Parameter identification and estimation of dynamical systems from a single trajectory

Parameter identification and estimation from experimental data is an important problem in mathematical model development. Motivated from disease and immunological study, where repeated collection of data from a single subject is impossible and data could be limited, we focus on the parameter identification/estimation of ODE dynamical systems from a single trajectory, on which there are small amount of exactly measured discrete data, particularly considered uniformly spaced in time. This is an inverse problem where we treat the solution map of initial value problem of the ODE system given the parameter as the corresponding forward problem. We put concentration on mathematical establishment of theories on the existence, uniqueness, robustness, uncertainty quantification about the inverse problem, and provide theoretical proofs on some results. Previous study has been completed for such inverse problems of linear systems. We generalize the results to affine dynamical systems, find sufficient and necessary conditions of existence and uniqueness from either full trajectory or discrete data, and then give bounds on maximal allowable uncertainty. There are also results on affine systems that require new proof techniques or never appear in linear systems. We apply the analysis of affine systems into parameter estimation of linear-in-parameters(LIP) systems, which can yield more accurate estimates than those based on finite differences. We then focus on a two-dimensional Lotka-Volterra system as the first example of LIP systems. We sketch diagrams that partially show the continuous dependence of qualitative behavior of inverse problem solution on data. Novel situations also appear for LV systems where data points are compatible with distinct parameter sets corresponding to same or different qualitative types. Finally, we design a numerical algorithm that estimate parameters faster and with less computation time than direct methods, apply to stiff and higher-dimensional linear-in-parameters systems as well as certain nonlinear systems