Variable projection methods for an optimized dynamic mode decomposition

Travis Askham, J. Nathan Kutz

Research output: Contribution to journalArticlepeer-review

116 Scopus citations


The dynamic mode decomposition (DMD) has become a leading tool for data-driven modeling of dynamical systems, providing a regression framework for fitting linear dynamical models to time-series measurement data. We present a simple algorithm for computing an optimized version of the DMD for data which may be collected at unevenly spaced sample times. By making use of the variable projection method for nonlinear least squares problems, the algorithm is capable of solving the underlying nonlinear optimization problem efficiently. We explore the performance of the algorithm with some numerical examples for synthetic and real data from dynamical systems and find that the resulting decomposition displays less bias in the presence of noise than standard DMD algorithms. Because of the flexibility of the algorithm, we also present some interesting new options for DMD-based analysis.

Original languageEnglish (US)
Pages (from-to)380-416
Number of pages37
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
StatePublished - 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modeling and Simulation


  • Dynamic mode decomposition
  • Inverse differential equations
  • Inverse linear systems
  • Variable projection algorithm


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