Robust and reliable software for the investigation of approximate solutions of systems of ordinary differential equations

用于研究常微分方程组近似解的强大而可靠的软件

• 批准号: 8644-2011
• 负责人: Enright, Wayne
• 金额: $ 1.46万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568609
• 关键词: Robust; reliable; software; investigation; approximate

In the proposed research we plan to analyze and improve the reliability, robustness and usefulness of numerical software for approximating the solution of differential equations. We will analyze and develop effective tools for the visualization and verification of approximate solutions and for investigating other issues of particular interest to practitioners developing mathematical models in a variety of different application areas. These areas include medicine, ecology, biology and finance. Our contribution and will not be to the development of new or better mathematical models, but rather we will provide a mechanism that will help practitioners understand and tune the parameters of their model to better fit the observed behaviour of the system they are modeling. The types of mathematical models we will focus on includes initial value problems (IVPs), delay differential equations (DDEs), boundary value problems (BVPs), differential algebraic equations (DAEs) and Volterra integro differential equations (VIDEs). For each class of problems, we plan to develop efficient procedures which can be used for error estimation, parameter fitting and sensitivity analysis that are valid even when the exact solution is discontinuous at a finite number of points. In our research we will identify and quantify the key trade-offs that inevitably arise between efficiency and reliability. Where possible we will develop software which allows the user to select from a small menu of available error control strategies. The more expensive strategies producing approximate solutions whose accuracy is much more likely to be within the prescribed error tolerance. We will present a rigorous asymptotic analysis of the approaches we implement. We will also perform sufficient testing on realistic problems to determine whether the behaviour predicted by our theoretical analysis is reflected on practical problems.

在提出的研究中，我们计划分析和改进用于近似微分方程解的数值软件的可靠性，鲁棒性和实用性。我们将分析和开发有效的工具，用于近似解的可视化和验证，以及研究在各种不同应用领域中开发数学模型的实践者特别感兴趣的其他问题。这些领域包括医学、生态学、生物学和金融学。我们的贡献并不是开发新的或更好的数学模型，而是我们将提供一种机制，帮助实践者理解和调整他们模型的参数，以更好地适应他们正在建模的系统的观察行为。