
-
Previous Article
Optimal control and stabilization of building maintenance units based on minimum principle
- JIMO Home
- This Issue
-
Next Article
Time-consistent multiperiod mean semivariance portfolio selection with the real constraints
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). |
[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. |
[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). |
[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. |
[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 |
[1] |
Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 |
[2] |
Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial and Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199 |
[3] |
Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 |
[4] |
Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004 |
[5] |
Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013 |
[6] |
Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 |
[7] |
Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems and Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291 |
[8] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems and Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
[9] |
Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 |
[10] |
Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3017-3025. doi: 10.3934/dcdss.2020465 |
[11] |
Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012 |
[12] |
Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic and Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038 |
[13] |
Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503 |
[14] |
Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083 |
[15] |
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 |
[16] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[17] |
Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053 |
[18] |
Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61 |
[19] |
Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure and Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49 |
[20] |
Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107 |
2021 Impact Factor: 1.411
Tools
Metrics
Other articles
by authors
[Back to Top]