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The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong SAR, China |
In this paper, we focus on a splitting method called the $ \theta $-scheme proposed by Glowinski and Le Tallec in [
References:
[1] |
M. V. Afonso, J. M. Bioucas-Dias and and M. A. Figueiredo,
Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356.
doi: 10.1109/TIP.2010.2047910. |
[2] |
N. Ahmed, T. Natarajan and and K. R. Rao,
Discrete cosine transform, IEEE Trans. Comput., C-23 (1974), 90-93.
doi: 10.1109/T-C.1974.223784. |
[3] |
J. B. Baillon and G. Haddad,
Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones, Israel J. Math., 26 (1977), 137-150.
doi: 10.1007/BF03007664. |
[4] |
H. H. Bauschke and P. L. Combettes, et al., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-48311-5. |
[5] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[6] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.
doi: 10.1137/080716542. |
[7] |
A. Beck and M. Teboulle,
A fast dual proximal gradient algorithm for convex minimization and applications, Oper. Res. Lett., 42 (2014), 1-6.
doi: 10.1016/j.orl.2013.10.007. |
[8] |
D. P. Bertsekas, Nonlinear Programming, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999. |
[9] |
S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein and et al.,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2010), 1-122.
doi: 10.1561/2200000016. |
[10] |
P. L. Combettes,
Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[11] |
P. L. Combettes and V. R. Wajs,
Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.
doi: 10.1137/050626090. |
[12] |
J. Douglas and H. H. Rachford Jr.,
On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[13] |
J. Eckstein and D. P. Bertsekas,
On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[14] |
J. Eckstein and W. Yao, Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results, RUTCOR Research Reports, 32 (2012). Google Scholar |
[15] |
R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Lecture Notes in Math., 506, Springer, Berlin, 1976, 73–89.
doi: 10.1007/BFb0080116. |
[16] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim., 2 (1992), 21-42.
doi: 10.1137/0802003. |
[17] |
R. Glowinski, Splitting methods for the numerical solution of the incompressible Navier-Stokes equations, in Vistas in Applied Mathematics, Optimization Software, New York, 1986, 57–95. |
[18] |
R. Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems, CBMS-NSF Regional Conference Series in Applied Mathematics, 86, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015.
doi: 10.1137/1.9781611973785.ch1. |
[19] |
R. Glowinski, P. G. Ciarlet and J.-L. Lions, Numerical Methods for Fluids: Finite Element Methods for Incompressible Viscous Flow, North Holland, 2003. Google Scholar |
[20] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970838. |
[21] |
R. Glowinski, S. Leung and J. L. Qian, Operator-splitting based fast sweeping methods for isotropic wave propagation in a moving fluid, SIAM J. Sci. Comput., 38 (2016), A1195–A1223.
doi: 10.1137/15M1043868. |
[22] |
R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et larésolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[23] |
R. Glowinski, S. J. Osher and W. T. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Scientific Computation, Springer, Cham, 2016.
doi: 10.1007/978-3-319-41589-5. |
[24] |
T. Goldstein, B. O'Donoghue, S. Setzer and R. Baraniuk,
Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2014), 1588-1623.
doi: 10.1137/120896219. |
[25] |
S. Haubruge, V. H. Nguyen and J. J. Strodiot,
Convergence analysis and applications of the Glowinski–Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 97 (1998), 645-673.
doi: 10.1023/A:1022646327085. |
[26] |
B. S. He and X. M. Yuan,
On the $O(1/n)$ convergence rate of the Douglas–Rachford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[27] |
P. Le Tallec, Numerical Analysis of Viscoelastic Problems, Research in Applied Mathematics, 15, Masson, Paris, 1990. |
[28] |
P. L. Lions and B. Mercier,
Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.
doi: 10.1137/0716071. |
[29] |
G. I. Marchuk, Splitting and alternating direction methods, in Handbook of Numerical Analysis, Vol. I, Handb. Numer. Anal., 1, North-Holland, Amsterdam, 1990, 197–462. |
[30] |
C. C. Paige and M. A. Saunders,
LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), 43-71.
doi: 10.1145/355984.355989. |
[31] |
D. W. Peaceman and H. H. Rachford Jr.,
The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.
doi: 10.1137/0103003. |
[32] |
K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, Inc., Boston, MA, 1990.
doi: 10.1016/c2009-0-22279-3.![]() ![]() |
[33] |
R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.
![]() |
[34] |
R. Tibshirani,
Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.
doi: 10.1111/j.2517-6161.1996.tb02080.x. |
[35] |
P. Tseng,
Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), 119-138.
doi: 10.1137/0329006. |
[36] |
P. T. Vuong and J. J. Strodiot,
The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces, J. Global Optim., 70 (2018), 477-495.
doi: 10.1007/s10898-017-0575-0. |
[37] |
H. R. Yue, Q. Z. Yang, X. F. Wang and X. M. Yuan, Implementing the alternating direction method of multipliers for big datasets: A case study of least absolute shrinkage and selection operator, SIAM J. Sci. Comput., 40 (2018), A3121–A3156.
doi: 10.1137/17M1146567. |
[38] |
C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
doi: 10.1142/9789812777096. |
[39] |
E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. Ⅰ: Projections on convex sets; Ⅱ: Spectral theory, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 237-424. |
show all references
References:
[1] |
M. V. Afonso, J. M. Bioucas-Dias and and M. A. Figueiredo,
Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356.
doi: 10.1109/TIP.2010.2047910. |
[2] |
N. Ahmed, T. Natarajan and and K. R. Rao,
Discrete cosine transform, IEEE Trans. Comput., C-23 (1974), 90-93.
doi: 10.1109/T-C.1974.223784. |
[3] |
J. B. Baillon and G. Haddad,
Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones, Israel J. Math., 26 (1977), 137-150.
doi: 10.1007/BF03007664. |
[4] |
H. H. Bauschke and P. L. Combettes, et al., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-48311-5. |
[5] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[6] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.
doi: 10.1137/080716542. |
[7] |
A. Beck and M. Teboulle,
A fast dual proximal gradient algorithm for convex minimization and applications, Oper. Res. Lett., 42 (2014), 1-6.
doi: 10.1016/j.orl.2013.10.007. |
[8] |
D. P. Bertsekas, Nonlinear Programming, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999. |
[9] |
S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein and et al.,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2010), 1-122.
doi: 10.1561/2200000016. |
[10] |
P. L. Combettes,
Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[11] |
P. L. Combettes and V. R. Wajs,
Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.
doi: 10.1137/050626090. |
[12] |
J. Douglas and H. H. Rachford Jr.,
On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[13] |
J. Eckstein and D. P. Bertsekas,
On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[14] |
J. Eckstein and W. Yao, Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results, RUTCOR Research Reports, 32 (2012). Google Scholar |
[15] |
R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Lecture Notes in Math., 506, Springer, Berlin, 1976, 73–89.
doi: 10.1007/BFb0080116. |
[16] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim., 2 (1992), 21-42.
doi: 10.1137/0802003. |
[17] |
R. Glowinski, Splitting methods for the numerical solution of the incompressible Navier-Stokes equations, in Vistas in Applied Mathematics, Optimization Software, New York, 1986, 57–95. |
[18] |
R. Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems, CBMS-NSF Regional Conference Series in Applied Mathematics, 86, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015.
doi: 10.1137/1.9781611973785.ch1. |
[19] |
R. Glowinski, P. G. Ciarlet and J.-L. Lions, Numerical Methods for Fluids: Finite Element Methods for Incompressible Viscous Flow, North Holland, 2003. Google Scholar |
[20] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970838. |
[21] |
R. Glowinski, S. Leung and J. L. Qian, Operator-splitting based fast sweeping methods for isotropic wave propagation in a moving fluid, SIAM J. Sci. Comput., 38 (2016), A1195–A1223.
doi: 10.1137/15M1043868. |
[22] |
R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et larésolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[23] |
R. Glowinski, S. J. Osher and W. T. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Scientific Computation, Springer, Cham, 2016.
doi: 10.1007/978-3-319-41589-5. |
[24] |
T. Goldstein, B. O'Donoghue, S. Setzer and R. Baraniuk,
Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2014), 1588-1623.
doi: 10.1137/120896219. |
[25] |
S. Haubruge, V. H. Nguyen and J. J. Strodiot,
Convergence analysis and applications of the Glowinski–Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 97 (1998), 645-673.
doi: 10.1023/A:1022646327085. |
[26] |
B. S. He and X. M. Yuan,
On the $O(1/n)$ convergence rate of the Douglas–Rachford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[27] |
P. Le Tallec, Numerical Analysis of Viscoelastic Problems, Research in Applied Mathematics, 15, Masson, Paris, 1990. |
[28] |
P. L. Lions and B. Mercier,
Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.
doi: 10.1137/0716071. |
[29] |
G. I. Marchuk, Splitting and alternating direction methods, in Handbook of Numerical Analysis, Vol. I, Handb. Numer. Anal., 1, North-Holland, Amsterdam, 1990, 197–462. |
[30] |
C. C. Paige and M. A. Saunders,
LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), 43-71.
doi: 10.1145/355984.355989. |
[31] |
D. W. Peaceman and H. H. Rachford Jr.,
The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.
doi: 10.1137/0103003. |
[32] |
K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, Inc., Boston, MA, 1990.
doi: 10.1016/c2009-0-22279-3.![]() ![]() |
[33] |
R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.
![]() |
[34] |
R. Tibshirani,
Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.
doi: 10.1111/j.2517-6161.1996.tb02080.x. |
[35] |
P. Tseng,
Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), 119-138.
doi: 10.1137/0329006. |
[36] |
P. T. Vuong and J. J. Strodiot,
The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces, J. Global Optim., 70 (2018), 477-495.
doi: 10.1007/s10898-017-0575-0. |
[37] |
H. R. Yue, Q. Z. Yang, X. F. Wang and X. M. Yuan, Implementing the alternating direction method of multipliers for big datasets: A case study of least absolute shrinkage and selection operator, SIAM J. Sci. Comput., 40 (2018), A3121–A3156.
doi: 10.1137/17M1146567. |
[38] |
C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
doi: 10.1142/9789812777096. |
[39] |
E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. Ⅰ: Projections on convex sets; Ⅱ: Spectral theory, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 237-424. |




Iteration | Time (s) | Objective | MSE | ISNR | ||
0.1 | 0.1 | 0.6259 | 1.6135e-04 | 11.9101 | ||
0.3 | 0.2 | 2852 | 101.04 | 0.6259 | 1.5999e-04 | 11.9470 |
0.7 | 0.5 | 1202 | 42.45 | 0.6259 | 1.5999e-04 | 11.9470 |
0.8 | 0.5 | 1088 | 38.57 | 0.6259 | 1.5999e-04 | 11.9470 |
0.9 | 0.8 | 879 | 31.01 | 0.6259 | 1.5999e-04 | 11.9470 |
0.9 | 0.9 | 847 | 29.82 | 0.6259 | 1.5999e-04 | 11.9470 |
Iteration | Time (s) | Objective | MSE | ISNR | ||
0.1 | 0.1 | 0.6259 | 1.6135e-04 | 11.9101 | ||
0.3 | 0.2 | 2852 | 101.04 | 0.6259 | 1.5999e-04 | 11.9470 |
0.7 | 0.5 | 1202 | 42.45 | 0.6259 | 1.5999e-04 | 11.9470 |
0.8 | 0.5 | 1088 | 38.57 | 0.6259 | 1.5999e-04 | 11.9470 |
0.9 | 0.8 | 879 | 31.01 | 0.6259 | 1.5999e-04 | 11.9470 |
0.9 | 0.9 | 847 | 29.82 | 0.6259 | 1.5999e-04 | 11.9470 |
Inner | Algorithm | Iteration | Mean/Max | Time (s) | Objective | MSE | ISNR |
precision | FGP | ||||||
1e-02 | 763 | 4.00/4 | 29.00 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.00/4 | 30.00 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.00/2 | 33.43 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 2.00/2 | 53.92 | 0.6261 | 1.5992e-04 | 11.9489 | |
1e-04 | 763 | 4.00/6 | 28.84 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.00/6 | 29.99 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.00/3 | 33.13 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 2.00/3 | 53.70 | 0.6261 | 1.5992e-04 | 11.9487 | |
1e-06 | 763 | 4.60/20 | 28.31 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.61/20 | 29.25 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.27/10 | 32.42 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 5.45/10 | 82.36 | 0.6259 | 1.5996e-04 | 11.9476 | |
1e-08 | 763 | 8.65/20 | 38.37 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 8.65/20 | 39.50 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 4.32/10 | 42.45 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 1277 | 7.62/10 | 44.71 | 0.6259 | 1.5997e-04 | 11.9476 | |
1e-10 | 762 | 16.61/20 | 58.43 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 16.72/20 | 59.79 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1525 | 8.44/10 | 63.16 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 1214 | 9.87/10 | 51.49 | 0.6259 | 1.5996e-04 | 11.9476 |
Inner | Algorithm | Iteration | Mean/Max | Time (s) | Objective | MSE | ISNR |
precision | FGP | ||||||
1e-02 | 763 | 4.00/4 | 29.00 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.00/4 | 30.00 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.00/2 | 33.43 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 2.00/2 | 53.92 | 0.6261 | 1.5992e-04 | 11.9489 | |
1e-04 | 763 | 4.00/6 | 28.84 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.00/6 | 29.99 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.00/3 | 33.13 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 2.00/3 | 53.70 | 0.6261 | 1.5992e-04 | 11.9487 | |
1e-06 | 763 | 4.60/20 | 28.31 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 4.61/20 | 29.25 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 2.27/10 | 32.42 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 3000 | 5.45/10 | 82.36 | 0.6259 | 1.5996e-04 | 11.9476 | |
1e-08 | 763 | 8.65/20 | 38.37 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 8.65/20 | 39.50 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1526 | 4.32/10 | 42.45 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 1277 | 7.62/10 | 44.71 | 0.6259 | 1.5997e-04 | 11.9476 | |
1e-10 | 762 | 16.61/20 | 58.43 | 0.6259 | 1.5999e-04 | 11.9470 | |
ADMM+GPRSM | 763 | 16.72/20 | 59.79 | 0.6259 | 1.5999e-04 | 11.9470 | |
GADMM | 1525 | 8.44/10 | 63.16 | 0.6259 | 1.5999e-04 | 11.9470 | |
FISTA | 1214 | 9.87/10 | 51.49 | 0.6259 | 1.5996e-04 | 11.9476 |
Inner | Algorithm | Iteration | Mean/Max | Time (s) | Objective | MSE | ISNR |
precision | FGP | ||||||
1e-02 | 763 | 4.00/4 | 27.87 | 0.6259 | 1.5999e-04 | 11.9470 | |
1e-02 | ADMM+GPRSM | 763 | 4.00/4 | 30.51 | 0.6259 | 1.5999e-04 | 11.9470 |
1e-04 | GADMM | 1526 | 2.00/3 | 33.01 | 0.6259 | 1.5999e-04 | 11.9470 |
1e-08 | FISTA | 1277 | 7.62/10 | 47.88 | 0.6259 | 1.5997e-04 | 11.9476 |
Inner | Algorithm | Iteration | Mean/Max | Time (s) | Objective | MSE | ISNR |
precision | FGP | ||||||
1e-02 | 763 | 4.00/4 | 27.87 | 0.6259 | 1.5999e-04 | 11.9470 | |
1e-02 | ADMM+GPRSM | 763 | 4.00/4 | 30.51 | 0.6259 | 1.5999e-04 | 11.9470 |
1e-04 | GADMM | 1526 | 2.00/3 | 33.01 | 0.6259 | 1.5999e-04 | 11.9470 |
1e-08 | FISTA | 1277 | 7.62/10 | 47.88 | 0.6259 | 1.5997e-04 | 11.9476 |
(m, n, s) | L | ||
( |
878.32 | 1.8416e+04 | 1.0860e-04 |
( |
293.15 | 4.4668e+03 | 4.4775e-04 |
( |
174.12 | 3.2264e+03 | 6.1989e-04 |
( |
67.90 | 9.4158e+02 | 2.1241e-03 |
( |
58.27 | 9.0143e+02 | 2.2187e-03 |
( |
9.59 | 3.6101e+02 | 5.5400e-03 |
(m, n, s) | L | ||
( |
878.32 | 1.8416e+04 | 1.0860e-04 |
( |
293.15 | 4.4668e+03 | 4.4775e-04 |
( |
174.12 | 3.2264e+03 | 6.1989e-04 |
( |
67.90 | 9.4158e+02 | 2.1241e-03 |
( |
58.27 | 9.0143e+02 | 2.2187e-03 |
( |
9.59 | 3.6101e+02 | 5.5400e-03 |
(m, n, s) | Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
( |
IN- |
9 | 24.72 | 5.8157e+04 | |
IN- |
9 | 710.26 | 5.8157e+04 | ||
IN- |
10 | 7.00/10 | 27.00 | 5.8157e+04 | |
IN- |
9 | 5.44/8 | 20.46 | 5.8157e+04 | |
IN- |
9 | 3.44/6 | 15.51 | 5.8157e+04 | |
IN- |
500 | 0.03/4 | 319.16 | 5.8157e+04 | |
IN- |
500 | 0.01/2 | 314.86 | 5.8157e+04 | |
IN- |
10 | 0.90/1 | 11.27 | 5.8157e+04 | |
( |
IN- |
9 | 6.36 | 1.6892e+04 | |
IN- |
9 | 61.89 | 1.6892e+04 | ||
IN- |
9 | 7.11/10 | 6.17 | 1.6892e+04 | |
IN- |
9 | 5.44/8 | 5.11 | 1.6892e+04 | |
IN- |
9 | 3.44/6 | 3.87 | 1.6892e+04 | |
IN- |
500 | 0.03/4 | 77.57 | 1.6892e+04 | |
IN- |
500 | 0.01/2 | 79.92 | 1.6892e+04 | |
IN- |
9 | 0.89/1 | 2.68 | 1.6892e+04 | |
( |
IN- |
8 | 5.98 | 1.0405e+04 | |
IN- |
8 | 96.69 | 1.0405e+04 | ||
IN- |
9 | 7.11/10 | 6.57 | 1.0405e+04 | |
IN- |
8 | 5.63/8 | 4.97 | 1.0405e+04 | |
IN- |
8 | 3.63/6 | 3.85 | 1.0405e+04 | |
IN- |
500 | 0.03/4 | 84.29 | 1.0405e+04 | |
IN- |
500 | 0.01/2 | 83.01 | 1.0405e+04 | |
IN- |
9 | 0.89/1 | 2.81 | 1.0405e+04 | |
( |
IN- |
10 | 2.94 | 4.7993e+03 | |
IN- |
10 | 232.74 | 4.7993e+03 | ||
IN- |
10 | 7.10/10 | 3.24 | 4.7993e+03 | |
IN- |
11 | 4.82/8 | 2.73 | 4.7993e+03 | |
IN- |
10 | 3.30/6 | 1.96 | 4.7993e+03 | |
IN- |
500 | 0.03/4 | 36.20 | 4.7993e+03 | |
IN- |
500 | 0.01/2 | 35.20 | 4.7993e+03 | |
IN- |
10 | 0.90/1 | 1.74 | 4.7993e+03 | |
( |
IN- |
9 | 38.52 | 4.9942e+03 | |
IN- |
9 | 7.22/10 | 50.94 | 4.9942e+03 | |
IN- |
9 | 5.44/8 | 41.94 | 4.9942e+03 | |
IN- |
9 | 3.44/6 | 31.82 | 4.9942e+03 | |
IN- |
500 | 0.03/4 | 644.25 | 4.9942e+03 | |
IN- |
500 | 0.01/2 | 641.81 | 4.9942e+03 | |
IN- |
9 | 0.89/1 | 20.72 | 4.9942e+03 | |
( |
IN- |
38 | 159.14 | 2.1247e+03 | |
IN- |
37 | 5.49/8 | 180.66 | 2.1247e+03 | |
IN- |
37 | 3.95/7 | 146.45 | 2.1247e+03 | |
IN- |
37 | 2.43/5 | 113.97 | 2.1247e+03 | |
IN- |
500 | 0.08/4 | 707.33 | 2.1247e+03 | |
IN- |
500 | 0.02/2 | 693.12 | 2.1247e+03 | |
IN- |
37 | 0.97/1 | 91.95 | 2.1247e+03 |
(m, n, s) | Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
( |
IN- |
9 | 24.72 | 5.8157e+04 | |
IN- |
9 | 710.26 | 5.8157e+04 | ||
IN- |
10 | 7.00/10 | 27.00 | 5.8157e+04 | |
IN- |
9 | 5.44/8 | 20.46 | 5.8157e+04 | |
IN- |
9 | 3.44/6 | 15.51 | 5.8157e+04 | |
IN- |
500 | 0.03/4 | 319.16 | 5.8157e+04 | |
IN- |
500 | 0.01/2 | 314.86 | 5.8157e+04 | |
IN- |
10 | 0.90/1 | 11.27 | 5.8157e+04 | |
( |
IN- |
9 | 6.36 | 1.6892e+04 | |
IN- |
9 | 61.89 | 1.6892e+04 | ||
IN- |
9 | 7.11/10 | 6.17 | 1.6892e+04 | |
IN- |
9 | 5.44/8 | 5.11 | 1.6892e+04 | |
IN- |
9 | 3.44/6 | 3.87 | 1.6892e+04 | |
IN- |
500 | 0.03/4 | 77.57 | 1.6892e+04 | |
IN- |
500 | 0.01/2 | 79.92 | 1.6892e+04 | |
IN- |
9 | 0.89/1 | 2.68 | 1.6892e+04 | |
( |
IN- |
8 | 5.98 | 1.0405e+04 | |
IN- |
8 | 96.69 | 1.0405e+04 | ||
IN- |
9 | 7.11/10 | 6.57 | 1.0405e+04 | |
IN- |
8 | 5.63/8 | 4.97 | 1.0405e+04 | |
IN- |
8 | 3.63/6 | 3.85 | 1.0405e+04 | |
IN- |
500 | 0.03/4 | 84.29 | 1.0405e+04 | |
IN- |
500 | 0.01/2 | 83.01 | 1.0405e+04 | |
IN- |
9 | 0.89/1 | 2.81 | 1.0405e+04 | |
( |
IN- |
10 | 2.94 | 4.7993e+03 | |
IN- |
10 | 232.74 | 4.7993e+03 | ||
IN- |
10 | 7.10/10 | 3.24 | 4.7993e+03 | |
IN- |
11 | 4.82/8 | 2.73 | 4.7993e+03 | |
IN- |
10 | 3.30/6 | 1.96 | 4.7993e+03 | |
IN- |
500 | 0.03/4 | 36.20 | 4.7993e+03 | |
IN- |
500 | 0.01/2 | 35.20 | 4.7993e+03 | |
IN- |
10 | 0.90/1 | 1.74 | 4.7993e+03 | |
( |
IN- |
9 | 38.52 | 4.9942e+03 | |
IN- |
9 | 7.22/10 | 50.94 | 4.9942e+03 | |
IN- |
9 | 5.44/8 | 41.94 | 4.9942e+03 | |
IN- |
9 | 3.44/6 | 31.82 | 4.9942e+03 | |
IN- |
500 | 0.03/4 | 644.25 | 4.9942e+03 | |
IN- |
500 | 0.01/2 | 641.81 | 4.9942e+03 | |
IN- |
9 | 0.89/1 | 20.72 | 4.9942e+03 | |
( |
IN- |
38 | 159.14 | 2.1247e+03 | |
IN- |
37 | 5.49/8 | 180.66 | 2.1247e+03 | |
IN- |
37 | 3.95/7 | 146.45 | 2.1247e+03 | |
IN- |
37 | 2.43/5 | 113.97 | 2.1247e+03 | |
IN- |
500 | 0.08/4 | 707.33 | 2.1247e+03 | |
IN- |
500 | 0.02/2 | 693.12 | 2.1247e+03 | |
IN- |
37 | 0.97/1 | 91.95 | 2.1247e+03 |
Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
IN- |
500 | 26.31 | 6.5859e+05 | |
IN- |
44 | 4.11/6 | 2.68 | 6.5859e+05 |
IN- |
44 | 3.07/5 | 2.28 | 6.5859e+05 |
IN- |
43 | 1.91/4 | 1.76 | 6.5859e+05 |
IN- |
500 | 0.08/3 | 10.92 | 6.5859e+05 |
IN- |
500 | 0.02/2 | 10.39 | 6.5860e+05 |
IN- |
46 | 0.61/2 | 1.36 | 6.5859e+05 |
Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
IN- |
500 | 26.31 | 6.5859e+05 | |
IN- |
44 | 4.11/6 | 2.68 | 6.5859e+05 |
IN- |
44 | 3.07/5 | 2.28 | 6.5859e+05 |
IN- |
43 | 1.91/4 | 1.76 | 6.5859e+05 |
IN- |
500 | 0.08/3 | 10.92 | 6.5859e+05 |
IN- |
500 | 0.02/2 | 10.39 | 6.5860e+05 |
IN- |
46 | 0.61/2 | 1.36 | 6.5859e+05 |
Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
IN- |
29 | 1794.95 | 6.1097e+05 | |
IN- |
30 | 4.00/4 | 869.09 | 6.0859e+05 |
IN- |
30 | 3.13/4 | 757.60 | 6.0859e+05 |
IN- |
30 | 3.00/3 | 740.08 | 6.0859e+05 |
IN- |
29 | 2.00/2 | 592.07 | 6.1097e+05 |
IN- |
29 | 1.10/2 | 474.73 | 6.1074e+05 |
IN- |
29 | 0.90/2 | 454.99 | 6.1045e+05 |
Algorithm | Iteration | Mean/Max CG | Time (s) | Objective |
IN- |
29 | 1794.95 | 6.1097e+05 | |
IN- |
30 | 4.00/4 | 869.09 | 6.0859e+05 |
IN- |
30 | 3.13/4 | 757.60 | 6.0859e+05 |
IN- |
30 | 3.00/3 | 740.08 | 6.0859e+05 |
IN- |
29 | 2.00/2 | 592.07 | 6.1097e+05 |
IN- |
29 | 1.10/2 | 474.73 | 6.1074e+05 |
IN- |
29 | 0.90/2 | 454.99 | 6.1045e+05 |
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