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

许多物理、生物和社会过程被建模为一个或多个具有未知参数的常微分方程（ode）。通常，使用这种ODE有三个基本任务：(1)检查ODE的结构是否允许原则上估计这些参数，(2)如果允许，则对参数进行数值估计，(3)用参数的估计值求解ODE。因此，为这三个任务开发工具是至关重要的。由于它的重要性，人们在开发必要的数学理论、算法和软件工具方面进行了广泛的研究，取得了巨大的进展/成就。从广义上讲，有两种不同的方法：符号和数字，每种方法都有自己的目标、理论、算法和软件工具。粗略地说，符号方法优先考虑正确性而不是效率，而数值方法优先考虑效率而不是正确性。自然地，他们发展了（通常是戏剧性地）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，这是一个极大的需求，也是一个挑战：开发一种新的方法（理论，算法），可以产生既有效又正确的软件工具。在这个项目中，研究人员提出了一种新的方法，有可能满足参数化ode的效率和正确性的挑战。该方法可以用关键词“验证和软方法”来描述。可以尝试用两种方法开发经过验证的（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)使用修正了正确性概念的数值区间方法，例如，指定优先错误界限。这允许使用近似算术，提供效率，但这只适用于非奇异ode。对于奇异问题，存在一个隐式的“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克服了这一限制：非正式地，允许奇点边缘的输入具有不确定的输出。由此产生的问题的软公式允许人们利用和结合符号和数值方法的优势，从而产生正确（在修正意义上）和实用（高效）的算法。研究人员的初步研究表明，经过验证的软方法非常有前途。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.