Some Parallel AMG Methods And Their Applications In Radiation Diffusion Problems

Parallel AMG method is one of the most efficient iterative methods for solving the lin-ear systems arising from discretizations of partial differential equations. CPU+GPU hetero-geneous computer has become an important trend in the development of high-performancecomputers, and MPI/OpenMP+CUDA is the main parallel computing platform and pro-gramming model which can fully exploit its effective peak performance. Presently, thereexist many problems worth studying in parallel AMG and its implementation techniquesoriented toward heterogeneous computer systems. The main two contents of this thesis areas follows.In the first part, considering a class of linear algebraic equations with sparse and bandedcoefficient matrices, we improve the OpenMP version of BoomerAMG, which is the defaultAMG solver in HYPRE, by modifying the generation program modules of the parallel in-terpolation and the parallel coarse grid operator. Firstly, we obtain the estimation formulaon the length and offset of the corresponding auxiliary arrays. Then we gain the calculationformula and implement the corresponding program modules of these new auxiliary arrays,which are less memory space occupy than the original. At last, we do some contrastive ex-periments between the improved and the original. Numerical results show that the improvedexpands its solving ability and efficiency.In the second part, we work on several staple UA-AMG methods with regard to thedifferent characteristics of CPU and GPU. The concrete embodies are in the following.Considering serial UA-AMG method under CPU, we get some UA-AMG methodswith lower computational complexity by combining VMB aggregation algorithm with someconventional Cycles and properties of the transfer operators at first. Secondly we imple-ment the corresponding program modules for these algorithms. Lastly, we choose theseUA-AMG as preconditioners for the PCG algorithm to solve a class of two-dimensionalsingle-temperature radiative diffusion problem. Numerical results show that UA-NA-CG-sis the most robust and efficient one, and more efficient than the default AMG-CG solver inHYPRE, AGMG and Cusp. Considering parallel UA-AMG method under GPU, we gain some UA-AMG meth-ods with lower computational complexity by combining MIS(2) aggregation algorithm withweighted Jacobi smoother to start with. Secondly we also implement their correspondingprogram modules. Finally, we select these UA-AMG as preconditioners for PCG algo-rithm to solve the above model problem. Numerical results show that UA-W-CG-p winsthe best robustness and efficiency, and better than SA-AMG-CG in Cusp. Furthermore, westudy these UA-AMG methods mixing with some colored Gauss-Seidel smoothers on thestructured grid, and solve the above model problem using the corresponding PCG methods.Numerical results show that these PCG solvers are also robust, and more efficient.