Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1657–1687 | Cite as

Optimal approximation with exponential sums by a maximum likelihood modification of Prony’s method

  • Ran Zhang
  • Gerlind PlonkaEmail author


We consider a modification of Prony’s method to solve the problem of best approximation of a given data vector by a vector of equidistant samples of an exponential sum in the 2-norm. We survey the derivation of the corresponding non-convex minimization problem that needs to be solved and give its interpretation as a maximum likelihood method. We investigate numerical iteration schemes to solve this problem and give a summary of different numerical approaches. With the help of an explicitly derived Jacobian matrix, we review the Levenberg-Marquardt algorithm which is a regularized Gauss-Newton method and a new iterated gradient method (IGRA). We compare this approach with the iterative quadratic maximum likelihood (IQML). We propose two further iteration schemes based on simultaneous minimization (SIMI) approach. While being derived from a different model, the scheme SIMI-I appears to be equivalent to the Gradient Condition Reweighted Algorithm (GRA) by Osborne and Smyth. The second scheme SIMI-2 is more stable with regard to the choice of the initial vector. For parameter identification, we recommend a pre-filtering method to reduce the noise variance. We show that all considered iteration methods converge in numerical experiments.


Prony method Nonlinear eigenvalue problem Nonconvex optimization Structured matrices Nonlinear structured least squares problem 

Mathematics Subject Classification (2010)

65F15 62J02 15A18 41A30 


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The authors would like to thank the reviewers for many valuable suggestions to improve this manuscript essentially.

Funding information

This work was funded by the NSFC (11571078), by a CSC Scholarship, and by the German Research Foundation within the framework of the RTG 2088.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsShanghai University of Finance and EconomicsShanghaiPeople’s Republic of China
  2. 2.Institute for Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany

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