tailieunhanh - Acoustic Waves Part 6

Tham khảo tài liệu 'acoustic waves part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Frequency-Domain Numerical Modelling of Visco-Acoustic Waves with Finite-Difference and Finite-Element Discontinuous Galerkin Methods 139 resolution of the linear system is subdivided into 3 main tasks. The first one is an analysis phase or symbolic factorization. Reordering of the matrix coefficients is first performed in order to minimize fill-in. We used the METIS algorithm which is based on a hybrid multilevel nested-dissection and multiple minimum degree algorithm Karypis Kumar 1999 . Then the dependency graph which describes the order in which the matrix can be factored is estimated as well as the memory required to perform the subsequent numerical factorization. The second task is the numerical factorization. The third task is the solution phase performed by forward and backward substitutions. During the solution phase multiple-shot solutions can be computed simultaneously from the LU factors taking advantage of threaded BLAS3 Basic Linear Algebra Subprograms library and are either assembled on the host or kept distributed on the processors for subsequent parallel computations. We performed the factorization and the solutions phases in complex arithmetic single precision. To reduce the condition number of the matrix a row and column scaling is applied in MUMPS before factorization. The sparsity of the matrix and suitable equilibration have made single precision factorization accurate enough so far for the 2D and 3D problems we tackled. If single precision factorization would be considered not accurate enough for very large problems an alternative approach to double precision factorization may be the postprocessing of the solution by a simple and fast iterative refinement performed in double precision Demmel 1997 pages 60-61 and Langou et al. 2006 Kurzak Dongarra 2006 . The main two bottlenecks of sparse direct solver is the time and memory complexity and the limited scalability of the LU decomposition. By complexity is meant the increase of the computational