Design and Analysis of Algorithms for Structured Nonlinear Problems

结构化非线性问题的算法设计与分析

• 批准号: RGPIN-2015-05481
• 负责人: Corless, Robert
• 金额: $ 1.75万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679626
• 关键词: Design; Analysis; Algorithms; Structured; Nonlinear

Modern science relies very heavily on computer simulation of scenarios that behave according to theoretical models. If the simulations perform as expected in the "testing" phase, then confidence in the theory underlying the model is bolstered. If the simulations manage to predict real-world behavior, especially if it is surprising (or might potentially cost lives or money if unanticipated) then they are of social benefit, not just academic importance.  But carrying out such simulations on a computer often involves solving a potentially large number of nonlinear equations. Doing so reliably and efficiently enough that predictions can be made in real time can be (depending on the subject area) extremely challenging. This project aims at producing new algorithms to make such solutions both more reliable and more efficient.***      The nonlinear algebraic problems that these algorithms are intended to solve occur in very many branches of engineering and science. Examples include fluid-structure interaction, computer-aided geometric design (CAGD) for manufacture, and what is coming to be known as "Algebraic Biology." Fluid-structure interaction is important in wind engineering of tall buildings, large bridges, transmission wires, cables and pipes. CAGD is a principal tool for manufacture of automobiles, aircraft, ships and other structures. The economic impact of those industries in total is in the billions in Canada alone. Algebraic Biology might be even more important in the future; already the design of new organisms is worth billions.***    Specifically, this project seeks to create numerically stable and efficient algorithms for certain structured mathematical modelling problems. We term this general approach the creation of "Domain Specific Algorithms." The main tool to be used for these algorithms is the replacement of the problem with a structured eigenvalue problem to be solved by standard or special-purpose iterative numerical methods. The structured matrices to be used come from my earlier work on companion pencils and linearizations from the Lagrange, Hermite and Birkhoff interpolational bases.**

现代科学在很大程度上依赖于计算机模拟，这些场景根据理论上的模型行为。如果模拟在“测试”阶段按预期执行，则对模型基础理论的信心得到加强。如果模拟设法预测了现实世界的行为，尤其是如果令人惊讶（或者如果意外，可能会损失生命或金钱），那么它们具有社会利益，而不仅仅是学术上的重要性。但是在计算机上进行此类模拟通常涉及解决潜在的大量非线性方程。如此可靠，有效地做到这一点，可以实时做出预测（取决于主题领域）极为挑战。该项目旨在生产新的算法，以使这些解决方案更加可靠，更有效。示例包括流体结构的相互作用，用于制造的计算机辅助几何设计（CAGD）以及即将被称为“代数生物学”的内容。流体结构的相互作用在高大建筑，大桥，变速箱，电缆和管道的风工程中很重要。 CAGD是用于制造汽车，飞机，船只和其他结构的主要工具。这些行业总共的经济影响仅在加拿大数十亿美元。代数生物学在将来可能更重要。新组织的设计已经值得数十亿美元。我们将这种一般方法称为“特定域算法”的创建。这些算法要使用的主要工具是将问题替换为可以通过标准或特殊用途的迭代数值方法来解决的结构化特征值问题。从我较早的伴侣铅笔和拉格朗日，埃尔米特和伯克霍夫插入式基地的线性化的工作中使用的结构化材料。**