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July  2021, 17(4): 1681-1711. doi: 10.3934/jimo.2020040

The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis

Yaonan Ma and  Li-Zhi Liao

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong SAR, China

Received  August 2019 Revised  October 2019 Published  February 2020

Fund Project: The work of L.-Z. Liao was supported in part by grants from Hong Kong Baptist University (FRG) and General Research Fund (GRF) of Hong Kong

In this paper, we focus on a splitting method called the $ \theta $-scheme proposed by Glowinski and Le Tallec in [17,20,27]. First, we present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact $ \theta $-scheme. Second, for unconstrained problems, we prove the convergence of the $ \theta $-scheme and show a sublinear convergence rate in terms of objective value. Furthermore, a practical inexact $ \theta $-scheme is derived to solve $ l_2 $-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the $ \theta $-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the $ \theta $-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the $ \theta $-scheme and our proposed methods.

Keywords: Splitting method, the $\theta $-scheme, convergence, convergence rate, inexactness criterion.
Mathematics Subject Classification: Primary: 90C25; Secondary: 49M27.
Citation: Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. P. Bertsekas, Nonlinear Programming, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999.  Google Scholar

[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.  Google Scholar

[10]

P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.  doi: 10.1080/02331930412331327157.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. Le Tallec, Numerical Analysis of Viscoelastic Problems, Research in Applied Mathematics, 15, Masson, Paris, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[33] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. P. Bertsekas, Nonlinear Programming, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999.  Google Scholar

[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.  Google Scholar

[10]

P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.  doi: 10.1080/02331930412331327157.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. Le Tallec, Numerical Analysis of Viscoelastic Problems, Research in Applied Mathematics, 15, Masson, Paris, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[33] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.  Google Scholar

[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.  Google Scholar

Figure 1.  Left to right: original image, blurring image, and recovered image by $ \theta $-scheme on TV-based deblurring
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Figure 2.  Visualization of function values, MSE, and ISNR on TV-based deblurring
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Figure 3.  Objective value with respect to the computing time and residual of the linear equation (29) with respect to the outer loop iteration on "news20.scale''
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Figure 4.  Objective value with respect to the computing time and residual of the linear equation (29) with respect to the outer loop iteration on "url_combined"
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Table 1.  A sensitivity test of the $ \theta $-scheme on TV-based deblurring
$ \beta_1 $ $ \beta_2 $ Iteration Time (s) Objective MSE ISNR
0.1 0.1 $ 3000 $ $ 106.41 $ 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
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$ \beta_1 $ $ \beta_2 $ Iteration Time (s) Objective MSE ISNR
0.1 0.1 $ 3000 $ $ 106.41 $ 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
Table 2.  Comparison of $ \theta $-scheme, ADMM+GPRSM, GADMM, and FISTA on TV-based deblurring under same inner precision
Inner Algorithm Iteration Mean/Max Time (s) Objective MSE ISNR
precision FGP
1e-02 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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
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Inner Algorithm Iteration Mean/Max Time (s) Objective MSE ISNR
precision FGP
1e-02 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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 $ \theta $-scheme 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
Table 3.  Comparison of $ \theta $-scheme, ADMM+GPRSM, GADMM, and FISTA on TV-based deblurring under different inner precisions
Inner Algorithm Iteration Mean/Max Time (s) Objective MSE ISNR
precision FGP
1e-02 $ \theta $-scheme 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
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Inner Algorithm Iteration Mean/Max Time (s) Objective MSE ISNR
precision FGP
1e-02 $ \theta $-scheme 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
Table 4.  Values of $ \eta $, $ L $ and $ \beta_1 (\beta_2) $ for synthetic dataset
(m, n, s) $ \eta $ L $ \beta_1(\beta_2) $
($ 1\times10^4, 1.5\times10^4, 50\% $) 878.32 1.8416e+04 1.0860e-04
($ 1\times10^4, 1.5\times10^4, 10\% $) 293.15 4.4668e+03 4.4775e-04
($ 1.5\times10^4, 2\times10^4, 5\% $) 174.12 3.2264e+03 6.1989e-04
($ 2\times10^4, 3\times10^4, 1\% $) 67.90 9.4158e+02 2.1241e-03
($ 2\times10^5, 3\times10^5, 0.1\% $) 58.27 9.0143e+02 2.2187e-03
($ 3\times10^5, 2\times10^6, 0.01\% $) 9.59 3.6101e+02 5.5400e-03
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(m, n, s) $ \eta $ L $ \beta_1(\beta_2) $
($ 1\times10^4, 1.5\times10^4, 50\% $) 878.32 1.8416e+04 1.0860e-04
($ 1\times10^4, 1.5\times10^4, 10\% $) 293.15 4.4668e+03 4.4775e-04
($ 1.5\times10^4, 2\times10^4, 5\% $) 174.12 3.2264e+03 6.1989e-04
($ 2\times10^4, 3\times10^4, 1\% $) 67.90 9.4158e+02 2.1241e-03
($ 2\times10^5, 3\times10^5, 0.1\% $) 58.27 9.0143e+02 2.2187e-03
($ 3\times10^5, 2\times10^6, 0.01\% $) 9.59 3.6101e+02 5.5400e-03
Table 5.  Comparison of IN-$ \theta_{\text{LSQR}} $, IN-$ \theta_{\text{Cholesky}} $, IN-$ \theta_{1e-t} $ and IN-$ \theta $ on synthetic dataset
(m, n, s) Algorithm Iteration Mean/Max CG Time (s) Objective
($ 1\times10^4, 1.5\times10^4, 50\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 24.72 5.8157e+04
IN-$ \theta_{\text{Cholesky}} $ 9 $ \sim/\sim $ 710.26 5.8157e+04
IN-$ \theta_{1e-10} $ 10 7.00/10 27.00 5.8157e+04
IN-$ \theta_{1e-8} $ 9 5.44/8 20.46 5.8157e+04
IN-$ \theta_{1e-6} $ 9 3.44/6 15.51 5.8157e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 319.16 5.8157e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 314.86 5.8157e+04
IN-$ \theta $ 10 0.90/1 11.27 5.8157e+04
($ 1\times10^4, 1.5\times10^4, 10\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 6.36 1.6892e+04
IN-$ \theta_{\text{Cholesky}} $ 9 $ \sim/\sim $ 61.89 1.6892e+04
IN-$ \theta_{1e-10} $ 9 7.11/10 6.17 1.6892e+04
IN-$ \theta_{1e-8} $ 9 5.44/8 5.11 1.6892e+04
IN-$ \theta_{1e-6} $ 9 3.44/6 3.87 1.6892e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 77.57 1.6892e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 79.92 1.6892e+04
IN-$ \theta $ 9 0.89/1 2.68 1.6892e+04
($ 1.5\times10^4, 2\times10^4, 5\% $) IN-$ \theta_{\text{LSQR}} $ 8 $ \sim/\sim $ 5.98 1.0405e+04
IN-$ \theta_{\text{Cholesky}} $ 8 $ \sim/\sim $ 96.69 1.0405e+04
IN-$ \theta_{1e-10} $ 9 7.11/10 6.57 1.0405e+04
IN-$ \theta_{1e-8} $ 8 5.63/8 4.97 1.0405e+04
IN-$ \theta_{1e-6} $ 8 3.63/6 3.85 1.0405e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 84.29 1.0405e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 83.01 1.0405e+04
IN-$ \theta $ 9 0.89/1 2.81 1.0405e+04
($ 2\times10^4, 3\times10^4, 1\% $) IN-$ \theta_{\text{LSQR}} $ 10 $ \sim/\sim $ 2.94 4.7993e+03
IN-$ \theta_{\text{Cholesky}} $ 10 $ \sim/\sim $ 232.74 4.7993e+03
IN-$ \theta_{1e-10} $ 10 7.10/10 3.24 4.7993e+03
IN-$ \theta_{1e-8} $ 11 4.82/8 2.73 4.7993e+03
IN-$ \theta_{1e-6} $ 10 3.30/6 1.96 4.7993e+03
IN-$ \theta_{1e-4} $ 500 0.03/4 36.20 4.7993e+03
IN-$ \theta_{1e-2} $ 500 0.01/2 35.20 4.7993e+03
IN-$ \theta $ 10 0.90/1 1.74 4.7993e+03
($ 2\times10^5, 3\times10^5, 0.1\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 38.52 4.9942e+03
IN-$ \theta_{1e-10} $ 9 7.22/10 50.94 4.9942e+03
IN-$ \theta_{1e-8} $ 9 5.44/8 41.94 4.9942e+03
IN-$ \theta_{1e-6} $ 9 3.44/6 31.82 4.9942e+03
IN-$ \theta_{1e-4} $ 500 0.03/4 644.25 4.9942e+03
IN-$ \theta_{1e-2} $ 500 0.01/2 641.81 4.9942e+03
IN-$ \theta $ 9 0.89/1 20.72 4.9942e+03
($ 3\times10^5, 2\times10^6, 0.01\% $) IN-$ \theta_{\text{LSQR}} $ 38 $ \sim/\sim $ 159.14 2.1247e+03
IN-$ \theta_{1e-10} $ 37 5.49/8 180.66 2.1247e+03
IN-$ \theta_{1e-8} $ 37 3.95/7 146.45 2.1247e+03
IN-$ \theta_{1e-6} $ 37 2.43/5 113.97 2.1247e+03
IN-$ \theta_{1e-4} $ 500 0.08/4 707.33 2.1247e+03
IN-$ \theta_{1e-2} $ 500 0.02/2 693.12 2.1247e+03
IN-$ \theta $ 37 0.97/1 91.95 2.1247e+03
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(m, n, s) Algorithm Iteration Mean/Max CG Time (s) Objective
($ 1\times10^4, 1.5\times10^4, 50\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 24.72 5.8157e+04
IN-$ \theta_{\text{Cholesky}} $ 9 $ \sim/\sim $ 710.26 5.8157e+04
IN-$ \theta_{1e-10} $ 10 7.00/10 27.00 5.8157e+04
IN-$ \theta_{1e-8} $ 9 5.44/8 20.46 5.8157e+04
IN-$ \theta_{1e-6} $ 9 3.44/6 15.51 5.8157e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 319.16 5.8157e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 314.86 5.8157e+04
IN-$ \theta $ 10 0.90/1 11.27 5.8157e+04
($ 1\times10^4, 1.5\times10^4, 10\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 6.36 1.6892e+04
IN-$ \theta_{\text{Cholesky}} $ 9 $ \sim/\sim $ 61.89 1.6892e+04
IN-$ \theta_{1e-10} $ 9 7.11/10 6.17 1.6892e+04
IN-$ \theta_{1e-8} $ 9 5.44/8 5.11 1.6892e+04
IN-$ \theta_{1e-6} $ 9 3.44/6 3.87 1.6892e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 77.57 1.6892e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 79.92 1.6892e+04
IN-$ \theta $ 9 0.89/1 2.68 1.6892e+04
($ 1.5\times10^4, 2\times10^4, 5\% $) IN-$ \theta_{\text{LSQR}} $ 8 $ \sim/\sim $ 5.98 1.0405e+04
IN-$ \theta_{\text{Cholesky}} $ 8 $ \sim/\sim $ 96.69 1.0405e+04
IN-$ \theta_{1e-10} $ 9 7.11/10 6.57 1.0405e+04
IN-$ \theta_{1e-8} $ 8 5.63/8 4.97 1.0405e+04
IN-$ \theta_{1e-6} $ 8 3.63/6 3.85 1.0405e+04
IN-$ \theta_{1e-4} $ 500 0.03/4 84.29 1.0405e+04
IN-$ \theta_{1e-2} $ 500 0.01/2 83.01 1.0405e+04
IN-$ \theta $ 9 0.89/1 2.81 1.0405e+04
($ 2\times10^4, 3\times10^4, 1\% $) IN-$ \theta_{\text{LSQR}} $ 10 $ \sim/\sim $ 2.94 4.7993e+03
IN-$ \theta_{\text{Cholesky}} $ 10 $ \sim/\sim $ 232.74 4.7993e+03
IN-$ \theta_{1e-10} $ 10 7.10/10 3.24 4.7993e+03
IN-$ \theta_{1e-8} $ 11 4.82/8 2.73 4.7993e+03
IN-$ \theta_{1e-6} $ 10 3.30/6 1.96 4.7993e+03
IN-$ \theta_{1e-4} $ 500 0.03/4 36.20 4.7993e+03
IN-$ \theta_{1e-2} $ 500 0.01/2 35.20 4.7993e+03
IN-$ \theta $ 10 0.90/1 1.74 4.7993e+03
($ 2\times10^5, 3\times10^5, 0.1\% $) IN-$ \theta_{\text{LSQR}} $ 9 $ \sim/\sim $ 38.52 4.9942e+03
IN-$ \theta_{1e-10} $ 9 7.22/10 50.94 4.9942e+03
IN-$ \theta_{1e-8} $ 9 5.44/8 41.94 4.9942e+03
IN-$ \theta_{1e-6} $ 9 3.44/6 31.82 4.9942e+03
IN-$ \theta_{1e-4} $ 500 0.03/4 644.25 4.9942e+03
IN-$ \theta_{1e-2} $ 500 0.01/2 641.81 4.9942e+03
IN-$ \theta $ 9 0.89/1 20.72 4.9942e+03
($ 3\times10^5, 2\times10^6, 0.01\% $) IN-$ \theta_{\text{LSQR}} $ 38 $ \sim/\sim $ 159.14 2.1247e+03
IN-$ \theta_{1e-10} $ 37 5.49/8 180.66 2.1247e+03
IN-$ \theta_{1e-8} $ 37 3.95/7 146.45 2.1247e+03
IN-$ \theta_{1e-6} $ 37 2.43/5 113.97 2.1247e+03
IN-$ \theta_{1e-4} $ 500 0.08/4 707.33 2.1247e+03
IN-$ \theta_{1e-2} $ 500 0.02/2 693.12 2.1247e+03
IN-$ \theta $ 37 0.97/1 91.95 2.1247e+03
Table 6.  Comparison of IN-$ \theta_{\text{LSQR}} $, IN-$ \theta_{1e-t} $ and IN-$ \theta_{\text{R}} $ on "news20.scale''
Algorithm Iteration Mean/Max CG Time (s) Objective
IN-$ \theta_{\text{LSQR}} $ 500 $ \sim/\sim $ 26.31 6.5859e+05
IN-$ \theta_{1e-10} $ 44 4.11/6 2.68 6.5859e+05
IN-$ \theta_{1e-8} $ 44 3.07/5 2.28 6.5859e+05
IN-$ \theta_{1e-6} $ 43 1.91/4 1.76 6.5859e+05
IN-$ \theta_{1e-4} $ 500 0.08/3 10.92 6.5859e+05
IN-$ \theta_{1e-2} $ 500 0.02/2 10.39 6.5860e+05
IN-$ \theta_{\text{R}} $ 46 0.61/2 1.36 6.5859e+05
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Algorithm Iteration Mean/Max CG Time (s) Objective
IN-$ \theta_{\text{LSQR}} $ 500 $ \sim/\sim $ 26.31 6.5859e+05
IN-$ \theta_{1e-10} $ 44 4.11/6 2.68 6.5859e+05
IN-$ \theta_{1e-8} $ 44 3.07/5 2.28 6.5859e+05
IN-$ \theta_{1e-6} $ 43 1.91/4 1.76 6.5859e+05
IN-$ \theta_{1e-4} $ 500 0.08/3 10.92 6.5859e+05
IN-$ \theta_{1e-2} $ 500 0.02/2 10.39 6.5860e+05
IN-$ \theta_{\text{R}} $ 46 0.61/2 1.36 6.5859e+05
Table 7.  Comparison of IN-$ \theta_{\text{LSQR}} $, IN-$ \theta_{1e-t} $ and IN-$ \theta_{R} $ on "url_combined" dataset
Algorithm Iteration Mean/Max CG Time (s) Objective
IN-$ \theta_{\text{LSQR}} $ 29 $ \sim/\sim $ 1794.95 6.1097e+05
IN-$ \theta_{1e-10} $ 30 4.00/4 869.09 6.0859e+05
IN-$ \theta_{1e-8} $ 30 3.13/4 757.60 6.0859e+05
IN-$ \theta_{1e-6} $ 30 3.00/3 740.08 6.0859e+05
IN-$ \theta_{1e-4} $ 29 2.00/2 592.07 6.1097e+05
IN-$ \theta_{1e-2} $ 29 1.10/2 474.73 6.1074e+05
IN-$ \theta_{\text{R}} $ 29 0.90/2 454.99 6.1045e+05
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Algorithm Iteration Mean/Max CG Time (s) Objective
IN-$ \theta_{\text{LSQR}} $ 29 $ \sim/\sim $ 1794.95 6.1097e+05
IN-$ \theta_{1e-10} $ 30 4.00/4 869.09 6.0859e+05
IN-$ \theta_{1e-8} $ 30 3.13/4 757.60 6.0859e+05
IN-$ \theta_{1e-6} $ 30 3.00/3 740.08 6.0859e+05
IN-$ \theta_{1e-4} $ 29 2.00/2 592.07 6.1097e+05
IN-$ \theta_{1e-2} $ 29 1.10/2 474.73 6.1074e+05
IN-$ \theta_{\text{R}} $ 29 0.90/2 454.99 6.1045e+05
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