  Parameter estimation problems of mathematical models can often be formulated
as nonlinear least squares problems. Typically these problems are solved
numerically using iterative methods. The local minimiserobtained using these
iterative methods usually depends on the choice of the initial iterate. Thus,
the estimated parameter and subsequent analyses using it depend on the choice
of the initial iterate. One way to reduce the analysis bias due to the choice
of the initial iterate is to repeat the algorithm from multiple initial
iterates (i.e. use a multi-start method). However, the procedure can be
computationally intensive and is not always used in practice. To overcome this
problem, we propose the Cluster Gauss-Newton (CGN) method, an efficient
algorithm for finding multiple approximate minimisers of nonlinear-least
squares problems. CGN simultaneously solves the nonlinear least squares problem
from multiple initial iterates. Then, CGN iteratively improves the solutions
from these initial iterates similarly to the Gauss-Newton method. However, it
uses a global linear approximation instead of the Jacobian. The global linear
approximations are computed collectively among all the iterates to minimise the
computational cost. We use physiologically based pharmacokinetic (PBPK) models
used in pharmaceutical drug development to demonstrate its use and show that
CGN is computationally more efficient and more robust against local minima
compared to the standard Levenberg-Marquardt method, as well as state-of-the
art multi-start and derivative-free methods.
