Frontal Solvers for Process Engineering: Local Row Ordering Strategies

The solution of chemical process simulation and optimization problems on today's high performance supercomputers requires algorithms that can take advantage of vector and parallel processing when solving the large, sparse matrices that arise. The frontal method can be highly e cient in this context due to its ability to make use of vectorizable dense matrix kernels on a relatively small frontal matrix in the innermost loop of the computation. However, the ordering of the rows in the coe cient matrix strongly a ects size of the frontal matrix and thus the solution time. If a poor row ordering is used it may make the frontal method uncompetitive with other methods. We describe here a graph theoretical framework for identifying suitable row orderings that speci cally addresses the issue of frontal matrix size. This leads to local, heuristic methods which aim to limit frontal matrix growth in the row and/or column dimensions. Results on a wide range of test problems indicate that improvements in frontal solver performance can often be obtained by the use of a restricted minimum column degree heuristic, which can be viewed as a variation of the minimum degree heuristic used in other contexts. Results also indicate that the natural unit-block structure of process simulation problems provides a quite reasonable ordering.