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

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

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  • 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.

    Mathematics Subject Classification: Primary: 90C25; Secondary: 49M27.

    Citation:

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  • Figure 1.  Left to right: original image, blurring image, and recovered image by $ \theta $-scheme on TV-based deblurring

    Figure 2.  Visualization of function values, MSE, and ISNR on TV-based deblurring

    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''

    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"

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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