CAREER: CISE-CCF-AF-Algebra: DMS-Algebra: Computational Differential Algebra

职业：CISE-CCF-AF-代数：DMS-代数：计算微分代数

• 批准号: 0952591
• 负责人: Alexey Ovchinnikov
• 金额: $ 56.16万
• 依托单位: CUNY Queens College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0952591&HistoricalAwards=false
• 关键词: CAREER; CISE; CCF; AF; Algebra

Partial differential algebraic equations (PDAEs) have many important applications, such as those in cellular biology, chemistry, mathematical physics, mechanics and dynamical systems, control theory, differential geometry, and analysis. Algorithms on PDAEs have been instrumental in simplifying and solving practical problems but are increasingly taxed as the size of problems increases. Improvements in efficiency will reduce the time for scientists to find properties, and perhaps solutions, to the systems they are working on. The research consists of developing a computational theory for systems of PDAEs, including those with parameters, and to design algorithms that provide efficient descriptions of their solutions. For non-parameterized systems, the PI will first obtain complexity estimates for existing differential elimination algorithms. For systems with parameters, he will study the structure of their differential Galois groups, which are linear groups consisting of solutions to PDAEs themselves, via representation theory. Better understanding of them can then be applied to improve the efficiency of existing algorithms or create new ones. Despite over a century of numerous studies on PDAEs, as pioneered by Holder, Janet, Riquer, Ritt, Kolchin, and recently furthered by Singer, Boulier, and Hubert, among many others, there do not yet exist methods computationally efficient enough to explore and understand the differential algebraic behavior of solutions of these systems. As a result of the proposed research activities, new improved complexity upper bounds will highlight some bottlenecks of existing algorithms, allowing deeper exploration of designs for more efficient algorithms. The research activities will further develop the theory of radical differential ideals which is essential for any substantial progress in theoretical and algorithmic developments. Among the open problems this project will investigate is the Ritt Problem, which if solved, would provide unique, irredundant, and effective representations of radical differential ideals.

偏微分代数方程（PDAE）在细胞生物学、化学、数学物理、力学和动力系统、控制理论、微分几何和分析等领域有着重要的应用。PDAE上的算法在简化和解决实际问题方面发挥了重要作用，但随着问题规模的增加，其负担越来越重。效率的提高将减少科学家的时间来寻找属性，也许解决方案，他们正在研究的系统，该研究包括发展的PDAE系统的计算理论，包括那些参数，并设计算法，提供有效的描述他们的解决方案。对于非参数化系统，PI将首先获得现有差分消除算法的复杂性估计。对于具有参数的系统，他将研究其微分伽罗瓦群的结构，这些群是由PDAE本身的解组成的线性群，通过表示论。对它们的更好理解可以应用于提高现有算法的效率或创建新算法。尽管在过去的世纪的众多研究PDAEs，率先由保持器，珍妮特，Riquer，里特，Kolchin，最近进一步由辛格，Boulier和休伯特，其中许多人，还没有存在的方法计算效率足以探索和理解这些系统的解的微分代数行为。作为所提出的研究活动的结果，新的改进的复杂性上限将突出现有算法的一些瓶颈，允许更深入地探索更有效的算法的设计。研究活动将进一步发展激进的微分理想的理论，这是必不可少的任何实质性进展的理论和算法的发展。在开放的问题，这个项目将调查是里特问题，如果解决，将提供独特的，无冗余的，有效的表示激进的微分理想。