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”阶段的表现与预期一致，那么对模型基础理论的信心就会得到加强。如果模拟能够预测现实世界的行为，特别是如果它是令人惊讶的（或者如果意外可能会导致生命或金钱的损失），那么它们就具有社会效益，而不仅仅是学术重要性。但是在计算机上进行这样的模拟通常需要解决潜在的大量非线性方程。足够可靠和有效地做到这一点，以便能够实时做出预测（取决于主题领域），这是极具挑战性的。该项目旨在产生新的算法，使这些解决方案更可靠，更有效。这些算法要解决的非线性代数问题出现在工程和科学的许多分支中。例子包括流体结构相互作用，计算机辅助几何设计（CAGD）的制造，以及即将被称为“代数生物学”。流固耦合在高层建筑、大型桥梁、输电线路、电缆和管道的风力工程中具有重要意义。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；新生物的设计已经价值数十亿美元。具体来说，该项目旨在为某些结构化数学建模问题创建数字稳定和有效的算法。我们将这种通用方法称为“领域特定算法”的创建。这些算法使用的主要工具是将问题替换为结构化特征值问题，并用标准或专用迭代数值方法求解。所使用的结构化矩阵来自我早期关于同伴铅笔和线性化的工作，这些线性化来自拉格朗日、赫米特和伯克霍夫插值基

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.**