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Parameter estimation problems of mathematical models can often be formulatedas nonlinear least squares problems. Typically these problems are solvednumerically using iterative methods. The local minimiserobtained using theseiterative methods usually depends on the choice of the initial iterate. Thus,the estimated parameter and subsequent analyses using it depend on the choiceof the initial iterate. One way to reduce the analysis bias due to the choiceof the initial iterate is to repeat the algorithm from multiple initialiterates (i.e. use a multi-start method). However, the procedure can becomputationally intensive and is not always used in practice. To overcome thisproblem, we propose the Cluster Gauss-Newton (CGN) method, an efficientalgorithm for finding multiple approximate minimisers of nonlinear-leastsquares problems. CGN simultaneously solves the nonlinear least squares problemfrom multiple initial iterates. Then, CGN iteratively improves the solutionsfrom these initial iterates similarly to the Gauss-Newton method. However, ituses a global linear approximation instead of the Jacobian. The global linearapproximations are computed collectively among all the iterates to minimise thecomputational cost. We use physiologically based pharmacokinetic (PBPK) modelsused in pharmaceutical drug development to demonstrate its use and show thatCGN is computationally more efficient and more robust against local minimacompared to the standard Levenberg-Marquardt method, as well as state-of-theart multi-start and derivative-free methods.

Cluster Gauss-Newton method for finding multiple approximate minimisers  of nonlinear least squares problems with applications to parameter estimation  of pharmacokinetic models