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.

在拟议的研究中，我们计划分析和提高数值软件的可靠性，鲁棒性和实用性，以近似微分方程的解决方案。我们将分析和开发有效的工具，以可视化和验证近似解决方案，并向从业人员在各种不同的应用领域中开发数学模型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。我们的贡献，不会是开发新的或更好的数学模型，而是我们将提供一种机制，可以帮助从业者理解和调整其模型的参数，以更好地适应他们正在建模的系统的观察到的行为。 我们将重点关注的数学模型的类型包括初始值问题（IVP），延迟微分方程（DDE），边界价值问题（BVP），微分代数方程（DAE）和Volterra Integro Integro微分方程（VIDE）。对于每个类别的问题，我们计划开发有效的过程，这些过程可用于误差估计，参数拟合和灵敏度分析，即使确切的解决方案在有限数量的点上不连续，它们也是有效的。 在我们的研究中，我们将确定并量化效率和可靠性之间不可避免地出现的关键权衡。在可能的情况下，我们将开发软件，该软件允许用户从可用错误控制策略的小菜单中进行选择。产生近似解决方案的更昂贵的策略，其准确性更可能是 在规定的误差公差内。我们将对我们实施的方法进行严格的渐近分析。我们还将对现实问题进行足够的测试，以确定我们的理论分析预测的行为是否反映在实际问题上。