Symbolic Factorisation of Sparse Matrix Using Elimination Trees

Many problems in science and engineering require the solving of linear systems of equations. As the problems get larger it becomes increasingly important to exploit the sparsity inherent in many such linear systems. It is well recognized that finding a fill-reducing ordering is crucial to the success of the numerical solution of sparse linear systems. The use of hybrid ordering partitioner is expected to improve significantly the fill-in of the factorized matrices, and also the scalability of the elimination tree obtained by symbolic factorization. The most obvious way to get the required increase in performance would be to use parallel algorithms. For dense symmetric matrices, there are quite a few well-known parallel algorithms that scale well and can be implemented efficiently on parallel computers. On the other hand, there are not many efficient, scalable parallel formulation for the sparse matrix factorization using elimination tree. A well-known sparse matrix ordering scheme PORD (Paderborn Ordering tool) uses the last element to compute the present element computation, so this prevents to do the parallelization part globally for the whole algorithm. PORD spend most of its time in splitting the graph into two parts and coloring it. So this thesis tries to parallelize the most occurring part of the factorization algorithm for getting the better result in symbolic factorization step. The given approach in this thesis might be useful in some of the parallel graph computation which uses sequential ordering step. By some estimates, more than 90% of the eigenvalue problems are real symmetric or complex Hermitian problems. This gives us flexibility to use the ordering step parallely with many other parallel algorithm of matrix computation. Several simple modifications to the minimum local fill-in ordering strategy has also been presented in this thesis such that these strategies exploit readily available information about node adjacencies to improve the fill bounds used to select a node for elimination. This thesis describes two simple modifications to the well known node selection strategy AMMF(Approximate minimum local fill-in) that further improve ordering quality. It is demonstrated that different types of node selection strategies give less amount of number of fronts which gives us better ordering followed by better subsequent factorization complexity.