A13 x1 b1 ax 4 0 a22 a23 5 4 x2 5 4 b2 5 5 at at a33

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: into LDLT 3 2 3 2 D1 I L=4 I 5 D = 4 D2 5 D3 !T !T I 23 13 6 Domain Decomposition Factorization procedure function L D] = dd(A) % dd: block factorization of a matrix A into LDLT for k = 1:2 Dk = Akk Solve Dk !k3 = Ak3 for !k3 end D3 = A33 ; !T D1!13 ; !T D2!23 13 23 Note: i. D1 and D2 can be factored without knowledge of each other or D3, ! 13, and ! 23 ii. The factorizations of D1 and D2 can be done in parallel iii. Design modi cations only change those matrices a ected iv. The procedure can be done recursively v. D3 is called the Schur complement of A { For N N grid problems, George (1973) 1 shows that factorization takes O(N 3) FLOPs { Banded and pro le techniques require O(N 4) FLOPs { Nested dissection has the optimal order of operations J.A. George (1973), Nested dissection of a regular nite element mesh, SIAM J. Numer. Anal., 10, 345-363 1 7 Nested Dissection Two-dimensional nested dissection { Bisect the domain { Number unknowns on the ne level rst { Number unknowns on the interface between two regions next { Number unknowns at the juncture of four regions last Nested ordering of a 4 4 mesh 12 1 3 8 4 X X 2 9 7 5 6 X 8 9 8 X XX 7 8 9 X X 4 2 67 X X 5 1 5 XX 3 6 34 XX X X X X X X X XX X X X XX XXXXX...
View Full Document

This document was uploaded on 03/16/2014 for the course CSCI 6800 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online