the operational forces from a set of measured operational responses and corresponding matrix of frequency response functions. In common with many such inverse problems, matrix ill-conditioning can affect the reliability of the results. Methods such as pseudo-inversion of over-determined matrices, singular value rejection, and Tikhonov regularization have been used previously to overcome this and it has been found that Tikhonov regularization generally performs well in reducing the errors in the reconstructed forces. However, full-rank pseudo-inversion (unregularized solution) gives better results than Tikhonov regularization in some cases, particularly with low condition numbers. Since the need for regularization is greatest when the matrix is ill-conditioned, this suggests the introduction of a threshold above which Tikhonov regularization is used and below which pseudo-inversion is used. In this study, the extent to which response errors are amplified in the force estimates is considered and plotted against the matrix condition number. This allows a threshold condition number to be identified above which Tikhonov regularization gives improved results. It is found that the threshold is related not only to the condition number but also to the matrix dimensions including the extent of over-determination. A simple empirical formula is obtained for this threshold that is usable for matrices in a wide range of matrix dimensions.
inverse problems, force determination, matrix regularization, tikhonov regularization, Ill-conditioning
0003-682X
700-719
Choi, H.G.
7825da15-7307-4f60-90e5-623c38efe7dd
Thite, A.N.
c3db753e-656c-4efe-9195-398ac5e7f6eb
Thompson, D.J.
bca37fd3-d692-4779-b663-5916b01edae5
Choi, H.G.
7825da15-7307-4f60-90e5-623c38efe7dd
Thite, A.N.
c3db753e-656c-4efe-9195-398ac5e7f6eb
Thompson, D.J.
bca37fd3-d692-4779-b663-5916b01edae5

Choi, H.G., Thite, A.N. and Thompson, D.J. (2006) A threshold for the use of Tikhonov regularization in inverse force determination. Applied Acoustics, 67 (7), 700-719. (doi:10.1016/j.apacoust.2005.11.003).

Record type: Article

Abstract

In the analysis of structure-borne sound from installed machinery, it is important to be able to estimate the operational forces. Assuming that their location is known, indirect approaches based on matrix inversion can be used to reconstruct the operational forces from a set of measured operational responses and corresponding matrix of frequency response functions. In common with many such inverse problems, matrix ill-conditioning can affect the reliability of the results. Methods such as pseudo-inversion of over-determined matrices, singular value rejection, and Tikhonov regularization have been used previously to overcome this and it has been found that Tikhonov regularization generally performs well in reducing the errors in the reconstructed forces. However, full-rank pseudo-inversion (unregularized solution) gives better results than Tikhonov regularization in some cases, particularly with low condition numbers. Since the need for regularization is greatest when the matrix is ill-conditioned, this suggests the introduction of a threshold above which Tikhonov regularization is used and below which pseudo-inversion is used. In this study, the extent to which response errors are amplified in the force estimates is considered and plotted against the matrix condition number. This allows a threshold condition number to be identified above which Tikhonov regularization gives improved results. It is found that the threshold is related not only to the condition number but also to the matrix dimensions including the extent of over-determination. A simple empirical formula is obtained for this threshold that is usable for matrices in a wide range of matrix dimensions.

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More information

Published date: 2006
Keywords: inverse problems, force determination, matrix regularization, tikhonov regularization, Ill-conditioning

Identifiers

Local EPrints ID: 28433
URI: https://eprints.soton.ac.uk/id/eprint/28433
ISSN: 0003-682X
PURE UUID: 59a7e70c-f6bb-437c-b2b4-6a33db52f51f
ORCID for D.J. Thompson: ORCID iD orcid.org/0000-0002-7964-5906

Catalogue record

Date deposited: 28 Apr 2006
Last modified: 06 Jun 2018 13:02

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