H. Hafsteinsson, Parallel Sparse Cholesky Factorization, PhD Thesis, Department of Computer Science, Cornell University, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for large problems. It is also worth mentioning that for symmetric problems preordered with Minimum Degree, a small but worthwhile reduction in the time for the set up phase was achieved by applying equivalent postorderings to the computed inverse factors, using techniques described in [21], 26] 27] In the interest of brevity, these results will be reported elsewhere. Finally, we performed some experiments using the METIS graph partitioning package [22] Graph partitioning affords a natural way to parallelize sparse matrix computations, and should be a valuable tool for ....

H. Hafsteinsson, Parallel Sparse Cholesky Factorization, PhD Thesis, Department of Computer Science, Cornell University, 1988.

....separator. 3.1 Construction of the Simplicial Tree We find the simplicial tree representation of G by first obtaining a clique tree of G 0 , and then converting it to a clique tree. We can find G 0 by using the PEO we found for G 0 to calculate the fill it induces on G. Hafsteinsson s [Ha 88] algorithm does this in O(log n) time with O(n 2 ) processors. Next we get a clique tree representation of G 0 in O(log n) time using O(jE [ F j) O(n 2 ) processors. Folklore says we can convert Klein s elimination tree into a clique tree within these bounds. Then check if the ....

Hafsteinsson, H. Parallel Sparse Cholesky Factorization, PhD Thesis, Cornell University (1988).

....which minimize the height of the elimination tree, which is important in the context of parallel sparse elimination. Previous work in this field was targeted at structurally symmetric problems and attempted to restructure the elimination tree so as to reduce its height as much as possible; see [23], 26] 30] and [31] The techniques proposed in these papers provide as a side effect a decrease in the overlap between closures of the graph vertices. Note that the problem of finding orderings which result in elimination trees of minimum height for general graphs is NP hard; see [35] 0 B B B ....

....for large problems. It is also worth mentioning that for symmetric problems preordered with Minimum Degree, a small but worthwhile reduction in the time for the set up phase was achieved by applying equivalent postorderings to the computed inverse factors, using techniques described in [23], 29] 30] In the interest of brevity, we do not report these results here. Finally, we performed some experiments using the METIS graph partitioning package [24] Graph partitioning affords a natural way to parallelize sparse matrix computations, and should be a valuable tool for parallelizing ....

H. Hafsteinsson, Parallel Sparse Cholesky Factorization, PhD Thesis, Department of Computer Science, Cornell University, 1988.

....The problem of finding a minimum height elimination tree is motivated from solving a large sparse symmetric positive definite linear system of the form Ax = b using parallel Cholesky factorization where A is an nn matrix. Various algorithms for doing this exist (Liu [9] and Hafsteinsson [4]) They all have in common that their speed depends on the height of the elimination tree of A. However it is known that this problem is NP hard (Pothen [12] The other important parameter for Cholesky factorization is the amount of fill in the Cholesky factor L. It is also known that minimizing ....

....e f T b d c f a e c f b e a d c f d a e b c f b d a e Fig. 4. The height of the elimination tree is now reduced from 4 to 2. 6. Tree Rotations and Chain Reorderings. We will now study two algorithms for reducing the height of an elimination tree by respectively Liu [8] and Hafsteinsson [4] and see how they compare with the Minimal Cutset al..gorithm. Liu s algorithm performs local rotations in the elimination tree that do not increase the number of fill edges. The algorithm is based upon the following idea: Let x be a node such that its ancestors in T induce a clique in G and let ....

H. Hafsteinsson, Parallel Sparse Cholesky Factorization, PhD thesis, Cornell University, 1988.

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