Optimization and Bayesian Techniques for the parameter estimation of Differential Equations

Differential equation models are widely used in many scientific fields that include engineering, physics and biomedical sciences. The so-called “forward problem” that is the problem of solving differential equations for given parameter values in the differential equation models has been extensively studied by mathematicians, physicists, and engineers. However, the “inverse problem”, the problem of parameter estimation based on the measurements of output variables, has not been well explored using modern optimization and statistical methods. Parameter estimation aims to find the unknown parameters of the model which give the best fit to a set of experimental data. In this way, parameters which cannot be measured directly will be determined in order to ensure the best fit of the model with the experimental results. This will be done by globally minimizing an objective function which measures the quality of the fit. This inverse problem usually considers a cost function to be optimized (such as maximum likelihood). This problem has applications in systems biology, HIV-AIDS study.