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.

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。 在这个拟议的研究计划中，我们专注于开发有效的软件工具，易于使用，并且比目前正在开发和使用现实数学模型的从业者所使用的替代工具更可靠和强大。我们的方法是开发和实施工具，自适应地确定最有效的方式来提供一个准确的近似解决方案，指定的任务。要做到这一点，我们必须可靠地估计和控制的截断误差和roundening错误的贡献，不可避免地出现时，一个近似的解决方案，以确定一个任务。我们在过去几年中开发的软件工具不能准确地监测和控制计算过程中出现的所有错误，我们研究计划的最重要贡献之一将是设计和实现我们已经实现的工具集的改进版本，能够更好地可靠地估计和控制当任务被“解决”到用户指定的准确度时出现的所有错误。我们研究的一个主要目标是开发最可靠和准确的软件来确定模型的未知参数的值，这些参数“最适合”底层模型的一些观察到的行为。也就是说，对于依赖于未知常数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“逆”问题已经在几个应用领域得到了很好的研究，我们的方法将提供一种有效的技术，可以应用于涉及几个参数和大量观测数据点的问题。