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

North Carolina State University, Raleigh NC

Investigators

Abstract

Many physical, biological, and social processes are modeled as one or more 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 to be estimated in principle, (2) if it does, numerically estimating the parameters, and (3) solving ODEs with the estimated values of the parameters. Thus, it is crucial to develop tools for the three tasks. Due to its importance, there has been extensive research on developing necessary mathematical theories, algorithms and software tools, with tremendous progress/achievements. Broadly, there have been two different approaches: symbolic and numeric, each with its own objective, theory, algorithms, and software tools. Roughly put, the symbolic approaches prioritize correctness over efficiency, while the numeric approaches prioritize efficiency over correctness. Naturally, they developed (often dramatically) different sets of theories and algorithms. Consequently, there are currently two kinds of software tools: one correct but often inefficient, the other efficient but often incorrect. Hence, there is an utmost need and thus a challenge: develop a new approach (theory, algorithms) that can yield software tools that are both efficient and correct. In this project, the investigators propose a novel approach that has a potential to meet the challenges of efficiency and correctness for parametric 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 always produces correct output, but is inefficient. (2) Use a numerical interval approach with modified notion of correctness, e.g., specifying a priori error bounds. This allows the use of approximate arithmetic, providing efficiency, but this is only true for non-singular ODEs. For singular problems, there is an implicit ``Zero Problem'' that does not yield to numerical approximations, and may not even be Turing-computable. The soft approach overcomes this limitation by allowing indeterminacy for certain inputs: informally, inputs on the verge of singularity are allowed to have indeterminate outputs. The resulting soft formulations of the problems allow one to exploit and combine strengths of both symbolic and numeric approaches, resulting in algorithms that are correct (in the modified sense) and practical (efficient). The investigators' preliminary research indicates 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.