多群辐射扩散问题特征驱动的并行AMG法

doi:10.19596/j.cnki.1001-246x.8768

摘要/Abstract

摘要：

对求解多群辐射扩散(MGRD)方程组的大规模离散系统的已有快速算法进行分类, 给出相应的综述。基于近年来所设计的关于并行代数多重网格(AMG)方面的工作, 形成基于物理量的近似Schur补型与基于物理和代数特征的组合型预条件算法和理论框架, 并对这些工作在该框架下的要素进行了刻画。利用上述框架, 设计一种具有基本逼近性和低计算复杂度的近似Schur补型预条件子, 并建立相应的谱等价理论; 对比数值实验表明: 新预条件子具有更好的稳健性和计算效率。最后提出需要进一步解决的若干问题。

关键词: 多群辐射扩散方程组, 特征驱动, 并行代数多重网格法, 预条件子, 近似Schur补

Abstract:

Firstly, a review is given by classifying the existing fast algorithms for solving large-scale discrete linear systems arising from the Multi-Group Radiation Diffusion (MGRD) equations. Secondly, based on our recent work on parallel algebraic multigrid (AMG), two preconditioning algorithms and related theoretical frameworks are developed on a higher level. One is the approximate Schur complement type based on physical quantities and the other is the combined type based on physical and algebraic features, and the relevant components of these works are portrayed within these frameworks. Based on the above framework, a approximate Schur complement preconditioner with fundamental approximation property and low computational complexity is designed, and the corresponding spectral equivalence theory is established. Numerical experiments show that the new preconditioner has better robustness and computational efficiency. Finally, several issues that need to be further addressed are presented.

Key words: multi-group radiation diffusion equations, feature-driven, parallel algebraic multigrid method, preconditioner, approximate Schur complement

中图分类号:

舒适, 岳孝强, 何剑萌, 徐小文, 莫则尧. 多群辐射扩散问题特征驱动的并行AMG法[J]. 计算物理, 2024, 41(1): 87-97.

Shi SHU, Xiaoqiang YUE, Jianmeng HE, Xiaowen XU, Zeyao MO. Feature-driven Parallel Algebraic Multigrid Methods for Multi-group Radiation Diffusion Problems[J]. Chinese Journal of Computational Physics, 2024, 41(1): 87-97.

图/表 4

参考文献 41

1	MIHALAS D , MIHALAS B W . Foundations of radiation hydrodynamics[M]. New York: Oxford University Press, 1984.
2	PORMRANING G C . The equations of radiation hydrodynamics[M]. Oxford: Pergamon Press, 1973.
3	CASTOR J I . Radiation hydrodynamics[M]. New York: Cambridge University Press, 2004.
4	AMESTOY P R , BUTTARI A , L'EXCELLENT J Y , et al. Performance and scalability of the block low-rank multifrontal factorization on multicore architectures[J]. ACM Transactions on Mathematical Software, 2019, 45 (1): 1- 26.
5	SCHENK O , GÄRTNER K , FICHTNER W , et al. PARDISO: A high-performance serial and parallel sparse linear solver in semiconductor device simulation[J]. Future Generation Computer Systems, 2001, 18 (1): 69- 78. DOI
6	LI X S . An overview of SuperLU: Algorithms, implementation, and user interface[J]. ACM Transactions on Mathematical Software, 2005, 31 (3): 302- 325. DOI
7	XIA Jianlin , CHANDRASEKARAN S , GU Ming , et al. Superfast multifrontal method for large structured linear systems of equations[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31 (3): 1382- 1411. DOI
8	DAVIS T A . Algorithm 832: UMFPACK V4.3——an unsymmetric-pattern multifrontal method[J]. ACM Transactions on Mathematical Software, 2004, 30 (2): 196- 199. DOI
9	VAN DER VORST H A . Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1992, 13 (2): 631- 644. DOI
10	HESTENES M R , STIEFEL E . Methods of conjugate gradients for solving linear systems[J]. Journal of Research of the National Bureau of Standards, 1952, 49 (6): 409- 436. DOI
11	SAAD Y , SCHULTZ M H . GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7 (3): 856- 869. DOI
12	PAIGE C C , SAUNDERS M A . Solution of sparse indefinite systems of linear equations[J]. SIAM Journal on Numerical Analysis, 1975, 12 (4): 617- 629. DOI
13	CLEARY A J , FALGOUT R D , HENSON V E , et al. Robustness and scalability of algebraic multigrid[J]. SIAM Journal on Scientific Computing, 2000, 21 (5): 1886- 1908. DOI
14	XU Xiaowen , MO Zeyao . Algebraic interface-based coarsening AMG preconditioner for multi-scale sparse matrices with applications to radiation hydrodynamics computation[J]. Numerical Linear Algebra with Applications, 2017, 24 (2): e2078. DOI
15	徐小文, 莫则尧, 安恒斌. 求解大规模稀疏线性代数方程组序列的自适应AMG预条件策略[J]. 中国科学(信息科学), 2016, 46 (10): 1411- 1420.
16	XU Xiaowen , MO Zeyao , YUE Xiaoqiang , et al. αSetup-AMG: An adaptive-setup-based parallel AMG solver for sequence of sparse linear systems[J]. CCF Transactions on High Performance Computing, 2020, 2 (2): 98- 110. DOI
17	BALDWIN C , BROWN P N , FALGOUT R , et al. Iterative linear solvers in a 2D radiation-hydrodynamics code: methods and performance[J]. Journal of Computational Physics, 1999, 154 (1): 1- 40. DOI
18	肖映雄, 舒适, 张平文, 等. 求解二维三温能量方程的半粗化代数多重网格法[J]. 数值计算与计算机应用, 2003, 24 (4): 293- 303.
19	JIANG Jun , HUANG Yunqing , SHU Shi , et al. Some new discretization and adaptation and multigrid methods for 2-D 3-T diffusion equations[J]. Journal of Computational Physics, 2007, 224 (1): 168- 181. DOI
20	舒适, 岳孝强, 周志阳, 等. SAMR网格上扩散方程有限体格式的逼近性与两层网格算法[J]. 计算物理, 2014, 31 (4): 390- 402.
21	MO Zeyao , SHEN Longjun , WITTUM G . Parallel adaptive multigrid algorithm for 2-d 3-t diffusion equations[J]. International Journal of Computer Mathematics, 2004, 81 (3): 361- 374. DOI
22	RIDER W J , KNOLL D A , OLSON G L . A multigrid Newton-Krylov method for multimaterial equilibrium radiation diffusion[J]. Journal of Computational Physics, 1999, 152 (1): 164- 191. DOI
23	BROWN P N , WOODWARD C S . Preconditioning strategies for fully implicit radiation diffusion with material-energy transfer[J]. SIAM Journal on Scientific Computing, 2001, 23 (2): 499- 516. DOI
24	GLOWINSKI R , TOIVANEN J . A multigrid preconditioner and automatic differentiation for non-equilibrium radiation diffusion problems[J]. Journal of Computational Physics, 2005, 207 (1): 354- 374. DOI
25	MOUSSEAU V A , KNOLL D A , RIDER W J . Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion[J]. Journal of Computational Physics, 2000, 160 (2): 743- 765. DOI
26	MOUSSEAU V A , KNOLL D A . New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion[J]. Journal of Computational Physics, 2003, 190 (1): 42- 51. DOI
27	FENG Tao , AN Hengbin , YU Xijun , et al. On linearization and preconditioning for radiation diffusion coupled to material thermal conduction equations[J]. Journal of Computational Physics, 2013, 236, 28- 40. DOI
28	FENG Tao , YU Xijun , AN Hengbin , et al. The preconditioned Jacobian-free Newton-Krylov methods for nonequilibrium radiation diffusion equations[J]. Journal of Computational and Applied Mathematics, 2014, 255, 60- 73. DOI
29	AN Hengbin , MO Zeyao , XU Xiaowen , et al. Operator-based preconditioning for the 2-D 3-T energy equations in radiation hydrodynamics simulations[J]. Journal of Computational Physics, 2019, 385, 51- 74. DOI
30	HU Qiya , ZHAO Lin . Domain decomposition preconditioners for the system generated by discontinuous Galerkin discretization of 2D-3T heat conduction equations[J]. Communications in Computational Physics, 2017, 22 (4): 1069- 1100. DOI
31	YUE Xiaoqiang , SHU Shi , WANG Junxian , et al. Substructuring preconditioners with a simple coarse space for 2-D 3-T radiation diffusion equations[J]. Communications in Computational Physics, 2018, 23 (2): 540- 560.
32	SHU Shi , LIU Menghuan , XU Xiaowen , et al. Algebraic multigrid block triangular preconditioning for multidimensional three-temperature radiation diffusion equations[J]. Advances in Applied Mathematics and Mechanics, 2021, 13 (5): 1203- 1226. DOI
33	徐小文, 莫则尧, 安恒斌. 求解二维三温辐射扩散方程组的一种代数两层迭代方法[J]. 计算物理, 2009, 26 (1): 1- 8. DOI
34	周志阳, 徐小文, 舒适, 等. 二维三温辐射扩散方程组两层预条件子的自适应求解[J]. 计算物理, 2012, 29 (4): 475- 483. DOI
35	HUANG Silu , YUE Xiaoqiang , XU Xiaowen . αSetup-PCTL: An adaptive setup-based two-level preconditioner for sequence of linear systems of three-temperature energy equations[J]. Communications in Computational Physics, 2022, 32 (5): 1287- 1309. DOI
36	YUE Xiaoqiang , WANG Chunqing , XU Xiaowen , et al. A new relaxed splitting preconditioner for multidimensional multi-group radiation diffusion equations[J]. East Asian Journal on Applied Mathematics, 2022, 12 (1): 163- 184. DOI
37	YUE Xiaoqiang , HE Jianmeng , XU Xiaowen , et al. Two physics-based Schwarz preconditioners for three-temperature radiation diffusion equations in high dimensions[J]. Communications in Computational Physics, 2022, 32 (3): 829- 849. DOI
38	YUE Xiaoqiang ZHANG Shulei , XU Xiaowen , et al. Algebraic multigrid block preconditioning for multi-group radiation diffusion equations[J]. Communications in Computational Physics, 2021, 29 (3): 831- 852. DOI
39	YUE Xiaoqiang , SHU Shi , XU Xiaowen , et al. An adaptive combined preconditioner with applications in radiation diffusion equations[J]. Communications in Computational Physics, 2015, 18 (5): 1313- 1335. DOI
40	HU Xiaozhe , WU Shuhong , WU Xiaohui , et al. Combined preconditioning with applications in reservoir simulation[J]. Multiscale Modeling & Simulation, 2013, 11 (2): 507- 521.
41	XU Xiaowen , YUE Xiaoqiang , MAO Runzhang , et al. JXPAMG: A parallel algebraic multigrid solver for extreme-scale numerical simulations[J]. CCF Transactions on High Performance Computing, 2023, 5 (1): 72- 83. DOI

n	P	NT	NNL	NL	NL-Max	NL-Ave
1 575	B_U	1 644	6 349	16 414	5	2.58
1 575	B_P	1 657	6 412	19 737	200	3.07
6 093	B_U	1 598	8 666	22 475	5	2.55
6 093	B_P	1 616	8 784	28 390	200	3.23
23 961	B_U	1 637	18 768	46 200	5	2.46
23 961	B_P	1 678	19 695	60 923	200	3.09

n	P	NT	NNL	NL	NL-Max	NL-Ave
1 575	B_U	1 644	6 349	16 414	5	2.58
1 575	B_P	1 657	6 412	19 737	200	3.07
6 093	B_U	1 598	8 666	22 475	5	2.55
6 093	B_P	1 616	8 784	28 390	200	3.23
23 961	B_U	1 637	18 768	46 200	5	2.46
23 961	B_P	1 678	19 695	60 923	200	3.09

Mesh	np	BoomerAMG				B₃
Mesh	np	48	144	432	1 296	48	144	432	1 296
$\mathcal{M}$₂	T₃	26	28	28	31	3	3	3	4
$\mathcal{M}$₂	T₄	70	71	71	74	4	4	4	4
$\mathcal{M}$₃	T₅	42	45	46	44	4	4	5	5
$\mathcal{M}$₃	T₆	88	91	93	96	5	5	5	6

np	BoomerAMG					B₃
np	48	144		432	1 296	48	144	432	1 296
$\mathcal{M}$₂	100.0%	66.9%		52.1%	38.2%	100.0%	63.8%	49.3%	36.0%
$\mathcal{M}$₃	100.0%	70.1%		58.4%	47.0%	100.0%	69.1%	56.4%	44.5%

Mesh	np	BoomerAMG				B₃
Mesh	np	48	144	432	1 296	48	144	432	1 296
$\mathcal{M}$₂	T₃	26	28	28	31	3	3	3	4
$\mathcal{M}$₂	T₄	70	71	71	74	4	4	4	4
$\mathcal{M}$₃	T₅	42	45	46	44	4	4	5	5
$\mathcal{M}$₃	T₆	88	91	93	96	5	5	5	6

np	BoomerAMG					B₃
np	48	144		432	1 296	48	144	432	1 296
$\mathcal{M}$₂	100.0%	66.9%		52.1%	38.2%	100.0%	63.8%	49.3%	36.0%
$\mathcal{M}$₃	100.0%	70.1%		58.4%	47.0%	100.0%	69.1%	56.4%	44.5%

np	BoomerAMG				Euclid(1)				B₂
np	352	704	1 408	2 816	352	704	1 408	2 816	352	704	1 408	2 816
M₁	47	49	49	51	3	3	3	3	3	4	3	3
M₂	31	33	34	33	21	21	21	21	4	4	4	4
M₃	29	31	31	30	2	2	3	2	4	4	4	4
M₄	58	59	57	43	4	4	4	4	5	5	5	5
M₅	41	42	41	43	>200	>200	>200	>200	4	4	5	4
M₆	44	43	46	46	>200	>200	>200	>200	5	5	5	5

np	BoomerAMG				Euclid(1)				B₂
np	352	704	1 408	2 816	352	704	1 408	2 816	352	704	1 408	2 816
M₁	18.4	16.1	12.7	9.5	1.8	1.2	0.8	0.5	6.6	5.2	2.3	1.4
M₂	12.2	10.9	8.9	6.6	12.7	7.7	4.8	2.9	9.2	5.4	3.2	1.9
M₃	11.0	9.8	8.1	6.1	1.3	0.8	0.5	0.3	9.1	5.3	3.1	1.8
M₄	20.1	17.6	14.0	10.3	2.6	1.6	0.9	0.6	11.1	6.5	3.8	2.2
M₅	14.3	12.7	10.2	7.7	-	-	-	-	9.2	5.4	4.0	1.9
M₆	16.2	14.2	11.1	8.3	-	-	-	-	11.2	6.6	3.9	2.3