$ h $ | $ \mu_1 $ | $ \mu_2 $ | $ \mu_3 $ |
0.5 | 0.4004 | 1.0000 | 1.1990 |
0.1 | 0.3714 | 1.0000 | 1.2570 |
0.05 | 0.3681 | 1.0000 | 1.2640 |
0.02 | 0.3661 | 1.0000 | 1.2580 |
0.01 | 0.3655 | 1.0000 | 1.2690 |
0.005 | 0.3651 | 1.0000 | 1.2700 |
0.001 | 0.3649 | 1.0000 | 1.2700 |
An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator $ T $ on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation
$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $
to the solution set $ T^{-1}(0) $ of a generalized equation $ 0 \in T(x) $. With smart choices of parameters $ \lambda(t) $ and $ \alpha(t) $, we prove the weak convergence of the trajectory to some point of $ T^{-1}(0) $ with $ \|\dot{x}(t)\|\leq {\rm O}(1/t) $ as $ t\rightarrow +\infty $. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory $ x^{h}(t) \rightarrow x(t) $ on interval $ [0,+\infty) $ is demonstrated when the step size $ h \rightarrow 0 $.
Citation: |
Table 1.
Convergence of multipliers with different
$ h $ | $ \mu_1 $ | $ \mu_2 $ | $ \mu_3 $ |
0.5 | 0.4004 | 1.0000 | 1.1990 |
0.1 | 0.3714 | 1.0000 | 1.2570 |
0.05 | 0.3681 | 1.0000 | 1.2640 |
0.02 | 0.3661 | 1.0000 | 1.2580 |
0.01 | 0.3655 | 1.0000 | 1.2690 |
0.005 | 0.3651 | 1.0000 | 1.2700 |
0.001 | 0.3649 | 1.0000 | 1.2700 |
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