Grant: Discovery Grants Program - Individual

Many scientific and engineering applications are studied in part by constructing computational models and simulations. These models, which often use advanced mathematics to describe real situations of practical interest, then need to be 'solved' in order to find design parameters that create the desired behaviour in the model or simulation. Examples of particular interest to me include computational models of the dynamics of various diseases; some caused by parasites, like malaria, and others caused by hereditary defects, like Stargardt's disease (which is a rare condition causing blindness, starting at about the age of puberty, and affecting about ten thousand Canadians). Other applications I am interested in include problems in computer-aided geometric design, which is used for example in robotics and in manufacturing. The nonlinear differential equations that underlie many of these models give rise to nonlinear algebraic equations whose solutions describe the location and stability (that is, sensitivity or lack thereof to small changes in the data or model equations) of the design situation. This project examines new, numerically stable ways of looking at a wide variety of these kinds of problems, and proposes computationally robust methods for solving many of them. These solutions are intended to be useful in addressing the application problems discussed above.