A twin error gauge for Kaczmarz's iterations

Bart S. van Lith; Per Christian Hansen; Michiel E. Hochstenbach

doi:10.1137/20M1349011

A twin error gauge for Kaczmarz's iterations

Bart S. van Lith, Per Christian Hansen, Michiel E. Hochstenbach

Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review

5 Citaten (Scopus)

Samenvatting

We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error-and is therefore able to stop at the best possible iterate-our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

Originele taal-2	Engels
Pagina's (van-tot)	S173-S199
Aantal pagina's	27
Tijdschrift	SIAM Journal on Scientific Computing
Volume	43
Nummer van het tijdschrift	5
DOI's	https://doi.org/10.1137/20M1349011
Status	Gepubliceerd - 2021

Bibliografische nota

Funding Information:
\ast Received by the editors June 30, 2020; accepted for publication (in revised form) January 26, 2021; published electronically May 3, 2021. https://doi.org/10.1137/20M1349011 Funding: The work of the first author was supported by the EuroTech Postdoc Programme, co-funded by the European Commission under its framework programme Horizon 2020 through grant agreement 754462. \dagger Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected]). \ddagger Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, 5600 MB, The Netherlands ([email protected], http://www.win.tue.nl/\sim hochsten/).

Publisher Copyright:
Copyright © by SIAM.

Financiering

\ast Received by the editors June 30, 2020; accepted for publication (in revised form) January 26, 2021; published electronically May 3, 2021. https://doi.org/10.1137/20M1349011 Funding: The work of the first author was supported by the EuroTech Postdoc Programme, co-funded by the European Commission under its framework programme Horizon 2020 through grant agreement 754462. \dagger Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected]). \ddagger Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, 5600 MB, The Netherlands ([email protected], http://www.win.tue.nl/\sim hochsten/).

Financiers	Financiernummer
European Union’s Horizon Europe research and innovation programme
European Commission
European Union’s Horizon Europe research and innovation programme	754462

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10.1137/20M1349011

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title = "A twin error gauge for Kaczmarz's iterations",

abstract = "We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error-and is therefore able to stop at the best possible iterate-our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule. ",

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PY - 2021

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N2 - We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error-and is therefore able to stop at the best possible iterate-our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

AB - We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error-and is therefore able to stop at the best possible iterate-our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

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A twin error gauge for Kaczmarz's iterations

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