Graph models and their efficient implementation for sparse Jacobian matrix determination

Large-scale combinatorial scientific computing problems arising in sparse or otherwise structured matrix computation are often expressed using an appropriate graph model, and sometimes the same problem can be given in more than one graph model with similar asymptotic computational complexity. The relative merits of different graph models for the same problem can then be expressed in terms of factors such as generality of the model or ease of computer implementation. We review contemporary graph formulations for large-scale sparse Jacobian matrix determination problem (JMDP) and suggest the pattern graph as a unifying framework for methods that exploit sparsity by matrix compression: row compression, column compression, or a combination of the two. We argue that with the pattern graph, which is structurally close to the underlying matrix, exploitable sparsity and structures in the matrix are unlikely to be lost in “translation” from a matrix problem to a graph problem. From an algorithmic view point the structural correspondence between the matrix and its graph, as we demonstrate in this paper, leads to a better exposition of the compression heuristics and their efficient computer realization. Array-based data structures are suggested as the basic building-blocks for the efficient implementation of relevant graph algorithms. Results from the numerical testing of a subset of implemented algorithms on a suite of test instances drawn from the standard test matrix collection are presented.