Bi-Directional Determination of Sparse Jacobian Matrices: Approaches and Algorithms

Abstract

Abstract We consider bi-directional determination of sparse Jacobian matrices via row-and-column compression. When the sparsity pattern of the Jacobian matrix is known a priori, the nonzero entries can be restored directly from the product of the Jacobian with suitable “seed” matrices. The underlying combinatorial optimization problem, formulated as a graph coloring problem [T.F. Coleman, A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, SIAM J. Sci. Comput. 19 (4) (1998) 1210–1233, A.S. Hossain, T. Steihaug, Computing a sparse Jacobian matrix by rows and columns, Optimization Methods and Software 10 (1998) 33–48], is computationally hard. A bi-directional coloring of the associated graph defines two seed matrices such that the nonzero unknowns of the Jacobian matrix can be restored directly (i.e., without any arithmetic operations) from the product of the seed matrices with the Jacobian which is obtained via Finite Difference approximations (FD) [A.R. Curtis, M.J.D. Powell, J.K. Reid, On the estimation of sparse Jacobian matrices, J. Inst. Math. Appl. 13 (1974) 117–119] or via Automatic or Algorithmic Differentiation (AD) [A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Number 19 in Frontiers in Appl. Math. SIAM, Philadelphia, Penn., 2000]. The goal is to minimize the number of matrix-vector products. We investigate well-known ordering techniques together with a greedy color assignment that have been successfully applied to one-directional problems [T.F. Coleman, J.J. More, Estimation of sparse Jacobian matrices and graph coloring problems, SIAM J. Numer. Anal. 20 (1) (1983) 187–209], and extend them to bi-directional determination. Furthermore, we propose an integer linear programming (ILP) formulation of the bi-directional coloring and discuss implementation issues. The computational results are encouraging as compared with the existing techniques.