A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations |
sep-tree
Figure 3. A separator-based supernodal elimination tree (right) over a quasi-2D subdomain (left). |
---|
subteam
Figure 4. Overlay of the process ranks (in binary) of the owning subteams of each supernode from the elimination tree in Fig. 3 when the tree is assigned to eight processes using a subtree-to-subteam mapping; a `*' is used to denote both 0 and 1, so that ` ' represents processes 0 and 1, ` ' represents processes 2 and 3, and ` ' represents all eight processes. |
---|
Roughly speaking, the analysis in (26) shows that, if processes are used in the multifrontal factorization of our quasi-2D subdomain problems, then we must have in order to maintain constant efficiency as is increased; similarly, if processes are used in the multifrontal triangular solves for a subdomain, then we must have (where we use to denote that the equality holds within logarithmic factors). Since we can simultaneously factor the subdomain matrices, we denote the total number of processes as and set and ; then the subdomain factorizations only require that , while the subdomain solves have the much stronger constraint that . This last constraint should be considered unacceptable, as we have the conflicting requirement that in order to store the factorizations in memory. It is therefore advantageous to consider more scalable alternatives to standard multifrontal triangular solves, even if they require additional computation.
A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations |