The development of reliable numerical software for the investigation of systems of differential equations

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 we assume that the mathematical model being investigated is well posed and the computational task is to approximate the solution and/or key properties of the mathematical model. 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 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 solution to the task that is specified. To do this we 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. An example of our research is our development of the most reliable and accurate method available to determine the values of unknown parameters of the model that 'best fits' some observed behaviour of the 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 arises in several application areas and our approach provides an efficient technique that can be applied to problems that involve several parameters and a large number of (possibly noisy) observations.