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Zbl 1022.65014
Golub, Gene; Pereyra, Victor
Separable nonlinear least squares: The variable projection method and its applications. (English)
[J] Inverse Probl. 19, No.2, R1-R26 (2003). ISSN 0266-5611

Summary: We review 30 years of developments and applications of the variable projection method for solving separable nonlinear least-squares problems. These are problems for which the model function is a linear combination of nonlinear functions. Taking advantage of this special structure, the method of variable projections eliminates the linear variables obtaining a somewhat more complicated function that involves only the nonlinear parameters. This procedure not only reduces the dimension of the parameter space but also results in a better-conditioned problem. The same optimization method applied to the original and reduced problems will always converge faster for the latter.\par We present first a historical account of the basic theoretical work and its various computer implementations, and then report on a variety of applications from electrical engineering, medical and biological imaging, chemistry, robotics, vision, and environmental sciences. An extensive bibliography is included. The method is particularly well suited for solving real and complex exponential model fitting problems, which are pervasive in their applications and are notoriously hard to solve.

MSC 2000:: *65D10 Smoothing
65-03 Historical (numerical analysis)
65-02 Research monographs (numerical analysis)
65C60 Computational problems in statistics
62J02 General nonlinear regression
01A60 Mathematics in the 20th century

Keywords: research survey; historical survey; nonlinear data fitting; variable projection method; separable nonlinear least-squares problems; optimization method; bibliography; exponential model fitting problems

Cited in: Zbl 1166.65345 Zbl 1174.65021 Zbl 1121.65057 Zbl 1104.65060

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