TY - JOUR
T1 - Squared polynomial extrapolation methods with cycling
T2 - An application to the positron emission tomography problem
AU - Roland, Ch
AU - Varadhan, R.
AU - Frangakis, C. E.
N1 - Funding Information:
Acknowledgements The authors thank Professor Claude Brezinski and Alain Berlinet for their useful comments on the manuscript. The second author (R.V.) was supported by the National Institute on Aging, Claude D. Pepper Older Americans Independence Centers, Grant P30 AG021334, and NIH-NIA Grant R37 AG19905 at the Johns Hopkins University School of Medicine during the conduct of this research.
PY - 2007/2
Y1 - 2007/2
N2 - Roland and Varadhan (Appl. Numer. Math., 55:215-226, 2005) presented a new idea called "squaring" to improve the convergence of Lemaréchal's scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, http://www.bepress.com/jhubiostat/paper63, 2004) noted that Lemaréchal's scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first- and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, "unsquared" vector polynomial methods.
AB - Roland and Varadhan (Appl. Numer. Math., 55:215-226, 2005) presented a new idea called "squaring" to improve the convergence of Lemaréchal's scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, http://www.bepress.com/jhubiostat/paper63, 2004) noted that Lemaréchal's scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first- and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, "unsquared" vector polynomial methods.
KW - Fixed-point methods
KW - Linear systems
KW - Nonlinear systems
KW - Polynomial extrapolation methods
KW - Squaring
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U2 - 10.1007/s11075-007-9094-2
DO - 10.1007/s11075-007-9094-2
M3 - Article
AN - SCOPUS:34250862061
SN - 1017-1398
VL - 44
SP - 159
EP - 172
JO - Numerical Algorithms
JF - Numerical Algorithms
IS - 2
ER -