In this paper we investigate the problem of extracting information about chemical reactions involving multiple species from the time history of the concentration of each species. The mathematical model of the kinetic system leads to a system of ordinary differential equations. Our focus is to examine whether the species' concentrations as functions of time are sufficient to determine what chemical reactions, and at what reaction rates, have occurred. We show that within the limitation of our model, there may be many candidate reaction systems that could explain the data. Using the notion of sparsity, we provide a quantitative assessment of the question of distinguishability. We further demonstrate that sparsity enforcing approaches, such as minimizing the ℓ1 or the ℓ0 norms are not reliable. Our conclusion is that additional knowledge about the kinetic system will be necessary to reliably solve this inverse problem.
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Acknowledgments This work was made possible by an REU Supplement from the National Science Foundation. Ben Weitz is grateful for the opportunity to conduct this research and thanks the Institute for Mathematics and its Applications for hosting his visit. The authors thank Dr. Srividhya Jeyaraman from whom the authors learn about the problem investigated in this work, and for helpful conversations on the subject. Dr. Mark Iwen is also acknowledged for useful discussions during the course of this work.
- Inverse problems
- Reaction kinetics