On the Preconditioned Delayed Weighted Gradient Method

ALEIXO, R.; LARA URDANETA, H.

doi:10.5540/tcam.2023.024.03.00437

R. ALEIXO ^**Corresponding author: Rafael Aleixo - E-mail: rafael.aleixo@ufsc.br

Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Blumenau, R. João Pessoa, 2514, 89036-004, Blumenau, SC, Brazil - E-mail: rafael.aleixo@ufsc.br

http://orcid.org/0000-0002-7865-6929

H. LARA URDANETA

Departamento de Engenharia de Controle, Automação e Computação, Universidade Federal de Santa Catarina, Campus Blumenau, R. João Pessoa, 2514, 89036-004, Blumenau, SC, Brazil - E-mail: hugo.lara.urdaneta@ufsc.br

http://orcid.org/0000-0003-1670-5075

*Corresponding author: Rafael Aleixo - E-mail: rafael.aleixo@ufsc.br

	mesh3em5				Muu
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	13	2.7597e-005	4.1695e-005	0.0010	84	2.1429e-004	9.6983e-009	0.0849
PCG	13	1.8247e-005	3.5884e-005	0.0009	30	1.0021e-004	7.8161e-009	0.0329
DWGM	13	2.9586e-005	4.0226e-005	0.0009	83	3.3700e-004	7.6887e-009	0.1019
PDWGM	13	2.0587e-005	3.4235e-005	0.0019	30	1.2917e-004	6.8319e-009	0.0369
	Chem97ZtZ					bodyy4
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	137	3.1782e-010	9.6090e-009	0.0199	273	1.4529e-009	9.7342e-009	0.274857
PCG	32	4.5905e-010	6.6680e-009	0.0059	214	3.7600e-009	9.7342e-009	0.229854
DWGM	135	9.0344e-010	7.5092e-009	0.0239	262	5.7226e-009	9.3859e-009	0.336092
PDWGM	34	1.0241e-010	3.1926e-009	0.0069	212	5.7226e-009	9.7141e-009	0.318831
	bundle1					nasa2146
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	242	4.1278e-019	9.3649e-009	0.9597	429	3.9752e-014	9.2289e-009	0.1979
PCG	66	1.8484e-019	7.1148e-009	0.2953	323	4.1317e-014	9.7221e-009	0.1479
DWGM	241	8.7440e-019	9.4834e-009	0.9444	414	1.9318e-013	9.4622e-009	0.1971
PDWGM	65	3.7965e-019	8.3751e-009	0.2758	312	1.3832e-013	9.4843e-009	0.1559
	wathen120					msc00726
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	413	2.2624e-009	9.2956e-009	1.4831	1512	9.5563e-014	7.8397e-009	0.3557
PCG	51	2.0597e-010	6.2655e-009	0.1848	126	2.9107e-015	7.0725e-009	0.0339
DWGM	405	6.9872e-009	9.2236e-009	1.6300	1520	3.4609e-013	1.9936e-008	0.3957
PDWGM	50	4.4364e-010	8.1356e-009	0.2158	122	1.7645e-014	7.0100e-009	0.0309
	bcsstm12					Pres Poisson
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	4313	1.5650e-005	6.9687e-009	0.8158	2243	5.3538e-005	9.8029e-007	8.7070
PCG	504	7.2098e-006	7.1205e-009	0.0999	767	2.8998e-005	9.8692e-007	3.1102
DWGM	3396	3.0353e-004	1.9472e-008	0.8019	2119	5.7822e-004	4.8639e-006	8.5900
PDWGM	464	9.3194e-005	8.9432e-009	0.1019	758	7.2819e-005	1.2784e-006	3.2261
	msc04515					1138 bus
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	4812	4.8974e-014	9.6160e-007	2.9952	2000	4.1944e-005	8.8203e-005	0.2198
PCG	3825	1.4561e-012	8.6307e-007	2.4625	970	1.7307e-005	9.6794e-005	0.1249
DWGM	4698	1.0657e-012	6.0433e-005	3.1961	1966	2.5406e-004	1.0520e-004	0.2188
PDWGM	3771	1.7856e-012	6.8977e-005	2.4694	975	3.1397e-005	6.8587e-005	0.1329
	cbuckle					cvxbqp1
	niter	e _k	r _k	CPU(s)	niter	e _k	r _k	CPU(s)
CG	4944	7.4180e-005	8.7746e-005	17.4160	8148	8.6930e-006	9.8426e-005	23.7433
PCG	970	7.9553e-005	9.7548e-005	3.7778	5103	8.6930e-006	9.9575e-005	16.9593
DWGM	3960	9.6921e-004	9.9202e-005	15.3422	5942	2.7696e-004	9.9960e-005	22.8188
PDWGM	785	0.0021630	9.6522e-005	3.0412	3793	3.1636e-004	9.9852e-005	16.0228

	n=7 and p=3
	niter	e _k	r _k	CPU(s)
DWGM	7 ± 0	2.8042e-013 ± 8.5845e-013	1.7165e-013 ± 5.2349e-013	1.0973e-003 ± 1.0944e-003
PDWGM	3 ± 0	4.6075e-013 ± 1.4180e-012	1.5011e-013 ± 4.6798e-013	6.9978e-004 ± 9.4906e-004
	n=100 and p=11
	niter	e _k	r _k	CPU(s)
DWGM	50.70 ± 10.79	1.3530e-005 ± 1.4383e-005	7.7019e-007 ± 1.7123e-007	1.1192e-002 ± 2.3468e-003
PDWGM	10.90 ± 0.31	1.2297e-007 ± 3.8885e-007	7.0609e-008 ± 2.2328e-007	4.4976e-003 ± 7.0674e-004
	n=1000 and p=51
	niter	e _k	r _k	CPU(s)
DWGM	143.50 ± 26.69	1.0365e-004 ± 4.9420e-005	9.4354e-007 ± 4.3212e-008	0.0785 ± 0.0138
PDWGM	33.00 ± 4.24	1.6387e-006 ± 8.6370e-007	5.6995e-007 ± 1.5505e-007	0.0620 ± 0.0101
	n=7000 and p=623
	niter	e _k	r _k	CPU(s)
DWGM	373.40 ± 73.30	1.1075e-003 ± 7.3672e-004	9.6649e-007 ± 1.7105e-008	14.9948 ± 2.9115
PDWGM	121.40 ± 31.77	2.3420e-006 ± 7.2454e-007	9.1388e-007 ± 5.6455e-008	12.2710 ± 3.2368

a_{k}^{Y} = 2 {(\frac{1}{α_{k - 1}^{SD}} + \frac{1}{α_{k}^{S D}} + \sqrt{{(\frac{1}{α_{k - 1}^{SD}} - \frac{1}{α_{k}^{S D}})}^{2} + \frac{4 {||g_{k}||}_{2}^{2}}{(α_{k - 1}^{SD} {||g_{k - 1}||}_{2}^{2})^{2}}})}^{- 1} .

{||g_{k + 1}||}_{M^{- 1 / 2}} \leq (\frac{λ_{1} (A^{1 / 2}) λ_{1} (M^{- 1 / 2}) - λ_{n} (A^{1 / 2}) λ_{n} (M^{- 1 / 2})}{(λ_{1} (A^{1 / 2}) + λ_{n} (A^{1 / 2})) λ_{n} (M^{- 1 / 2})}) {||g_{k}||}_{M^{- 1 / 2}} .

z_{k + 1}^{T} (v_{k} - g_{k - 1}) = g_{k + 1}^{T} M^{- 1} (v_{k} - g_{k - 1}) = (g_{k - 1} + β_{k} (v_{k} - g_{k - 1}))^{T} M^{- 1} (v_{k} - g_{k - 1}) = g_{k - 1}^{T} M^{- 1} (v_{k} - g_{k - 1}) + \frac{g_{k - 1}^{T} M^{- 1} (g_{k - 1} - v_{k})}{(g_{k - 1} - v_{k})^{T} M^{- 1} (g_{k - 1} - v_{k})} (v_{k} - g_{k - 1})^{T} M^{- 1} (v_{k} - g_{k - 1}) = 0 .

z_{k + 1}^{T} (g_{k} - g_{k - 1}) = g_{k + 1}^{T} M^{- 1} (g_{k} - g_{k - 1}) = g_{k + 1}^{T} M^{- 1} (v_{k} + α_{k} A z_{k} - g_{k - 1}) = g_{k + 1}^{T} M^{- 1} (v_{k} - g_{k - 1}) + α_{k} g_{k + 1}^{T} M^{- 1} A M^{- 1} g_{k} = 0 .

z_{k + 1}^{T} g_{k + 1} = g_{k + 1}^{T} M^{- 1} g_{k + 1} = g_{k + 1}^{T} M^{- 1} (g_{k - 1} + β_{k} (v_{k} - g_{k - 1})) = g_{k + 1}^{T} M^{- 1} (g_{k - 1} + β_{k} (g_{k} - g_{k - 1}) - α_{k} β_{k} A z_{k}) = g_{k + 1}^{T} M^{- 1} g_{k - 1} + β_{k} g_{k + 1}^{T} M^{- 1} (g_{k} - g_{k - 1}) - α_{k} β_{k} g_{k + 1}^{T} M^{- 1} A M^{- 1} g_{k} .

{||g_{k - 1} - v_{k}||}_{M^{- 1 / 2}}^{2} = {||g_{k - 1} - g_{k} + α_{k} A z_{k}||}_{M^{- 1 / 2}}^{2} = {||g_{k - 1} - g_{k}||}_{M^{- 1 / 2}}^{2} + 2 α_{k} z_{k}^{T} A M^{- 1} (g_{k - 1} - g_{k}) + α_{k}^{2} z_{k}^{T} A M^{- 1} A z_{k} = {||g_{k - 1} - g_{k}||}_{M^{- 1 / 2}}^{2} + 2 α_{k} (z_{k}^{T} A z_{k - 1} - z_{k}^{T} A z_{k}) + α_{k}^{2} z_{k}^{T} A M^{- 1} A z_{k} = {||g_{k - 1} - g_{k}||}_{M^{- 1 / 2}}^{2} - 2 (\frac{z_{k}^{T} A z_{k}}{z_{k} A M^{- 1} A z_{k}}) z_{k}^{T} A z_{k} + {(\frac{z_{k}^{T} A z_{k}}{z_{k}^{T} A M^{- 1} A z_{k}})}^{2} z_{k}^{T} A M^{- 1} A z_{k} = {||g_{k - 1} - g_{k}||}_{M^{- 1 / 2}}^{2} - \frac{(z_{k}^{T} A z_{k})^{2}}{z_{k}^{T} A M^{- 1} A z_{k}} .

z_{\hat{k} + 1}^{T} (g_{j} - g_{j - 1}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{j} - g_{j - 1}) = [g_{\hat{k} - 1} + β_{\hat{k}} (v_{\hat{k}} - g_{\hat{k} - 1})]^{T} M^{- 1} (g_{j} - g_{j - 1}) = [(1 - β_{\hat{k}}) g_{\hat{k} - 1} + β_{k} v_{\hat{k}}]^{T} M^{- 1} (g_{j} - g_{j - 1}) = β_{\hat{k}} v_{\hat{k}}^{T} M^{- 1} (g_{j} - g_{j - 1}) = β_{\hat{k}} (g_{\hat{k}} - α_{\hat{k}} A z_{\hat{k}})^{T} M^{- 1} (g_{j} - g_{j - 1}) = - β_{\hat{k}} α_{\hat{k}} g_{\hat{k}}^{T} M^{- 1} A M^{- 1} (g_{j} - g_{j - 1}) = 0 .

z_{\hat{k} + 1}^{T} (g_{\hat{k} - 1} - g_{\hat{k} - 2}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{\hat{k} - 1} - g_{\hat{k} - 2}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{\hat{k} - 1} - v_{k} + v_{k} - g_{\hat{k} - 2}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{\hat{k}} - α_{\hat{k}} A z_{\hat{k}} - g_{\hat{k} - 2}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{\hat{k}} - g_{\hat{k} - 2}) = g_{\hat{k} + 1}^{T} M^{- 1} (g_{\hat{k} - 2} + β_{\hat{k} - 1} (v_{\hat{k} - 1} - g_{\hat{k} - 2}) - g_{\hat{k} - 2}) = β_{\hat{k} - 1} g_{\hat{k} + 1}^{T} M^{- 1} (v_{\hat{k} - 1} - g_{\hat{k} - 2}) = β_{\hat{k} - 1} z_{\hat{k} + 1}^{T} (v_{\hat{k} - 1} - g_{\hat{k} - 2}) .

[1] *Corresponding author: Rafael Aleixo - E-mail: rafael.aleixo@ufsc.br

Brasil

Brasil

On the Preconditioned Delayed Weighted Gradient Method

ABSTRACT