J Sci Comput (2018) 74:1314–1324
A Homotopy Method for Parameter Estimation of
Nonlinear Differential Equations with Multiple Optima
Received: 23 March 2017 / Revised: 19 June 2017 / Accepted: 25 July 2017 /
Published online: 19 August 2017
© Springer Science+Business Media, LLC 2017
Abstract 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.
Keywords Biological systems · Parameter estimation · Homotopy continuation method
The biological systems are often described by nonlinear ordinary differential equations
(ODEs) or partial differential equations (PDEs) with uncertain and unknown parameters
[5,9,10,15,18,29–31,35]. The estimations of these unknown parameters are required for
accurate descriptions of the biological processes. Biologically speaking, how to treat the
uncertainty involved in these unknown parameters reﬂects measurement error, non-stringent
experimental design, the ﬂexibility of the processes themselves, etc. Mathematically speak-
ing, the parameter estimation also involves some advanced computational and statistical
methods in optimization problems. Different methodologies have been developed to estimate
parameters such as the Bayesian framework [3,19,20], the combination of the polynomial
chaos theory and the Extended Kalman Filter theory , and the inverse problem theory .
The polynomial chaos approach has been shown to be an efﬁcient method for quantifying
the effects of such uncertainties on nonlinear systems of differential equations [22,32–34].
Department of Mathematics, The Penn State University, University Park, PA 16802, USA