Design and Analysis of Algorithms for Structured Nonlinear Problems

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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.