Improving model fitting by inverse problem analysis

Parameter fitting of mathematical models is a difficult problem, especially in biological and medical contexts. High dimensionality and nonlinearity of realistic models combined with general paucity of data makes it difficult to judge whether the model and data are compatible, find reasonable parametrizations, evaluate their accuracy and uniqueness, and quantify the predictive power of the model. Some of these problems can be related to gaps in understanding of the relationship between the trajectories of a model and its parameters. Presented will be an overview of research that attempts to shed light on this relationship via the analysis of the inverse problem for ODE systems. So far, such analysis has provided geometric conditions on trajectories that lead to parameter identifiability, descriptions of regions in data space that correspond to dynamical systems with particular behavior, and estimates of maximal permissible uncertainty on data that retains the qualitative system properties. These results are available for linear, affine, and, most recently, the Lotka-Volterra dynamical systems.