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

许多物理、生物和社会过程都是由一个或多个参数未知的常微分方程（ODEs）来模拟的，处理这些常微分方程通常有三个基本任务：（1）检查常微分方程的结构是否允许这些参数在原则上被估计，（2）如果允许，数值估计参数，（3）用参数的估计值求解常微分方程。因此，为这三项任务开发工具至关重要。由于其重要性，人们对开发必要的数学理论、算法和软件工具进行了广泛的研究，并取得了巨大的进展/成就。广义上，有两种不同的方法：符号和数字，每一种都有自己的目标，理论，算法和软件工具。粗略地说，符号方法将正确性置于效率之上，而数值方法将效率置于正确性之上。自然，他们开发了（通常是戏剧性的）不同的理论和算法。因此，目前有两种软件工具：一种是正确的，但往往效率低下，另一种是有效的，但往往不正确。因此，有一个最大的需求，因此是一个挑战：开发一种新的方法（理论，算法），可以产生软件工具，既有效又正确。 在这个项目中，研究人员提出了一种新的方法，有可能满足效率和正确性的挑战参数常微分方程。这一方法可用“经过验证的软方法”这一关键短语来描述。人们可以尝试以两种方式开发验证（正确）的算法。(1)使用象征性的方法。它总是产生正确的输出，但效率低下。(2)使用数值区间方法，并修改正确性概念，例如，指定priorerror bounds。这允许使用近似算法，提供效率，但这仅适用于非奇异常微分方程。对于奇异问题，有一个隐含的“零问题”，它不能产生数值近似，甚至可能不是图灵可计算的。软方法通过允许某些输入的不确定性来克服这一限制：非正式地，允许奇点边缘的输入具有不确定的输出。由此产生的软配方的问题alllow一个利用和联合收割机的优势，符号和numericapproaches，导致算法是正确的（在修改的意义上）和实用的（有效的）。调查人员的初步研究表明，经过验证的软方法是非常有前途的。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.