Constructing Polyhedral Homotopies on Grid-of-Clusters

Polyhedral homotopy continuation is a promising approach to solving large-scale polynomial systems. However, the method faces a computational bottleneck in constructing the homotopy functions because we need to find all mixed cells for polynomial systems. The search for mixed cells can be formulated as a vertex enumeration problem for polyhedra. Among the search methods for solving the problem efficiently, the dynamic enumeration method by Mizutani, Takeda and Kojima appears to have a speed advantage. However, no matter how we improve the efficiency of the search methods, we may not avoid the fact that the search cost rises rapidly as the size of the polynomial system grows. In this paper, we propose a parallel dynamic enumeration that can deal with mixed cell enumeration for large-scale polynomial systems. The introduction of data redistribution technique in the parallel implementation enables us to enhance parallelization efficiency. Numerical results show that the parallel method can enumerate all mixed cells of polynomial systems whose scale exceeds the capabilities of other methods.