The Development of Reliable Numerical Software for the Investigation of Systems of Differential Equations

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Mathematical modeling of real world systems is pervasive in all areas of research. In applications in physical sciences, biological sciences, engineering, medicine and finance, the mathematical models that are employed often involve systems of differential equations and the 'exact' solution of these models can, at best, only be approximated. In our research program we assume that the mathematical model being investigated is well posed and the computational (or numerical) task is to approximate the solution and key properties of the solution. In the past I have been involved, with my students and colleagues, in the development of effective numerical methods for approximating the solutions of systems of ordinary differential equations (ODEs) and the development of software tools to visualize important properties of the solution. Some of the properties of the solution, that we have developed effective tools to investigate, include tools to estimate and quantify the conditioning, tools to quantify the stability and tools to quantify the sensitivity of the solution (to changes in the parameters that define the model). In this proposed research program we focus on developing effective software tools that are easy to use and are much more reliable and robust than alternative tools that are currently employed by practitioners who are developing and using realistic mathematical models in their research. Our approach is to develop and implement tools that adaptively determine the most effective way to deliver an accurate approximate solution to the task that is specified. To do this we must reliably estimate and control the contribution of both the truncation errors and the roundoff errors that inevitably arise when an approximate solution to a task is determined. The software tools we have developed over the last few years do not accurately monitor and control all the errors that arise during a computation and one of the most significant contributions of our research program will be to design and implement an improved version of the set of tools we have already implemented, that are better able to estimate reliably and control all the errors that arise when the task is 'solved' to an accuracy specified by the user. A major goal of our research is to develop the most reliable and accurate software to determine the values of unknown parameters of a model that 'best fits' some observed behaviour of the underlying model. That is, for mathematical models that depend on a vector of unknown constants p, the associated task is to determine the optimum choice for p. This 'inverse' problem has been well-studied in several application areas and our approach will provide an efficient technique that can be applied to problems that involve several parameters and a large number of observed data points.