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

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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.