دانلود رایگان مقاله ارزش محاسباتی حل کننده مولتی فرونتال در ماشین با حافظه توزیع شده موازی

Abstract

This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the C p−1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O(log(N)p 2 ) for the one dimensional (1D) case, O(N p2 ) for the two dimensional (2D) case, and O(N 4/3 p 2 ) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N. For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 1283 unknowns and linear B-splines with C 0 global continuity, and 15% for a 3D example with 643 unknowns and quartic B-splines with C 3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited.

5. Conclusions

In this paper, we analyzed theoretically and experimentally the performance of multi-frontal direct solvers on distributed memory parallel machines. The theoretical estimates assume sufficiently large number of processors to perform concurrent row subtractions during the local factorizations. We show that the computational cost of the parallel direct solvers grows as p 2 when increasing the global continuity and N is fixed. Additionally, for fixed p, we show that the 2D parallel direct solver delivers linear computational and communication costs. In 3D, the computational and communication costs of the parallel solvers grow in terms of the problem size N as O(N 4/3 ) when executed on distributed memory parallel machines. The obtained computational cost estimates for the distributed memory parallel direct solver are similar to those obtained for shared memory parallel machines [21]. The difference in the derivation is that here it appears an additional term related to the communication cost. The theoretical estimates are verified with numerical experiments.