On the second-order asymptotical regularization of linear ill-posed inverse problems

Figures & data

Figure 1. The behaviour of $\chi \left(T\right)$ from (34(34) $\chi \left(T\right):=\parallel Ax\left(T\right)-y^{\delta }\parallel -\tau \delta =0,$(34) ) with different damping parameters η.

Table 1. The dependence of the solution accuracy and the convergence rate on the initial data $x_0$. $\mathrm{\Delta }t=19.4946,\eta =2.5648\times {10}^{-4},{\dot{x}}_0=0,\tau =2,p=0.1125$.

		DP		TEDP
δ (${\delta }^{\prime }$)	$x_0$	$k^{\ast }\left(\delta \right)$	L2Err	$k^{\ast }\left(\delta \right)$	L2Err
7.1191e−04 (1e−03)	0	1295	0.2214	2033	0.2894
7.1337e−04 (1e−03)	$-1$	1354	0.2284	2210	0.2825
7.0838e−04 (1e−03)	1	1108	0.1965	1777	0.1414
7.1602e−05 (1e−04)	0	2397	0.0688	2958	0.0653
8.0523e−05 (1e−04)	$-1$	2525	0.0983	3076	0.0920
7.8903e−05 (1e−04)	1	1600	0.0449	2643	0.0431

Table 2. The dependence of the solution accuracy and the convergence rate on the initial data ${\dot{x}}_0$. $\mathrm{\Delta }t=19.4946,\eta =0.0154,x_0=1,\tau =2,p=0.1125$.

		DP		TEDP
δ (${\delta }^{\prime }$)	${\dot{x}}_0$	$k^{\ast }\left(\delta \right)$	L2Err	$k^{\ast }\left(\delta \right)$	L2Err
6.3541e−04 (1e−03)	0	155	0.1062	156	0.1061
7.3673e−04 (1e−03)	$-0.01$	374	0.1576	376	0.1574
7.1484e−04 (1e−03)	0.01	39	0.0443	39	0.0443
8.0763e−05 (1e−04)	0	4559	0.0675	4561	0.0675
7.3174e−05 (1e−04)	$-0.01$	12466	0.0953	12467	0.0953
7.6200e−05 (1e−04)	0.01	982	0.0293	983	0.0293

Table 3. The dependence of the solution accuracy and the convergence rate on the damping parameter η. $\mathrm{\Delta }t=19.4946,x_0=1,{\dot{x}}_0=0,\tau =2,p=0.1125$.

		DP		TEDP
δ (${\delta }^{\prime }$)	η	$k^{\ast }\left(\delta \right)$	L2Err	$k^{\ast }\left(\delta \right)$	L2Err
8.7715e−04 (1e−03)	2.5648e−05	6943	0.8319	17393	0.7744
7.3673e−04 (1e−03)	2.5648e−04	1108	0.1310	1728	0.1080
7.1484e−04 (1e−03)	2.5648e−03	124	0.0901	183	0.0870
8.1011e−05 (1e−04)	2.5648e−02	205	0.1101	206	0.1100
7.3174e−05 (1e−04)	0.0513	398	0.1108	398	0.1108
7.2721e−05 (1e−04)	0.1026 ($=2/\mathrm{\Delta }t$)	848	0.1096	848	0.1096

Table 4. Comparisons with the Landweber method, the Nesterov's method, the Chebyshev method, and the CGNE method.

	DP			TEDP			Landweber
δ	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err
Example 1
1.0400e−2	70	0.2023	0.1494	70	0.1995	0.1494	20438	52.6428	0.2639
1.0380e−3	2872	8.9239	0.0945	2871	1.9105	0.0945	$k_{max}$	1.6899e3	0.1807
1.0445e−4	56246	177.3387	0.0597	56246	177.9603	0.0473	$k_{max}$	1.8048e3	0.1807
Example 2
1.0761e−2	15	0.0554	0.3676	25	0.0753	0.1987	1457	6.3672	0.7032
1.0703e−3	93	0.2488	0.0637	132	0.3567	0.0581	23790	67.6082	0.1767
1.1006e−4	453	1.4931	0.0195	531	1.8000	0.0184	187188	679.8274	0.0509
	Nesterov			Chebyshev			CGNE
δ	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err	$k^{\ast }\left(\delta \right)$	CPU (s)	L2Err
Example 1
1.0400e−2	419	1.1384	0.2590	264	0.7378	0.2553	6	0.0351	0.2213
1.0380e−3	2813	8.5986	0.1600	2229	7.6512	0.1496	18	0.0904	0.1383
1.0445e−4	16642	60.1179	0.1025	17443	52.2056	0.0897	39	0.2013	0.0894
Example 2
1.0761e−2	102	0.3018	0.7043	62	0.1768	0.7102	6	0.0148	0.4835
1.0703e−3	416	1.1732	0.1676	415	1.1908	0.1190	12	0.0261	0.1514
1.1006e−4	1805	6.0793	0.0280	2226	7.6967	0.0196	15	0.0309	0.0447