Exploitation Of Sparsity Research Articles

On the efficient computation of the nonlinear full-information maximum-likelihood estimator

This paper examines important elements in calculating the nonlinear full-information maximum-likelihood (NLFIML) estimator which produce substantial reductions (80 percent or more) in computational cost. It examines the (i) choice of optimization algorithm, (ii) method of Hessian approximation, (iii) choice of stopping criterion, and (iv) exploitation of sparsity. We find that the Newton-Raphson algorithm employing an analytically-computed Hessian is computationally much more efficient (up to 75 percent) in this context than its oft-employed competitors, such as DFP. Additional gains (up to 30 percent) result from using a weighted-gradient stopping criterion. Exploitation of matrix sparsity adds further gains.

Hypermatrix solution of large sets of symmetric positive-definite linear equations

Direct methods are described for the computer solution of large sets of symmetric positive-definite linear equations, such as those arising from application of the Matrix Displacement Method. Several important criteria are outlined which should be met by efficient equation solution packages for modern high-speed computers. Well-known existing methods including the band algorithm, the wavefront method, the Cholesky sparse scheme and optimally ordered triangular factorization are reviewed as to their potential fulfillment of these criteria. In particular, the hypermatrix Cholesky approach is presented, as implemented by the authors and their colleagues as a part of the ASKA structural analysis package. The main topics emphasized comprise the exploitation of sparsity within the structural stiffness matrix and optimal ordering of the decomposition, in order to reduce both the time spent in performing the arithmetical operations as well as the required number of data transfers between core and external storage.

Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization

The compensation theorem is applied in conjunction with ordered triangular factorization of the nodal admittance matrix to simulate the effect of changes in the passive elements of the network on the solution of a problem without changing the factorization. The scheme includes network elements with mutual impedances. Compared with impedance matrix methods which are ordinarily used for power system applications, this method, which permits exploitation of matrix sparsity, always requires less computer storage and, with few exceptions, is much faster.