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

用于研究微分方程组的可靠数值软件的开发

• 批准号: RGPIN-2016-05595
• 负责人: Enright, Wayne
• 金额: $ 1.89万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615794
• 关键词: Development; Reliable; Numerical; Software; Investigation

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.

现实世界系统的数学建模在所有研究领域都普遍存在。在物理科学，生物科学，工程，医学和金融中的应用中，使用的数学模型通常涉及微分方程的系统，并且这些模型的“精确”解决方案充其量只能近似。在我们的研究计划中，我们假设正在研究的数学模型是很好的，并且计算（或数值）任务是近似解决方案的解决方案和关键特性。过去，我与我的学生和同事一起参与了有效的数值方法的开发，以近似于普通微分方程（ODE）的系统解决方案以及开发软件工具以可视化解决方案的重要属性。解决方案的某些属性，我们已经开发了有效的工具来调查，包括估算和量化条件的工具，量化稳定性和工具以量化解决方案的灵敏度（以更改定义模型的参数的变化）。 在此拟议的研究计划中，我们专注于开发易于使用的有效软件工具，并且比目前正在开发和使用现实的数学模型的从业人员使用的替代工具更可靠和健壮。我们的方法是开发和实施工具，这些工具可以适应地确定为指定任务提供准确的近似解决方案的最有效方法。为此，我们必须可靠地估计和控制截断误差的贡献和当确定任务的近似解决方案时不可避免地会出现的循环误差的贡献。我们在过去几年中开发的软件工具无法准确监控和控制计算过程中出现的所有错误，并且我们的研究计划的最重要贡献之一将是设计和实施我们已经实施的一组工具的改进版本，这些工具可以可靠地估算并控制任务“ SOLVED”在“ solved”中出现的所有错误，以确定served'的精确措施。我们研究的主要目标是开发最可靠和准确的软件，以确定“最适合”基础模型的某些观察到的行为的模型的未知参数的值。也就是说，对于取决于未知常数P的向量的数学模型，相关的任务是确定p的最佳选择。这个“反向”问题在几个应用领域都进行了充分研究，我们的方法将提供有效的技术，该技术可以应用于涉及多个参数和大量观察到的数据点的问题。