Efficiency, Structure and Robustness in Algebraic Computation

这项建议描述了一项关于符号数学计算中基本问题的算法的设计、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和数学这样的商业软件以目前无法企及的规模处理新的应用。它将允许计算机准确地操纵和求解大型和复杂的方程式集合，即使一些量是未知的，并作为变量留在那里。我的研究将集中在三个特定的领域：稀疏多项式的算法，具有稀疏矩阵的符号和精确线性代数，以及近似多项式和矩阵的符号-数值算法。我的项目中的一个共同主线是关注利用稀疏性(数据描述中的许多零或“间隙”)和简洁的表示法。我们将开发更快的方法来解决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大量问题实例的快速实现。 多项式是描述简单但基本的数学函数的关键工具。发现稀疏表示，其中系数为零的项不被表示，是稀疏内插的经典问题。方法已经知道300年了，但仍有很大的改进空间。其他使用稀疏多项式的操作，如因式分解和分解，都停留在什么是实用的，什么是难处理的前沿。我们的目标是将稀疏内插的成本降低到接近最优，并设计处理稀疏多项式的高效算法。 求大型稀疏线性方程组精确解的方法正在成为符号计算的中心工具，许多应用现在都简化为这一点。我们将寻求比任何已知的算法都快的新算法，用于求解系统和通过对角化(史密斯形式)对所有可能的解进行分类。 我们将解决符号-数值问题，允许稀疏内插和线性多项式方程系统中的不精确数据和解。我们将结合现代计算机代数方法与稀疏重构和优化方面的最新进展，为科学计算和计算代数的结合点问题实现更快和可证明稳健的算法。 我将继续我过去十年来的记录，培养非常有才华和高素质的硕士和博士人才，他们将继续担任学术界和工业界的顶级职位。我们的算法进展将在顶级科学场所发表，并在符号代数软件如Maple、Sage和LinBox中实现，并将向所有人开放。

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.