Cluster Newton Method for Sampling Multiple Solutions of Underdetermined Inverse Problems: Application to a Parameter Identification Problem in Pharmacokinetics

A new algorithm is proposed for simultaneously finding multiple solutions of an underdetermined inverse problem. The algorithm was developed for an ODE parameter identification problem in pharmacokinetics for which multiple solutions are of interest. The algorithm proceeds by computing a cluster of solutions simultaneously, and is more efficient than algorithms that compute multiple solutions one-by-one because it fits the Jacobian in a collective way using a least squares approach. It is demonstrated numerically that the algorithm finds accurate solutions that are suitably distributed, guided by a priori information on which part of the solution set is of interest, and that it does so much more efficiently than a baseline Levenberg--Marquardt method that computes solutions one-by-one. It is also demonstrated that the algorithm benefits from improved robustness due to an inherent smoothing provided by the least-squares fitting.