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

真实的世界系统的数学建模在所有研究领域都是普遍存在的。在物理科学、生物科学、工程学、医学和金融学的应用中，所采用的数学模型通常涉及微分方程系统，这些模型的“精确”解充其量只能近似。在我们的研究计划中，我们假设正在研究的数学模型是适定的，计算（或数值）任务是近似解和解的关键属性。在过去，我一直参与，与我的学生和同事，在有效的数值方法的发展，用于逼近常微分方程（ODE）的系统的解决方案和软件工具的开发，以可视化的解决方案的重要属性。我们已经开发了有效的工具来调查解决方案的一些属性，包括估计和量化条件的工具，量化稳定性的工具和量化解决方案的灵敏度（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).