Optimal Choice of the Regularization Parameter for Direct Identification of Polymers Relaxation Time and Frequency Spectra.
| Title: | Optimal Choice of the Regularization Parameter for Direct Identification of Polymers Relaxation Time and Frequency Spectra. |
|---|---|
| Authors: | Stankiewicz, Anna1 (AUTHOR) anna.m.stankiewicz@gmail.com, Bojanowska, Monika2 (AUTHOR) monika.bojanowska@up.lublin.pl |
| Source: | Polymers (20734360). Jan2025, Vol. 17 Issue 1, p31. 38p. |
| Subject Terms: | *REGULARIZATION parameter, *TIKHONOV regularization, *INVERSE problems, *FREQUENCY spectra, *VISCOELASTICITY |
| Abstract: | Recovering the relaxation spectrum, a fundamental rheological characteristic of polymers, from experiment data requires special identification methods since it is a difficult ill-posed inverse problem. Recently, a new approach relating the identification index directly with a completely unknown real relaxation spectrum has been proposed. The integral square error of the relaxation spectrum model was applied. This paper concerns regularization aspects of the linear-quadratic optimization task that arise from applying Tikhonov regularization to relaxation spectra direct identification problem. An influence of the regularization parameter on the norms of the optimal relaxation spectra models and on the fit of the related relaxation modulus model to the experimental data was investigated. The trade-off between the integral square norms of the spectra models and the mean square error of the relaxation modulus model, parameterized by varying regularization parameter, motivated the definition of two new multiplicative indices for choosing the appropriate regularization parameter. Two new problems of the regularization parameter optimal selection were formulated and solved. The first and second order optimality conditions were derived and expressed in the matrix-vector form and, alternatively, in finite series terms. A complete identification algorithm is presented. The usefulness of the new regularization parameter selection rules is demonstrated by three examples concerning the Kohlrausch–Williams–Watts spectrum with short relaxation times and uni- and double-mode Gauss-like spectra with middle and short relaxation times. [ABSTRACT FROM AUTHOR] |
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| Items | – Name: Title Label: Title Group: Ti Data: Optimal Choice of the Regularization Parameter for Direct Identification of Polymers Relaxation Time and Frequency Spectra. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Stankiewicz%2C+Anna%22">Stankiewicz, Anna</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> anna.m.stankiewicz@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Bojanowska%2C+Monika%22">Bojanowska, Monika</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> monika.bojanowska@up.lublin.pl</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Polymers+%2820734360%29%22">Polymers (20734360)</searchLink>. Jan2025, Vol. 17 Issue 1, p31. 38p. – Name: Subject Label: Subject Terms Group: Su Data: *<searchLink fieldCode="DE" term="%22REGULARIZATION+parameter%22">REGULARIZATION parameter</searchLink><br />*<searchLink fieldCode="DE" term="%22TIKHONOV+regularization%22">TIKHONOV regularization</searchLink><br />*<searchLink fieldCode="DE" term="%22INVERSE+problems%22">INVERSE problems</searchLink><br />*<searchLink fieldCode="DE" term="%22FREQUENCY+spectra%22">FREQUENCY spectra</searchLink><br />*<searchLink fieldCode="DE" term="%22VISCOELASTICITY%22">VISCOELASTICITY</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Recovering the relaxation spectrum, a fundamental rheological characteristic of polymers, from experiment data requires special identification methods since it is a difficult ill-posed inverse problem. Recently, a new approach relating the identification index directly with a completely unknown real relaxation spectrum has been proposed. The integral square error of the relaxation spectrum model was applied. This paper concerns regularization aspects of the linear-quadratic optimization task that arise from applying Tikhonov regularization to relaxation spectra direct identification problem. An influence of the regularization parameter on the norms of the optimal relaxation spectra models and on the fit of the related relaxation modulus model to the experimental data was investigated. The trade-off between the integral square norms of the spectra models and the mean square error of the relaxation modulus model, parameterized by varying regularization parameter, motivated the definition of two new multiplicative indices for choosing the appropriate regularization parameter. Two new problems of the regularization parameter optimal selection were formulated and solved. The first and second order optimality conditions were derived and expressed in the matrix-vector form and, alternatively, in finite series terms. A complete identification algorithm is presented. The usefulness of the new regularization parameter selection rules is demonstrated by three examples concerning the Kohlrausch–Williams–Watts spectrum with short relaxation times and uni- and double-mode Gauss-like spectra with middle and short relaxation times. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Polymers (20734360) is the property of MDPI and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.3390/polym17010031 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 38 StartPage: 31 Subjects: – SubjectFull: REGULARIZATION parameter Type: general – SubjectFull: TIKHONOV regularization Type: general – SubjectFull: INVERSE problems Type: general – SubjectFull: FREQUENCY spectra Type: general – SubjectFull: VISCOELASTICITY Type: general Titles: – TitleFull: Optimal Choice of the Regularization Parameter for Direct Identification of Polymers Relaxation Time and Frequency Spectra. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Stankiewicz, Anna – PersonEntity: Name: NameFull: Bojanowska, Monika IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Text: Jan2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 20734360 Numbering: – Type: volume Value: 17 – Type: issue Value: 1 Titles: – TitleFull: Polymers (20734360) Type: main |
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