NIEVINSKI, I.C.L.; SOUZA, M.; GOLDFELD, P.; AUGUSTO, D.A.; RODRIGUES, J.R.P.; CARVALHO, L.M.

doi:10.5540/tema.2018.019.01.0059

I.C.L. NIEVINSKI ^**Corresponding author: Ítalo Nievinski - E-mail: italonievinski@gmail.com.

Faculdade de Engenharia Mecânica, PPGEM, UERJ - Universidade do Estado do Rio de Janeiro, 20550-900 Rio de Janeiro, RJ, Brasil.

M. SOUZA

Departamento de Estatística e Matemática Aplicada DEMA UFC - Universidade Federal do Ceará, Campus do PICI, 60455-760, Fortaleza, CE, Brasil. E-mail: michael@ufc.br

P. GOLDFELD

Departamento de Matemática Aplicada, IM-UFRJ, Caixa Postal 68530, CEP 21941-909, Rio de Janeiro, RJ, Brasil. E-mail: goldfeld@ufrj.br

D.A. AUGUSTO

Fundação Oswaldo Cruz, Fiocruz, Av. Brasil, 4365, 21040-360 Rio de Janeiro, RJ, Brasil. E-mail: daa@fiocruz.br

J.R.P. RODRIGUES

PETROBRAS/CENPES Av. Horácio Macedo 950, Cidade Universitária, 21941-915 Rio de Janeiro, RJ, Brasil. E-mail: jrprodrigues@petrobras.com.br

L.M. CARVALHO

Instituto de Matemática e Estatística, IME, UERJ - Univ do Estado do Rio de Janeiro, 20550-900 Rio de Janeiro, RJ, Brasil. E-mail: luizmc@ime.uerj.br

*Corresponding author: Ítalo Nievinski - E-mail: italonievinski@gmail.com.

Matrix	Problem Size	Matrix Size
F8	182558	730232
U26	408865	1635460
U31	617459	2469836
Psalt	765620	3062480
SPE10	1094421	4377684

Mat	Prec	Number of Iterations Processes					Total Time (s) Processes
Mat	Prec	1	2	4	8	16	1	2	4	8	16
F8	iS	104	0.9113	0.9106	0.992	0.9102	10.8	7.9	4.1	2.1	1.8
F8	bJ	104	119	109	125	135	10.8	0.96.1	0.93.0	0.91.9	0.91.7
U26	iS	92	0.9131	0.9199	0.9218	0.9303	22.0	0.919.8	0.914.5	0.99.1	9.9
U26	bJ	92	252	NC	404	349	22.0	28.2	NC	13.3	0.99.7
U31	iS	69	0.9110	0.9449	0.9517	0.9864	28.0	28.8	0.948.7	0.932.1	0.943.1
U31	bJ	69	124	593	639	NC	28.0	0.924.1	56.7	35.4	NC
Psalt	iS	34	0.968	0.957	0.972	0.954	17.8	24.3	11.9	8.2	5.6
Psalt	bJ	34	87	76	117	76	17.8	0.919.7	0.99.2	0.98.0	0.95.0
SPE10	iS	116	0.9135	0.9148	0.9148	0.9176	71.5	53.9	30.9	18.0	17.5
SPE10	bJ	116	142	149	164	198	71.5	0.943.6	0.924.1	0.915.4	0.916.1

Iterative Solver Time (s)
Mat	Prec	Processes
Mat	Prec	1	2	4	8	16
F8	iS	10.5	6.3	3.3	0.91.7	0.91.6
F8	bJ	10.5	0.96.1	0.93.0	1.9	1.7
U26	iS	21.7	0.916.2	0.912.7	0.98.2	0.99.4
U26	bJ	21.7	28.1	NC	13.3	9.7
U31	iS	27.5	0.922.8	0.945.6	0.930.5	0.942.2
U31	bJ	27.5	23.9	56.7	35.3	NC
Psalt	iS	17.3	0.917.5	0.98.4	0.96.4	0.94.6
Psalt	bJ	17.3	19.6	9.2	8.0	5.0
SPE10	iS	70.8	44.2	26.0	15.5	0.916.1
SPE10	bJ	70.8	0.943.4	0.924.0	0.915.3	16.1

A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}],

A = L U = [\begin{matrix} L_{1} \\ ⋱ \\ L_{P} \\ B_{1} & \dots & B_{P} & I \end{matrix}] [\begin{matrix} U_{1} & C_{1} \\ ⋱ & ⋮ \\ U_{P} & C_{P} \\ S \end{matrix}],

A^{- 1} = [\begin{matrix} U_{1}^{- 1} \\ ⋱ \\ U_{P}^{- 1} \\ I \end{matrix}] [\begin{matrix} C_{1} \\ I & ⋮ \\ C_{P} \\ - I \end{matrix}] [\begin{matrix} I \\ S^{- 1} \end{matrix}] [\begin{matrix} I \\ B_{1} & \dots & B_{P} & - I \end{matrix}] [\begin{matrix} L_{1}^{- 1} \\ ⋱ \\ L_{P}^{- 1} \\ I \end{matrix}] .

M_{F} = [\begin{matrix} {\tilde{U}}_{1}^{- 1} \\ ⋱ \\ {\tilde{U}}_{P}^{- 1} \\ I \end{matrix}] [\begin{matrix} {\tilde{C}}_{1} \\ I & ⋮ \\ {\tilde{C}}_{P} \\ - I \end{matrix}] [\begin{matrix} I \\ S_{F}^{- 1} \end{matrix}] [\begin{matrix} I \\ {\tilde{B}}_{1} & \dots & {\tilde{B}}_{P} & - I \end{matrix}] [\begin{matrix} {\tilde{L}}_{1}^{- 1} \\ ⋱ \\ {\tilde{L}}_{P}^{- 1} \\ I \end{matrix}] .

L e v (w_{l_{k}}) = \min {L e v (v_{l_{k}}), \min_{1 \leq i \leq (k - 1)} {L e v ({\tilde{L}}_{J_{k i}}) + L e v (w l_{i}) + 1}},

L e v (F_{J_{k l}}) = \min {L e v (A_{Γ Γ_{k l}}), \min_{1 \leq i \leq m} {L e v ({\tilde{B}}_{J_{k i}}) + L e v ({\tilde{C}}_{J_{i l}}) + 1}},

S_{F}^{- 1} = \sum_{J = 1}^{P} T_{J} {(L_{{\tilde{S}}_{J}} U_{{\tilde{S}}_{J}})}^{- 1} R_{J} \approx \sum_{J = 1}^{P} T_{J} {\tilde{S}}_{J}^{- 1} R_{J},

[1] *Corresponding author: Ítalo Nievinski - E-mail: italonievinski@gmail.com.

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Parallel Implementation of a Two-level Algebraic ILU(k)-based Domain Decomposition Preconditioner ^† † This article is based on work presented at CNMAC2016.

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Parallel Implementation of a Two-level Algebraic ILU(k)-based Domain Decomposition Preconditioner † † This article is based on work presented at CNMAC2016.

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Parallel Implementation of a Two-level Algebraic ILU(k)-based Domain Decomposition Preconditioner ^† † This article is based on work presented at CNMAC2016.