Solving Sparse Polynomial Systems by Polyhedral Homotopies

小行星9804846 稀疏多项式系统频繁地出现在科学和工程中的各种各样的应用中，其中所有它们的知识 孤立的解决方案是必要的。实践证明，同伦延拓算法在求多项式系统的孤立零点方面是高效、可靠和精确的。 此外，该方法本质上是并行的，因此是分布式处理架构的自然候选者。 本文的研究方向是沿着几条利用新出现的多面体同伦求解多项式系统的途径。该项目的实质是开发高效的计算算法，其中包括对该方法的理论研究，对算法的稳定性和效率的调查，以及对广泛问题的基准测试。该项目的最终目标是产生一个高质量的软件包，可用于解决各种先进架构上的大规模真实的世界应用程序。

9804846 Li Sparse polynomial systems occur frequently in a wide variety of applications in sciences and engineering where the knowledge of all their isolated solutions is essential. Practical evidence has shown that homotopy continuation algorithms are efficient, reliable and accurate in finding all the isolated zeros of polynomial systems. Moreover, the method is inherently parallel and is therefore a natural candidate for distributed processing architectures. The proposed research is to be directed along several avenues in solving polynomial systems by the newly emerged polyhedral homotopies. The essence of the project is to develop efficient computational algorithms, which involves a theoretical study of the method, investigation of the stability and efficiency of the algorithm, and benchmark testing for a wide range of problems. The ultimate goal of the project is to produce a high-quality software package that can be used to solve large-scale real world applications on a wide variety of advanced architectures.