Modified Dai-Yuan iterative scheme for nonlinear systems and its application

Mohammed Yusuf Waziri; Kabiru Ahmed; Abubakar Sani Halilu; Aliyu Mohammed Awwal

doi:10.3934/naco.2021044

Article Contents

2023, Volume 13, Issue 1: 53-80. Doi: 10.3934/naco.2021044

This issue Previous Article Existence and well-posedness for excess demand equilibrium problems Next Article Nonlinear algebra and applications

Modified Dai-Yuan iterative scheme for nonlinear systems and its application

1.
Department of Mathematical Sciences, Bayero University, Kano, Nigeria
2.
Department of Mathematics, Sule Lamido University, Kafin Hausa, Nigeria
3.
Department of Mathematics, Gombe state University, Gombe, Nigeria
4.
Numerical Optimization Research Group, Bayero University, Kano, Nigeria

^* Corresponding author: Mohammed Yusuf Waziri
^* Corresponding author: Mohammed Yusuf Waziri

Received: February 2021

Revised: September 2021

Early access: November 16, 2021

Published online: November 16, 2021

Abstract / Introduction Full Text(HTML) Figure(4) / Table(10) Related Papers Cited by

Abstract

By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Performance profile of Dolan and Moré for number of iterations

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Figure 2. Performance profile of Dolan and Moré for function evaluations

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Figure 3. Performance profile of Dolan and Moré for processing time

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Figure 4. Original and blurred images (First and second columns from the left). Restored images by the two methods (Third and Fourth columns)

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Table 1. Test results of the four methods for problems 4.1

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	6	12	0.27234	2.0475E-09	16	35	0.30469	4.1222E-09	11	25	0.13273	2.4989E-09	26	106	0.36405	0.0000E+00
5000	x2	6	12	0.07889	4.0428E-09	16	35	0.05510	8.2456E-09	12	28	0.05936	2.2971E-09	26	106	0.12459	0.0000E+00
5000	x3	3	5	0.01436	8.4985E-09	18	39	0.06173	4.5974E-09	12	28	0.07119	5.7248E-09	30	122	0.17739	0.0000E+00
5000	x4	5	9	0.02079	1.8997E-09	20	43	0.06536	4.6462E-09	14	34	0.05971	2.5758E-09	34	138	0.16181	0.0000E+00
5000	x5	11	23	0.03935	3.9754E-10	20	43	0.07339	8.5913E-09	15	37	0.05510	2.6441E-09	36	146	0.18173	0.0000E+00
5000	x6	20	45	0.08341	8.4879E-09	21	45	0.07166	3.4438E-09	15	36	0.06630	5.1003E-09	35	141	0.17659	0.0000E+00
5000	x7	12	25	0.05233	5.4026E-10	21	46	0.08474	4.3326E-09	15	39	0.09094	5.0139E-09	35	142	0.16383	0.0000E+00
5000	x8	10	21	0.04351	9.5806E-09	21	46	0.07816	5.2005E-09	15	41	0.07043	4.5835E-09	35	142	0.18567	0.0000E+00
10000	x1	6	12	0.04366	2.8956E-09	16	35	0.11302	5.8297E-09	10	23	0.05868	8.5342E-09	26	106	0.22574	0.0000E+00
10000	x2	6	12	0.04652	5.7173E-09	17	37	0.10115	3.8700E-09	12	28	0.13233	3.1978E-09	26	106	0.25446	0.0000E+00
10000	x3	4	6	0.02748	5.9571E-10	18	39	0.10617	6.5017E-09	12	28	0.08322	3.7136E-09	30	122	0.25620	0.0000E+00
10000	x4	5	9	0.03605	2.6866E-09	20	43	0.12137	6.5707E-09	14	33	0.10267	4.4531E-09	34	138	0.26421	0.0000E+00
10000	x5	11	23	0.09120	5.6221E-10	21	45	0.12382	4.0482E-09	15	37	0.11103	5.1673E-09	36	146	0.31887	0.0000E+00
10000	x6	21	48	0.14197	6.1811E-09	20	44	0.15292	5.5936E-09	16	39	0.12905	3.0303E-09	35	141	0.31019	0.0000E+00
10000	x7	12	25	0.08928	7.6404E-10	21	46	0.13180	6.1272E-09	16	43	0.12267	3.0885E-09	35	142	0.29184	0.0000E+00
10000	x8	11	22	0.07167	3.5449E-10	21	47	0.12698	7.3547E-09	15	41	0.14901	2.9293E-09	35	142	0.29874	0.0000E+00
50000	x1	6	12	0.18477	6.4748E-09	17	37	0.43468	4.3261E-09	12	28	0.31329	3.4966E-09	26	106	1.02635	0.0000E+00
50000	x2	7	13	0.18950	4.9344E-10	17	37	0.39080	8.6535E-09	12	28	0.32265	5.2020E-09	26	106	0.99223	0.0000E+00
50000	x3	4	6	0.09771	1.3320E-09	19	41	0.47699	4.8342E-09	13	31	0.37285	4.4055E-09	30	122	1.16776	0.0000E+00
50000	x4	5	9	0.13988	6.0075E-09	21	45	0.48957	4.8925E-09	16	42	0.43939	2.3366E-09	34	138	1.26896	0.0000E+00
50000	x5	11	23	0.29815	1.2571E-09	23	53	0.54401	5.3731E-09	18	52	0.53469	2.9724E-09	36	146	1.33839	0.0000E+00
50000	x6	22	51	0.55331	7.5655E-09	23	54	0.54909	7.2056E-09	19	55	0.59269	3.2582E-09	35	141	1.28348	0.0000E+00
50000	x7	12	25	0.32319	1.7085E-09	23	54	0.58040	4.9765E-09	19	61	0.60155	3.3731E-09	35	142	1.33907	0.0000E+00
50000	x8	11	22	0.26948	7.9265E-10	23	54	0.56286	5.4177E-09	18	56	0.56586	2.7351E-09	35	142	1.31569	0.0000E+00

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Table 2. Test results of the four methods for problems 4.2

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	1	2	0.01078	0.0000E+00	1	2	0.01097	0.0000E+00	1	2	0.01014	0.0000E+00	1	2	0.01003	0.0000E+00
5000	x2	1	2	0.02040	0.0000E+00	1	2	0.00944	0.0000E+00	1	2	0.01005	0.0000E+00	1	2	0.02019	0.0000E+00
5000	x3	1	2	0.01013	0.0000E+00	1	2	0.00965	0.0000E+00	1	2	0.00947	0.0000E+00	1	2	0.01038	0.0000E+00
5000	x4	1	2	0.00951	0.0000E+00	1	2	0.00966	0.0000E+00	1	2	0.00928	0.0000E+00	**	**	**	**
5000	x5	1	2	0.01045	0.0000E+00	1	2	0.00748	0.0000E+00	1	2	0.00649	0.0000E+00	1	2	0.00520	0.0000E+00
5000	x6	1	2	0.00940	0.0000E+00	1	2	0.00732	0.0000E+00	1	2	0.00662	0.0000E+00	1	2	0.00842	0.0000E+00
5000	x7	1	2	0.00979	0.0000E+00	1	2	0.00660	0.0000E+00	1	2	0.00667	0.0000E+00	1	2	0.00757	0.0000E+00
5000	x8	1	2	0.01111	0.0000E+00	1	2	0.00795	0.0000E+00	1	2	0.00687	0.0000E+00	1	2	0.00779	0.0000E+00
10000	x1	1	2	0.01478	0.0000E+00	1	2	0.01622	0.0000E+00	1	2	0.01650	0.0000E+00	1	2	0.01449	0.0000E+00
10000	x2	1	2	0.01571	0.0000E+00	1	2	0.01633	0.0000E+00	1	2	0.01488	0.0000E+00	1	2	0.01594	0.0000E+00
10000	x3	1	2	0.01490	0.0000E+00	1	2	0.02384	0.0000E+00	1	2	0.03357	0.0000E+00	1	2	0.01579	0.0000E+00
10000	x4	1	2	0.01739	0.0000E+00	1	2	0.01549	0.0000E+00	1	3	0.02037	0.0000E+00	**	**	**	**
10000	x5	1	2	0.01636	0.0000E+00	1	2	0.01000	0.0000E+00	1	2	0.01013	0.0000E+00	1	2	0.00759	0.0000E+00
10000	x6	1	2	0.01637	0.0000E+00	1	2	0.00939	0.0000E+00	1	2	0.01043	0.0000E+00	1	2	0.01118	0.0000E+00
10000	x7	1	2	0.01602	0.0000E+00	1	2	0.00893	0.0000E+00	1	2	0.00994	0.0000E+00	1	2	0.01012	0.0000E+00
10000	x8	1	2	0.01539	0.0000E+00	1	2	0.01026	0.0000E+00	1	2	0.01045	0.0000E+00	1	2	0.01100	0.0000E+00
50000	x1	1	2	0.05796	0.0000E+00	1	2	0.06032	0.0000E+00	1	2	0.06280	0.0000E+00	1	2	0.05899	0.0000E+00
50000	x2	1	2	0.06203	0.0000E+00	1	2	0.05772	0.0000E+00	1	2	0.06076	0.0000E+00	1	2	0.05922	0.0000E+00
50000	x3	1	2	0.05224	0.0000E+00	1	2	0.05857	0.0000E+00	1	2	0.06821	0.0000E+00	1	2	0.06629	0.0000E+00
50000	x4	1	2	0.06079	0.0000E+00	1	3	0.07697	0.0000E+00	1	5	0.11996	0.0000E+00	**	**	**	**
50000	x5	1	2	0.06053	0.0000E+00	1	2	0.03772	0.0000E+00	1	2	0.03890	0.0000E+00	1	2	0.03026	0.0000E+00
50000	x6	1	2	0.05694	0.0000E+00	1	2	0.03597	0.0000E+00	1	2	0.03897	0.0000E+00	1	2	0.03691	0.0000E+00
50000	x7	1	2	0.06281	0.0000E+00	1	2	0.03922	0.0000E+00	1	2	0.03959	0.0000E+00	1	2	0.03842	0.0000E+00
50000	x8	1	2	0.05664	0.0000E+00	1	2	0.03785	0.0000E+00	1	2	0.03866	0.0000E+00	1	2	0.04029	0.0000E+00

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Table 3. Test results of the four methods for problems 4.3

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	44	271	0.62935	3.1455E-09	42	258	0.60676	6.9529E-09	33	161	0.45597	7.0484E-09	**	**	**	**
5000	x2	39	243	0.63451	5.8143E-09	42	258	0.63349	6.8440E-09	49	253	0.65486	6.8393E-09	**	**	**	**
5000	x3	40	252	0.60480	7.5147E-09	41	252	0.58345	9.1694E-09	31	147	0.42323	8.9989E-09	**	**	**	**
5000	x4	37	233	0.57729	3.7953E-09	35	216	0.53482	9.0123E-09	40	194	0.51740	7.4457E-09	**	**	**	**
5000	x5	33	202	0.46893	9.9461E-09	35	217	0.53483	8.7841E-09	42	204	0.52178	7.4199E-09	**	**	**	**
5000	x6	34	212	0.52856	8.7562E-09	40	247	0.57437	6.6311E-09	64	358	0.89216	7.2966E-09	**	**	**	**
5000	x7	43	267	0.70223	9.1513E-09	37	228	0.55987	7.0629E-09	33	160	0.43868	9.8882E-09	**	**	**	**
5000	x8	39	244	0.62983	5.5424E-09	38	236	0.56374	9.7994E-09	39	212	0.51684	4.9792E-09	**	**	**	**
10000	x1	46	283	1.24261	8.9039E-09	41	252	1.06589	8.7049E-09	43	201	0.99828	6.2921E-09	**	**	**	**
10000	x2	43	270	1.20866	6.3134E-09	41	252	1.09452	8.5691E-09	33	125	0.65028	8.5659E-09	**	**	**	**
10000	x3	35	213	0.98029	7.5116E-09	41	252	1.11311	7.5636E-09	38	200	0.90475	9.7816E-09	**	**	**	**
10000	x4	41	256	1.13918	6.2659E-09	35	216	0.95719	7.4878E-09	**	**	**	**	**	**	**	**
10000	x5	38	232	1.09266	4.3520E-09	35	217	0.95539	7.2962E-09	32	165	0.78884	6.7343E-09	**	**	**	**
10000	x6	38	232	1.05984	8.2915E-09	39	241	1.02612	8.3641E-09	40	189	0.88094	8.8953E-09	**	**	**	**
10000	x7	41	262	1.15504	6.3728E-09	32	200	0.91508	9.2095E-09	42	207	1.01358	9.6843E-09	**	**	**	**
10000	x8	44	263	1.22024	4.4229E-09	45	278	1.23987	7.9323E-09	**	**	**	**	**	**	**	**
50000	x1	51	316	6.68143	6.4806E-09	42	260	5.45556	7.9101E-09	38	187	4.12196	7.2817E-09	**	**	**	**
50000	x2	48	301	6.33527	8.4813E-09	42	260	5.42795	7.8102E-09	40	181	4.31980	5.7514E-09	**	**	**	**
50000	x3	36	218	4.66628	7.4951E-09	42	260	5.36750	7.0264E-09	44	190	4.36631	3.5019E-09	**	**	**	**
50000	x4	33	206	4.39647	3.1500E-09	34	210	4.33667	7.4626E-09	34	143	3.36253	3.8370E-09	**	**	**	**
50000	x5	36	226	4.71243	9.0560E-09	34	211	4.36821	7.2827E-09	32	132	3.13728	9.7006E-09	**	**	**	**
50000	x6	38	235	4.97988	9.5093E-09	40	249	5.25738	8.0717E-09	36	164	3.76700	5.7396E-09	**	**	**	**
50000	x7	44	282	5.86905	6.3230E-09	37	234	4.87686	6.5499E-09	40	200	4.41362	5.0952E-09	**	**	**	**
50000	x8	37	231	4.86337	9.4485E-09	33	210	4.50616	9.9184E-09	36	138	3.29200	6.6412E-09	**	**	**	**

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Table 4. Test results of the four methods for problems 4.4

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	4	6	0.01613	1.1694E-09	16	35	0.06067	4.1367E-09	11	25	0.04204	2.7254E-09	13	22	0.05005	2.5649E-09
5000	x2	4	6	0.01681	8.3703E-09	16	35	0.06441	8.3035E-09	6	14	0.02475	3.7553E-09	15	25	0.05655	1.3159E-09
5000	x3	3	5	0.01304	3.3286E-09	18	39	0.05937	4.7509E-09	12	28	0.04195	4.8432E-09	17	28	0.05855	5.2831E-09
5000	x4	15	33	0.05422	1.7766E-09	20	43	0.06423	3.6850E-09	13	30	0.05059	2.4951E-09	21	35	0.09843	3.9220E-09
5000	x5	10	23	0.05208	1.0831E-09	16	35	0.05671	5.9886E-09	15	36	0.06009	5.0128E-09	21	37	0.07153	3.9833E-09
5000	x6	8	17	0.04271	6.9887E-09	20	44	0.06183	7.2667E-09	16	41	0.06403	4.8801E-09	22	42	0.09015	3.9830E-09
5000	x7	20	44	0.06693	1.8089E-09	20	46	0.07138	9.6847E-09	15	43	0.05725	3.8385E-09	22	43	0.08100	3.9830E-09
5000	x8	8	16	0.03136	2.2485E-09	20	46	0.06917	6.3318E-09	15	44	0.52479	4.4557E-09	22	44	0.08352	3.9830E-09
10000	x1	4	6	0.02494	1.6538E-09	16	35	0.08973	5.8502E-09	11	25	0.06575	2.1388E-09	13	22	0.07581	3.6273E-09
10000	x2	5	7	0.02770	1.0276E-10	17	37	0.08415	3.8974E-09	12	28	0.06498	3.1362E-09	15	25	0.12385	1.8610E-09
10000	x3	3	5	0.02101	4.7073E-09	18	39	0.08491	6.7188E-09	11	26	0.07826	6.4560E-09	17	28	0.12292	7.4714E-09
10000	x4	15	33	0.10815	2.5125E-09	20	43	0.09236	5.2114E-09	15	37	0.09462	2.7562E-09	21	35	0.13054	5.5465E-09
10000	x5	10	23	0.06283	1.5317E-09	17	38	0.09884	9.9575E-09	16	40	0.21861	3.1439E-09	21	37	0.13807	5.6333E-09
10000	x6	8	17	0.05651	9.8835E-09	21	48	0.11028	7.9275E-09	16	42	0.10622	4.2741E-09	22	42	0.13112	5.6328E-09
10000	x7	20	44	0.10776	2.5582E-09	21	49	0.12883	4.5754E-09	16	46	0.08559	5.0752E-09	22	43	0.12979	5.6328E-09
10000	x8	8	16	0.05015	3.1798E-09	20	47	0.11975	8.9545E-09	16	47	0.11307	5.1334E-09	22	44	0.12479	5.6328E-09
50000	x1	4	6	0.09720	3.6981E-09	17	37	0.36221	4.3415E-09	12	28	0.27787	3.4805E-09	13	22	0.29020	8.1110E-09
50000	x2	5	7	0.11296	2.2973E-10	17	37	0.32769	8.7148E-09	12	28	0.27745	5.2196E-09	15	25	0.48431	4.1613E-09
50000	x3	4	7	0.09722	5.7871E-09	19	41	0.37649	4.9966E-09	13	31	0.30385	4.6998E-09	19	31	0.40698	2.0700E-09
50000	x4	15	33	0.34967	5.6182E-09	22	49	0.42467	3.5662E-09	16	41	0.39497	5.2693E-09	23	38	0.48994	1.5367E-09
50000	x5	10	23	0.25464	3.4249E-09	22	52	0.44383	8.0423E-09	18	50	0.44577	4.6654E-09	23	40	0.49968	1.5607E-09
50000	x6	9	18	0.19377	6.5941E-10	23	55	0.46324	4.6940E-09	19	54	0.46182	5.3105E-09	24	45	0.53573	1.5606E-09
50000	x7	20	44	0.43657	5.7203E-09	23	58	0.53040	9.1580E-09	18	57	0.47142	3.8366E-09	24	46	0.54006	1.5606E-09
50000	x8	8	16	0.17111	7.1103E-09	23	58	0.49150	8.6064E-09	18	57	0.48898	3.4965E-09	24	47	0.56776	1.5606E-09

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Table 5. Test results of the four methods for problems 4.5

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	13	29	0.08409	6.1206E-09	21	45	0.11835	4.5793E-09	13	34	0.07926	8.7285E-09	27	57	0.14177	9.2581E-09
5000	x2	13	29	0.08607	6.0981E-09	21	45	0.11653	4.5624E-09	13	34	0.07905	8.6940E-09	27	57	0.14553	9.2256E-09
5000	x3	13	29	0.08117	5.9183E-09	21	45	0.09188	4.4271E-09	13	34	0.09186	8.4192E-09	27	57	0.13343	8.9623E-09
5000	x4	13	29	0.07941	4.4547E-09	20	43	0.14263	9.9982E-09	12	29	0.09527	8.3184E-09	27	57	0.14216	6.6834E-09
5000	x5	13	29	0.07700	3.3263E-09	20	43	0.11080	7.4583E-09	15	38	0.09559	3.4919E-09	27	57	0.12148	4.9324E-09
5000	x6	12	27	0.07471	8.0821E-09	20	43	0.12521	4.9185E-09	14	34	0.08074	3.5022E-09	26	55	0.12577	7.3575E-09
5000	x7	12	27	0.11041	3.8935E-09	19	41	0.11220	7.1229E-09	13	31	0.07748	4.0165E-09	25	53	0.13185	8.0804E-09
5000	x8	11	25	0.06216	4.7068E-09	18	39	0.10240	9.9853E-09	13	31	0.08459	3.3404E-09	24	51	0.51949	8.5601E-09
10000	x1	13	30	0.15350	2.3785E-09	23	53	0.19978	5.1107E-09	15	46	0.16562	8.1229E-09	28	58	0.31646	0.0000E+00
10000	x2	15	37	0.15897	5.1927E-09	23	53	0.19735	5.0918E-09	15	46	0.17351	8.0936E-09	28	58	0.24591	0.0000E+00
10000	x3	19	54	0.23207	2.7569E-09	23	53	0.21008	4.9409E-09	15	46	0.15890	7.8592E-09	28	58	0.23659	0.0000E+00
10000	x4	13	32	0.13566	2.2501E-09	21	45	0.18849	4.7069E-09	14	39	0.16351	6.5113E-09	27	57	0.37153	9.3333E-09
10000	x5	14	35	0.13436	7.6581E-09	21	45	0.16255	3.5112E-09	13	34	0.12313	6.3921E-09	27	57	0.23985	6.9447E-09
10000	x6	12	28	0.12876	7.1509E-09	20	43	0.18314	6.9563E-09	11	25	0.12049	8.3813E-09	27	56	0.25423	0.0000E+00
10000	x7	9	18	0.08808	5.6775E-09	20	43	0.21754	3.3642E-09	12	29	0.13141	4.5477E-09	26	54	0.21101	0.0000E+00
10000	x8	7	13	0.06663	5.1294E-09	19	41	0.16822	4.6958E-09	13	31	0.14696	4.3632E-09	25	52	0.23674	0.0000E+00
50000	x1	8	14	0.31637	9.7835E-10	26	67	1.02269	9.0422E-09	23	107	1.49375	8.1128E-09	24	50	0.92306	0.0000E+00
50000	x2	8	14	0.29982	9.7482E-10	26	67	1.06621	9.0088E-09	23	107	1.41950	8.0837E-09	24	50	0.92547	0.0000E+00
50000	x3	8	14	0.29072	9.4607E-10	26	67	1.07177	8.7417E-09	23	104	1.38072	6.5311E-09	24	50	0.89133	0.0000E+00
50000	x4	8	14	0.29903	7.1236E-10	25	63	1.00707	7.9619E-09	20	80	1.19303	6.3556E-09	24	50	0.94953	0.0000E+00
50000	x5	8	14	0.29453	5.3210E-10	23	53	0.87334	6.1957E-09	17	58	0.90448	6.6277E-09	24	50	0.88540	0.0000E+00
50000	x6	8	14	0.31612	3.5134E-10	22	49	0.86319	7.0213E-09	14	40	0.63377	7.7993E-09	22	46	0.86539	0.0000E+00
50000	x7	8	14	0.28685	1.7015E-10	20	43	0.74746	7.5227E-09	15	38	0.65625	3.5221E-09	22	46	0.85031	0.0000E+00
50000	x8	7	13	0.27290	5.4616E-09	20	43	0.74983	3.5066E-09	11	27	0.48111	6.0062E-09	22	46	0.85129	0.0000E+00

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Table 6. Test results of the four methods for problems 4.6

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	11	29	0.05830	5.9634E-09	21	66	0.09121	4.9608E-09	12	33	0.05388	5.3440E-09	60	245	0.30692	9.2918E-09
5000	x2	11	29	0.05493	4.3213E-09	21	66	0.07995	4.7837E-09	12	33	0.05358	5.8173E-09	60	245	0.29193	9.1558E-09
5000	x3	10	27	0.04793	1.1728E-09	20	63	0.08563	9.6409E-09	12	33	0.05520	4.2467E-09	60	245	0.30544	7.9678E-09
5000	x4	8	21	0.03816	8.8309E-09	6	15	0.02606	1.5409E-09	10	25	0.04814	5.5576E-09	59	241	0.27576	8.6175E-09
5000	x5	17	52	0.08657	6.7905E-09	22	69	0.10643	3.8403E-09	12	32	0.06375	3.6002E-09	61	248	0.29183	8.0471E-09
5000	x6	12	31	0.04854	6.6177E-09	22	68	0.11060	4.6406E-09	13	42	0.10346	5.9588E-09	62	250	0.29224	8.1876E-09
5000	x7	13	36	0.05865	5.1067E-09	23	72	0.11025	5.6953E-09	11	31	0.05111	2.8649E-09	62	250	0.27411	7.6296E-09
5000	x8	14	38	0.06893	9.5514E-10	7	18	0.03452	1.0412E-09	11	29	0.05033	7.4314E-09	62	250	0.31480	8.3682E-09
10000	x1	11	29	0.12935	8.4335E-09	21	66	0.14482	7.0156E-09	13	37	0.09583	1.4459E-09	61	249	0.49287	9.5333E-09
10000	x2	11	29	0.08446	6.1113E-09	21	66	0.18801	6.7652E-09	13	37	0.10481	1.7840E-09	61	249	0.50182	9.3937E-09
10000	x3	10	27	0.08571	1.6586E-09	21	66	0.15276	4.9435E-09	12	33	0.10223	4.7426E-09	61	249	0.48216	8.1748E-09
10000	x4	9	23	0.07085	1.8142E-09	6	15	0.05041	2.1791E-09	10	25	0.08289	9.5969E-09	60	245	0.46931	8.8414E-09
10000	x5	17	52	0.14664	9.6032E-09	22	69	0.18159	5.4309E-09	11	28	0.08064	8.7915E-09	62	252	0.50543	8.2562E-09
10000	x6	12	31	0.08264	9.3589E-09	23	73	0.15916	6.3474E-09	14	46	0.10846	1.7662E-09	63	254	0.51418	8.4004E-09
10000	x7	13	36	0.12906	7.2220E-09	23	73	0.25665	8.5323E-09	14	43	0.10606	2.1715E-09	63	254	0.51744	7.8279E-09
10000	x8	14	38	0.10402	1.3508E-09	7	18	0.07598	1.4725E-09	14	42	0.09957	2.1096E-09	63	254	0.49114	8.5857E-09
50000	x1	12	31	0.33042	9.3840E-10	22	69	0.62427	5.6816E-09	14	41	0.42098	2.3754E-09	64	261	2.32850	8.1397E-09
50000	x2	12	31	0.36379	5.6107E-10	22	69	0.62287	5.4788E-09	14	41	0.42743	2.1971E-09	64	261	2.27368	8.0206E-09
50000	x3	10	27	0.29782	3.7088E-09	22	69	0.64410	4.0033E-09	13	39	0.41428	8.9237E-09	63	257	2.26461	9.6210E-09
50000	x4	9	23	0.26084	4.0568E-09	6	15	0.17896	4.8726E-09	13	38	0.39586	3.2734E-09	63	257	2.21871	7.5490E-09
50000	x5	18	54	0.54905	3.5244E-09	24	77	0.69938	4.1819E-09	17	62	0.58483	2.4241E-09	64	260	2.27889	9.7167E-09
50000	x6	13	33	0.38900	1.9096E-09	24	77	0.74722	5.1440E-09	15	50	0.45349	5.4303E-09	65	262	2.28170	9.8864E-09
50000	x7	14	38	0.38756	1.1869E-09	24	77	0.67827	6.9131E-09	14	39	0.40820	1.4432E-09	65	262	2.33294	9.2126E-09
50000	x8	14	38	0.40306	3.0204E-09	24	77	0.69055	7.2256E-09	14	38	0.38383	1.4634E-09	66	266	2.30200	7.3306E-09

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Table 7. Test results of the four methods for problems 4.7

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	16	59	0.09432	1.3050E-09	30	124	0.14592	7.0705E-09	10	32	0.04996	7.5384E-09	57	349	0.32182	9.4550E-09
5000	x2	14	53	0.07568	4.0067E-09	30	124	0.13148	6.9353E-09	10	30	0.04617	9.1316E-09	57	349	0.35728	9.3912E-09
5000	x3	16	60	0.08765	6.4133E-09	30	124	0.13896	5.8827E-09	9	30	0.03916	7.4550E-09	57	349	0.30529	8.7821E-09
5000	x4	16	62	0.07775	5.5391E-09	11	37	0.06037	1.8485E-09	9	29	0.04321	4.7455E-09	53	325	0.31769	7.4711E-09
5000	x5	15	57	0.07999	9.8573E-09	12	40	0.05478	9.3263E-09	11	37	0.05279	5.2468E-09	57	346	0.31532	7.5276E-09
5000	x6	15	58	0.07809	7.8166E-09	13	43	0.06090	2.5833E-09	**	**	**	**	58	349	0.29898	8.0592E-09
5000	x7	16	59	0.08286	1.3795E-09	14	44	0.07650	3.3787E-09	**	**	**	**	56	333	0.32301	7.5313E-09
5000	x8	12	42	0.06033	2.4066E-10	14	43	0.06044	6.0314E-09	**	**	**	**	54	321	0.29418	8.5637E-09
10000	x1	16	59	0.14692	1.8456E-09	30	124	0.26347	9.9993E-09	11	33	0.06356	7.5740E-09	58	355	0.54602	9.3986E-09
10000	x2	14	53	0.10996	5.6664E-09	30	124	0.23092	9.8079E-09	11	34	0.07902	4.5397E-11	58	355	0.55024	9.3352E-09
10000	x3	16	60	0.13400	9.0698E-09	30	124	0.21597	8.3194E-09	13	43	0.10110	9.3470E-09	58	355	0.53853	8.7297E-09
10000	x4	16	62	0.13363	7.8334E-09	11	37	0.30166	2.6142E-09	6	19	0.04657	4.5079E-10	54	331	0.52338	7.4265E-09
10000	x5	16	61	0.12671	4.8938E-09	13	43	0.07998	1.9679E-09	11	39	0.08054	2.4455E-09	58	352	0.52326	7.4827E-09
10000	x6	16	62	0.12887	5.9649E-09	13	43	0.09948	3.6534E-09	**	**	**	**	59	355	0.61184	8.0112E-09
10000	x7	16	59	0.14343	1.9510E-09	14	44	0.07941	4.7783E-09	**	**	**	**	57	339	0.51324	7.4864E-09
10000	x8	12	42	0.09692	3.4034E-10	14	43	0.09317	8.5297E-09	167	1820	2.60568	4.7429E-11	55	327	0.52918	8.5127E-09
50000	x1	16	59	0.52241	4.1268E-09	32	132	0.95785	4.9313E-09	14	54	0.46701	1.1179E-09	61	373	2.61151	7.2981E-09
50000	x2	15	56	0.46085	5.5395E-09	32	132	0.98458	4.8369E-09	16	60	0.51123	4.0239E-10	61	373	2.58425	7.2489E-09
50000	x3	17	64	0.56293	7.8990E-09	31	128	0.97264	8.7364E-09	16	58	0.52341	3.8080E-09	60	367	2.53142	9.6440E-09
50000	x4	17	65	0.52033	1.7365E-09	11	37	0.33492	5.8455E-09	10	32	0.30108	3.4982E-09	56	343	2.30189	8.2044E-09
50000	x5	17	64	0.56208	1.1172E-09	13	43	0.40677	4.4003E-09	87	897	5.90688	5.5572E-09	60	364	2.46086	8.2665E-09
50000	x6	17	66	0.57559	4.9142E-09	13	43	0.38142	8.1692E-09	61	577	4.09402	3.9378E-09	61	367	2.51340	8.8502E-09
50000	x7	16	59	0.51799	4.3625E-09	17	54	0.49638	1.9299E-09	66	643	4.51278	7.7185E-09	59	351	2.42242	8.2705E-09
50000	x8	12	42	0.40991	7.6102E-10	35	137	1.04612	6.9307E-09	185	1959	12.83235	7.1829E-09	57	339	2.30068	9.4043E-09

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Table 8. Test results of the four methods for problems 4.8

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm	Niter	Nfev	Cpt	Norm
5000	x1	14	105	0.16757	0.0000E+00	**	**	**	**	8	76	0.08422	0.0000E+00	3	19	0.03360	0.0000E+00
5000	x2	14	102	0.14631	0.0000E+00	**	**	**	**	8	76	0.10084	0.0000E+00	5	32	0.05756	0.0000E+00
5000	x3	14	96	0.13992	0.0000E+00	4	34	0.03438	0.0000E+00	15	141	0.16698	0.0000E+00	4	20	0.06854	0.0000E+00
5000	x4	17	126	0.17229	0.0000E+00	5	44	0.06671	0.0000E+00	16	90	0.14892	0.0000E+00	2	16	0.03029	0.0000E+00
5000	x5	7	55	0.08252	0.0000E+00	6	44	0.07206	0.0000E+00	22	125	0.19471	0.0000E+00	3	30	0.06496	0.0000E+00
5000	x6	4	30	0.04918	0.0000E+00	3	19	0.04112	0.0000E+00	13	95	0.14016	0.0000E+00	2	19	0.03237	0.0000E+00
5000	x7	4	30	0.05837	0.0000E+00	5	36	0.06107	0.0000E+00	19	111	0.18132	0.0000E+00	2	20	0.04827	0.0000E+00
5000	x8	6	55	0.08952	0.0000E+00	4	24	0.06101	0.0000E+00	68	499	0.65635	0.0000E+00	3	22	0.03871	0.0000E+00
10000	x1	10	73	1.18135	0.0000E+00	5	35	0.11482	0.0000E+00	8	76	0.16320	0.0000E+00	3	19	0.05802	0.0000E+00
10000	x2	11	75	0.20685	0.0000E+00	212	1703	3.57595	0.0000E+00	8	76	0.20896	0.0000E+00	3	19	0.06237	0.0000E+00
10000	x3	14	94	0.24852	0.0000E+00	5	37	0.11556	0.0000E+00	5	51	0.13497	0.0000E+00	4	20	0.06374	0.0000E+00
10000	x4	10	74	0.20559	0.0000E+00	3	20	0.05981	0.0000E+00	26	243	0.57624	0.0000E+00	2	16	0.05057	0.0000E+00
10000	x5	4	35	0.09897	0.0000E+00	5	31	0.09720	0.0000E+00	31	267	0.65314	0.0000E+00	3	30	0.08397	0.0000E+00
10000	x6	3	18	0.05165	0.0000E+00	5	34	0.10333	0.0000E+00	12	108	0.25737	0.0000E+00	2	19	0.05356	0.0000E+00
10000	x7	5	42	0.13123	0.0000E+00	4	23	0.07361	0.0000E+00	33	288	0.65692	0.0000E+00	2	20	0.05280	0.0000E+00
10000	x8	4	25	0.07246	0.0000E+00	4	23	0.08236	0.0000E+00	7	64	0.17944	0.0000E+00	3	22	0.06539	0.0000E+00
50000	x1	10	72	0.86301	0.0000E+00	7	42	0.49996	0.0000E+00	8	76	0.82304	0.0000E+00	3	19	0.24206	0.0000E+00
50000	x2	9	71	0.85537	0.0000E+00	7	41	0.49430	0.0000E+00	8	76	0.85630	0.0000E+00	3	19	0.24635	0.0000E+00
50000	x3	10	71	0.82141	0.0000E+00	7	39	0.46821	0.0000E+00	4	28	0.35046	0.0000E+00	3	19	0.23377	0.0000E+00
50000	x4	8	51	0.64090	0.0000E+00	**	**	**	**	7	60	0.69240	0.0000E+00	2	16	0.16730	0.0000E+00
50000	x5	3	24	0.29724	0.0000E+00	7	38	0.40553	0.0000E+00	9	69	0.73666	0.0000E+00	4	31	0.32924	0.0000E+00
50000	x6	3	18	0.25253	0.0000E+00	7	40	0.47899	0.0000E+00	10	73	0.88118	0.0000E+00	2	19	0.20408	0.0000E+00
50000	x7	3	18	0.22632	0.0000E+00	**	**	**	**	8	64	0.74632	0.0000E+00	2	20	0.23963	0.0000E+00
50000	x8	4	24	0.30785	0.0000E+00	6	32	0.33168	0.0000E+00	36	246	2.80129	0.0000E+00	3	22	0.27244	0.0000E+00

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Table 9. Summary of result from tables 1-4 showing the number of problems/percentage solved with least number of iteration, function values and processing time by each of the four methods

Method	Niter	Percentage	fev	Percentage	Ptime	Percentage	Fails
MDDYM	83	43.22%	91	47.40%	81	40.19%	-
PDYM	19	9.90%	14	7.29%	30	15.63%	4
MDY	36	18.75%	40	20.83%	55	28.65%	7
MFR	23	11.98%	20	10.42%	26	13.53%	27
Undecided	31	16.15%	27	14.06%	-	-	-

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Table 10. Performance results for MDDYM and MFRM methods based on objective function (ObjFun) value, mean square error (MSE), SNR and SSIM index

Image & size	ObjFun		MSE		SNR		SSIM
	MDDYM	MFRM	MDDYM	MFRM	MDDYM	MFRM	MDDYM	MFRM
Barbara $ 256\times256 $	$ 1.530\times 10^6 $	$ 1.585\times 10^6 $	$ 1.5967\times 10^2 $	$ 2.0627\times 10^2 $	20.66	19.55	0.80	0.75
Girl $ 256\times256 $	$ 6.224\times 10^6 $	$ 6.321\times 10^6 $	$ 1.4978\times 10^2 $	$ 1.6071\times 10^2 $	21.73	21.42	0.77	0.75
Lena $ 256\times256 $	$ 1.459\times 10^6 $	$ 1.513\times 10^6 $	$ 6.5691\times 10^1 $	$ 9.0026\times 10^1 $	24.29	22.93	0.90	0.87
Cameraman $ 256\times256 $	$ 1.415\times 10^6 $	$ 1.473\times 10^6 $	$ 1.2579\times 10^2 $	$ 1.7757\times 10^2 $	21.55	20.05	0.87	0.83

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Related Papers

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References

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