Graph coloring in sparse derivative matrix computation

viii, 83 leaves ; 29 cm.There has been extensive research activities in the last couple of years to efficiently determine large sparse Jacobian matrices. It is now well known that the estimation of Jacobian matrices can be posed as a graph coloring problem. Unidirectional coloring by Coleman and More [9] and bidirectional coloring independently proposed by Hossain and Steihaug [23] and Coleman and Verma [12] are techniques that employ graph theoretic ideas. In this thesis we present heuristic and exact bidirectional coloring techniques. For bidirectional heuristic techniques we have implemented variants of largest first ordering, smallest last ordering, and incidence degree ordering schemes followed by the sequential algorithm to determine the Jacobian matrices. A "good" lower bound given by the maximum number of nonzero entries in any row of the Jacobian matrix is readily obtained in an unidirectional determination. However, in a bidirectional determination no such "good" lower bound is known. A significant goal of this thesis is to ascertain the effectiveness of the existing heuristic techniques in terms of the number of matrix-vector products required to determine the Jacobian matrix. For exact bidirectional techniques we have proposed an integer linear program to solve the bidirectional coloring problem. Part of exact bidirectional coloring results were presented at the "Second International Workshop on Cominatorial Scientific Computing (CSC05), Toulouse, France.

Sparsity Issues in the Computation of Jacobian Matrices

The knowledge of sparsity information plays an important role in e#cient determination of sparse Jacobian matrices. In a recent work, we have proposed sparsity-exploiting substitution techniques to determine Jacobian matrices. In this paper, we take a closer look at the underlying combinatorial problem. We propose a column ordering heuristic to augment the "usable sparsity" in the Jacobian matrix. Furthermore, we present a new elimination technique based on merging of successive columns