Collaborative Research: CCF: AF: Medium: Validated Soft Approaches to Parametric ODE Solving

协作研究：CCF：AF：中：经过验证的参数 ODE 求解软方法

• 批准号: 2212462
• 负责人: Chee Yap
• 金额: $ 42.89万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2212462&HistoricalAwards=false
• 关键词: Collaborative; Research; CCF; AF; Medium

Many physical, biological, and social processes are modeled as one ormore ordinary differential equations (ODEs) with unknown parameters.Usually there are three fundamental tasks in working with such ODEs: (1)checking whether the structure of the ODE even allows these parameters tobe estimated in principle, (2) if it does, numerically estimating theparameters, and (3) solving ODEs with the estimated values of theparameters. Thus, it is crucial to develop tools for the three tasks. Dueto its importance, there has been extensive research on developingnecessary mathematical theories, algorithms and software tools, withtremendous progress/achievements. Broadly, there have been two differentapproaches: symbolic and numeric, each with its own objective, theory,algorithms, and software tools. Roughly put, the symbolic approachesprioritize correctness over efficiency, while the numeric approachesprioritize efficiency over correctness. Naturally, they developed (oftendramatically) different sets of theories and algorithms. Consequently,there are currently two kinds of software tools: one correct but ofteninefficient, the other efficient but often incorrect. Hence, there is anutmost need and thus a challenge: develop a new approach (theory,algorithms) that can yield software tools that are both efficient andcorrect. In this project, the investigators propose a novel approach that has apotential to meet the challenges of efficiency and correctness forparametric ODEs. The approach may be described by the key phrase``validated and soft approach''. One may try to develop validated(correct) algorithms in two ways. (1) Use a symbolic approach. It alwaysproduces correct output, but is inefficient. (2) Use a numerical intervalapproach with modified notion of correctness, e.g., specifying a priorierror bounds. This allows the use of approximate arithmetic, providingefficiency, but this is only true for non-singular ODEs. For singularproblems, there is an implicit ``Zero Problem'' that does not yield tonumerical approximations, and may not even be Turing-computable. The softapproach overcomes this limitation by allowing indeterminacy for certaininputs: informally, inputs on the verge of singularity are allowed to haveindeterminate outputs. The resulting soft formulations of the problemsallow one to exploit and combine strengths of both symbolic and numericapproaches, resulting in algorithms that are correct (in the modifiedsense) and practical (efficient). The investigators' preliminary researchindicates that the validated soft approach is quite promising.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多物理，生物学和社会过程被建模为具有未知参数的一个或更多的普通微分方程（ODE）。通常，使用这种ODES有三个基本任务：（1）检查ode的结构甚至允许这些参数允许以原则上的估计，（2）是否可以估算估算的估算值，即估算均值（3）soletvers和3）。因此，为这三个任务开发工具至关重要。 Dueto的重要性是关于开发不必要的数学理论，算法和软件工具，伴随进步/成就的广泛研究。从广义上讲，有两种不同的操作：符号和数字，每种都有自己的目标，理论，算法和软件工具。粗略地说，象征性的方法可以证明对效率的正确性，而数字将效率优于正确性。自然，他们（通常从刺激）开发了不同的理论和算法。因此，目前有两种软件工具：一种正确但经常启发的软件工具，另一种有效但通常不正确。因此，有最大的需求，因此是一个挑战：开发一种新方法（理论，算法），该方法可以产生既有高效又纠正的软件工具。 在这个项目中，研究人员提出了一种新颖的方法，该方法具有应对效率和正确性forparametric ODE的挑战。该方法可以通过关键短语“验证和软方法”来描述。一个人可能会尝试通过两种方式开发经过验证的（正确）算法。 （1）使用符号方法。它总是产生正确的输出，但效率低下。 （2）使用具有修改的正确性概念（例如指定PirorSierRor界限）的数值IntelaClacrach。这允许使用近似算术，提供效率，但这仅适用于非单个ODES。对于奇异问题，存在一个隐含的``零问题''，它不会产生音调近似值，甚至可能无法兼容。通过允许对某些输入的不确定性，软附加量克服了这一限制：非正式的奇异性的输入可以赋予分解输出。产生的问题的软配方可以利用和结合符号和数值的强度，从而产生正确的算法（在修改后）和实用性（有效）。调查人员的初步研究表明，经过验证的软方法是很有希望的。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。