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Global preordering for Newton equations using model hierarchy

Authors

  • Kirk A. Abbott,

    1. Dept. of Chemical Engineering and the Engineering Design Research Center, Carnegie Mellon University, Pittsburgh, PA 15213
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  • Benjamin A. Allan,

    Corresponding author
    1. Dept. of Chemical Engineering and the Engineering Design Research Center, Carnegie Mellon University, Pittsburgh, PA 15213
    • Dept. of Chemical Engineering, Carnegie Mellon Univ., Pittsburgh, PA 15213
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  • Arthur W. Westerberg

    1. Dept. of Chemical Engineering and the Engineering Design Research Center, Carnegie Mellon University, Pittsburgh, PA 15213
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Abstract

In solving large simulation problems using Newton's method, a large sparse linear system Ax = b has to be solved. The cost of solving its problems can dominate the overall solution cost of the problem. Current approaches of reducing these costs are reviewed, and then a new algorithm for preordering the sparse matrix A is described that is based on the hierarchical structure provided by an object-oriented description of many recent modeling systems such as ASCEND, gPROMS, DIVA, and Omola. Particularly, rapid preorderings are obtained to support interactive manipulation of models and efficient solutions in automatic process synthesis algorithms, two applications where the preordering cost will be spread over only a few factorizations. With a factorization routine that permits a-priori reorderings (LU1SOL), this algorithm produces order of magnitude reductions in analysis and factoring times as well as in fill and operation count over our previous experience. The time to factor the 50,000 Newton equations for a highly recycled ethylene plant model is of the order of a few seconds on a conventional workstation. Abstracting and applying the fundamental concepts of this algorithm made it possible to improve the performance of the ma28 code significantly. This approach makes solution speeds competitive with and generally more consistent than codes considered the state-of-the-art (ma48 and umfpack 1.0).

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