Linear least squares parameter estimation of nonlinear reaction diffusion equations

This paper concerns with the development of a direct parameter identification procedure for a class of nonlinear reaction–diffusion equations. We assume to know the model equations with the exception of a set of constant parameters, such as diffusivity and reaction term parameters. Using the finite element method the original partial differential equation is transformed into a set of ordinary differential equations. A linear least squares method is then applied to estimate the unknown parameters by using normal equations. The measurements errors obtained following this approach are significantly lower than the error obtained by a nonlinear least squares identification procedure. In order to better understand the differences between the two approaches, a sensitivity analysis with respect to initial conditions and mesh dimension is performed. The robustness of the method is tested on noise corrupted data, showing that the linear least square method may be sensitive to perturbations in the data. The procedure is applied to two ecological models describing the dynamics of population growth.