3D-VAR for parameterized partial differential equations: a certified reduced basis approach

Nicole Aretz-Nellesen, Martin A. Grepl (Corresponding author), Karen Veroy

Research output: Contribution to journalArticleAcademicpeer-review

13 Citations (Scopus)


In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parameterized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly, we present a modified formulation that penalizes the experimentally observable misfit in the measurement space. Furthermore, we include a model correction term that allows to obtain an improved state estimate. We begin by discussing the influence of the measurement space on the amplification of noise and prove a necessary and sufficient condition for the identification of a “good” measurement space. We then propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution, and the observable misfit with respect to the true state for real-time and many-query applications. A posteriori bounds are proposed for the error in each of these approximations. Finally, we introduce different approaches for the generation of the reduced basis spaces and the stability-based selection of measurement functionals. The 3D-VAR method and the associated certified reduced basis approximation are tested in a parameter and state estimation problem for a steady-state thermal conduction problem with unknown parameters and unknown Neumann boundary conditions.

Original languageEnglish
Pages (from-to)2369-2400
Number of pages32
JournalAdvances in Computational Mathematics
Issue number5-6
Publication statusPublished - 25 Jul 2019
Externally publishedYes


  • 3D-VAR
  • A posteriori error estimation
  • Model correction
  • Parameter estimation
  • Reduced basis method
  • State estimation
  • Variational data assimilation


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