A Homotopy Method for Parameter Estimation of Nonlinear Differential Equations with Multiple Optima

A numerical method for estimating multiple parameter values of nonlinear systems arising from biology is presented. The uncertain parameters are modeled as random variables. Then the solutions are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. Homotopy continuation method is employed to solve the resulting polynomial system, and more importantly, to compute the multiple optimal parameter values. Several numerical examples, from a single equation to problems with relatively complicated forms of governing equations, are used to demonstrate the robustness and effectiveness of this numerical method.