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Variable projection for nonlinear least squares problems

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Abstract

The variable projection algorithm of Golub and Pereyra (SIAM J. Numer. Anal. 10:413–432, 1973) has proven to be quite valuable in the solution of nonlinear least squares problems in which a substantial number of the parameters are linear. Its advantages are efficiency and, more importantly, a better likelihood of finding a global minimizer rather than a local one. The purpose of our work is to provide a more robust implementation of this algorithm, include constraints on the parameters, more clearly identify key ingredients so that improvements can be made, compute the Jacobian matrix more accurately, and make future implementations in other languages easy.

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Notes

  1. More general matrices W can be used to simplify the correlation structure in the errors, but in our work we assume that the matrix is diagonal.

  2. Use of the normal equations gives a less reliable solution, and we do not consider it.

  3. ϵ mach is the gap between 1 and the next larger floating-point number.

  4. For example, in (4), \(\varPhi_{1}(t) = e^{\alpha_{1}t}\) depends only on α 1, and \(\varPhi_{2}(t) = e^{\alpha_{2}t}\) depends only on α 2.

  5. The decomposition diagnoses any redundancy in the basis functions, so the user need not be concerned about it.

  6. This solution is unique if rank(Φ w )=n; otherwise we specify the minimum-norm solution. Again, for a “good” model, Φ w should have rank n, but we allow for rank deficiencies.

  7. This definition differs from that used in [25].

  8. We believe that the true values of the linear parameters should be [7.4334e-08, 2.0094e-11, 8.0493e-07] rather than the [2.0094e-11, 7.4334e-08, 3.1559e-05] reported by Krogh.

  9. We show 2 digits in the table, but the output indicates that to 6-digits, no progress is made during the stalls.

  10. This weighting is widely used (but seldom justified) when the data arises from counts. It seems appropriate when the dominant source of error is conversion of observations to numbers with a fixed number of significant digits, since the standard deviation of the error is then proportional to the observation. Other applications of this weighting are given in [1].

  11. varpro.m and lsqnonlin.m stop if the change in the parameters is less than 10−6 or if the change in the weighted sum of squared residuals is less than 10−6 times (1 plus the old value).

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Acknowledgements

We are grateful to Ronald F. Boisvert, Julianne Chung, David E. Gilsinn, Katharine M. Mullen, and the referee for helpful comments on the manuscript.

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Authors and Affiliations

  1. National Institute of Standards and Technology, Gaithersburg, MD, USA

    Dianne P. O’Leary

  2. Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742, USA

    Dianne P. O’Leary

  3. Applied and Computational Mathematics Division, National Institute of Standards and Technology, NIST Stop 8910, Gaithersburg, MD, 20899-8910, USA

    Bert W. Rust

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  1. Dianne P. O’Leary
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Correspondence to Dianne P. O’Leary.

Additional information

Certain commercial software products are identified in this paper in order to adequately specify the computational procedures. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology nor does it imply that the software products identified are necessarily the best available for the purpose. Part of this work was supported by the National Science Foundation under Grant NSF DMS 1016266.

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O’Leary, D.P., Rust, B.W. Variable projection for nonlinear least squares problems. Comput Optim Appl 54, 579–593 (2013). https://doi.org/10.1007/s10589-012-9492-9

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