Efficiency, Structure and Robustness in Algebraic Computation

点击翻译按钮获取中文摘要

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.