Separable Nonlinear Least Squares with Multiple Right-Hand Sides

One of the significant problems in modal analysis, that of extracting modal parameters from experimental data, can be written as a separable nonlinear least-squares problem in which a linear combination of nonlinear functions is fit to data in many data sets. The linear parameters are to be specific to each data set but the nonlinear parameters have to minimize the least-squares function for all the data sets. Golub and LeVeque have devised a method for problems like the modal analysis problem where there are multiple data sets. For determining the Jacobian J of the problem, their algorithm essentially computes a decomposition as if there were only one data set and then uses this decomposition for all the data sets. Many general nonlinear least-squares solvers compute an orthogonal decomposition of the Jacobian and after the Golub–LeVeque algorithm has been applied, this decomposition is the most time-consuming portion of the method. This paper proposes using the orthogonal decomposition of a specific derivative matrix for one data set to reduce the work in determining the orthogonal decomposition of the Jacobian for all the data sets. For general minimizers which require JT J, rather than J itself, the paper shows how to compute JT J quickly to take advantage of the structure of the Jacobian in the multiple data set case. For the modal analysis problem, the nonlinear variables in the model also separate, which gives the model additional structure that is exploited to further reduce the construction time. We find that with the economies of speed and storage described below, nonlinear parameter fitting to extract modal parameters from large data sets becomes computationally feasible.