A numerical method for solving a stochastic inverse problem for parameters

We review recent work (Briedt et al., 2011 and Butler et al., 2012) on a new approach to the formulation and solution of the stochastic inverse parameter determination problem, i.e. determine the random variation of input parameters to a map that matches specified random variation in the output of the map, and then apply the various aspects of this method to the interesting Brusselator model. In this approach, the problem is formulated as an inverse problem for an integral equation using the Law of Total Probability. The solution method employs two steps: (1) we construct a systematic method for approximating set-valued inverse solutions and (2) we construct a computational approach to compute a measure-theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem. In addition to convergence analysis, we carry out an a posteriori error analysis on the computed probability distribution that takes into account all sources of stochastic and deterministic error.

► We give an integral equation formulation of the inverse problem for parameters.
► We describe a method for solving the stochastic inverse problem for parameters.
► We present an a posteriori error estimate for stochastic inverse problems.
► We apply the methods to an interesting model in chemical reactions.