Efficiency, Structure and Robustness in Algebraic Computation

本提案描述了符号数学计算中基本问题的算法设计、分析和实现的综合研究计划。我的工作将探索基本的科学问题，并允许像Maple和Mathematica这样的商业软件在现有范围之外的规模上解决新的应用。它将允许计算机精确地处理和解决大量复杂的方程组，甚至当一些未知量作为变量时也是如此。我的研究将集中在三个特定领域：稀疏多项式的算法，稀疏矩阵的符号和精确线性代数，近似多项式和矩阵的符号-数值算法。在我的项目中，一个共同的思路是关注利用稀疏性（数据描述中的许多零或“空白”）和简洁的表示。我们将开发更快的方法来解决具有稀疏数据表示的关键数学问题，探索潜在的计算复杂性，并提供可应用于数据科学，密码学和控制系统中出现的巨大问题实例的快速实现。

This proposal describes a comprehensive program of research into the design, analysis and implementation of algorithms for foundational problems in symbolic mathematical computation. My work will both explore fundamental scientific problems and allow commercial software like Maple and Mathematica to address new applications at a scale beyond their current reach. It will allow computers to manipulate and solve large and complex sets of equations exactly, even when some quantities are unknown and left as variables. My research will centre on three specific areas: algorithms for sparse polynomials, symbolic and exact linear algebra with sparse matrices, and symbolic-numeric algorithms for approximate polynomials and matrices. A common thread in my projects is a focus on exploiting sparsity (many zeros or “gaps” in the descriptions of data) and succinct representations. We will develop faster methods to solve key mathematical problems with sparse data representations, explore the underlying computational complexities, and provide fast implementations which can be applied to huge problem instances arising in data science, cryptography and control systems. Polynomials are a key tool for describing simple but fundamental mathematical functions. Discovering sparse representations, where terms with a coefficient of zero are not represented, is the classical problem of sparse interpolation. Methods have been known for 300 years, but there is still much room for improvement. Other operations with sparse polynomials such as factorization and decomposition live on the frontier of what is practical and what is intractable. Our goal is to reduce costs for sparse interpolation to near optimal, and to design efficient algorithms for manipulating sparse polynomials. Methods for finding exact solutions to huge systems of sparse linear equations are becoming central tools for symbolic computation, and many applications are now reduced to this. We will seek new algorithms that are provably faster than any previously known, for both solving systems and classifying all possible solutions through diagonalization (Smith form). We will address symbolic-numeric problems, allowing for inexact data and solutions in sparse interpolation and linear systems of polynomial equations. We will combine modern computer algebra methods with recent advances in sparse reconstruction and optimization to achieve faster and provably robust algorithms for problems at the nexus of scientific computing and computational algebra. I will continue my record over the past decade of training exceptionally talented and highly qualified personnel at the Master's and PhD level, who go on to top-level positions in academia and industry. Our algorithmic advances will be published in top scientific venues and implemented in symbolic algebra software such as Maple, SAGE and LinBox, and will be openly available for all.