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

真实世界系统的数学建模在所有研究领域都很普遍。在物理科学、生物科学、工程、医学和金融领域的应用中，所使用的数学模型通常涉及微分方程系统，而这些模型的“精确”解充其量只能是近似值。在我们的研究计划中，我们假设所研究的数学模型是良好的，计算（或数值）任务是近似解和解的关键性质。在过去，我一直与我的学生和同事一起参与开发有效的数值方法来近似常微分方程（ode）系统的解，并开发软件工具来可视化解的重要性质。我们已经开发了一些有效的工具来研究解决方案的一些特性，包括用于估计和量化条件的工具，用于量化稳定性的工具和用于量化解决方案敏感性的工具（对定义模型的参数变化的工具）。

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.