Design and Analysis of Algorithms for Structured Nonlinear Problems

现代科学在很大程度上依赖于根据理论模型运行的情景的计算机模拟。如果模拟在“测试”阶段表现出预期的效果，那么人们对该模型背后的理论的信心就会增强。如果这些模拟能够预测真实世界的行为，特别是如果它令人惊讶(或者如果出乎意料可能会付出生命或金钱的代价)，那么它们就具有社会效益，而不仅仅是学术上的重要性。但在计算机上进行这样的模拟往往涉及求解潜在的大量非线性方程。要做到足够可靠和高效，以便能够实时做出预测，这可能是非常具有挑战性的(取决于主题领域)。这个项目旨在产生新的算法，使这样的解决方案更可靠和更高效。*这些算法旨在解决的非线性代数问题出现在工程和科学的许多分支中。例如流体-结构相互作用、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”。流固耦合是高层建筑、大型桥梁、输电线路、电缆、管道等风工程中的重要问题。CAGD是制造汽车、飞机、船舶和其他结构的主要工具。仅在加拿大，这些行业的经济影响就高达数十亿美元。代数生物学在未来可能会更加重要；新生物体的设计已经价值数十亿美元。*具体地说，这个项目寻求为某些结构化数学建模问题创建数字稳定和高效的算法。我们将这种通用方法称为“领域特定算法”的创建。用于这些算法的主要工具是用标准或特殊用途的迭代数值方法将问题替换为结构化的特征值问题。将使用的结构化矩阵来自于我早期关于伴铅笔的工作以及来自Lagrange、Hermite和Birkhoff插值基的线性化。**

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.**