Using the collage method to solve ODEs inverse problems with multiple data sets Original Research Article

We consider inverse problems for ordinary differential equations (ODEs) where multiple noisy observational data sets are available. The inverse problem for a single data set asks us to find an ODE or system thereof that admits the (interpolated) data set as an approximate solution. Such an inverse problem can be cast within the framework of fixed point equations, so that we seek to approximate the data values by a fixed point of an operator related to the ODEs. The collage coding method replaces the minimization of this approximation error by the minimization of the so-called collage distance. The purpose of this paper is to formulate and explore various methods for dealing with the collection of collage distances corresponding to multiple noisy data sets. We present three schemes: averaging the parameter estimates obtained by minimizing each of the separate collage distances, and two approaches arising from goal programming. Numerous examples are presented.