The variable projection algorithm in time-resolved spectroscopy, microscopy and mass spectrometry applications

Nonlinear least squares optimization problems in which the parameters can be partitioned into two sets such that optimal estimates of parameters in one set are easy to solve for given fixed values of the parameters in the other set are common in practice. Particularly ubiquitous are data fitting problems in which the model function is a linear combination of nonlinear functions, which may be addressed with the variable projection algorithm due to Golub and Pereyra. In this paper we review variable projection, with special emphasis on its application to matrix data. The generalization of the algorithm to separable problems in which the linear coefficients of the nonlinear functions are subject to constraints is also discussed. Variable projection has been instrumental for model-based data analysis in multi-way spectroscopy, time-resolved microscopy and gas or liquid chromatography mass spectrometry, and we give an overview of applications in these domains, illustrated by brief case studies.