WIAS Preprint No. 2929, (2022)

Reconstruction of flow domain boundaries from velocity data via multi-step optimization of distributed resistance


  • Pártl, Ondřej
  • Wilbrandt, Ulrich
  • Mura, Joaquín
  • Caiazzo, Alfonso
    ORCID: 0000-0002-7125-8645

2020 Mathematics Subject Classification

  • 49M41 76D55 76D07


  • Brinkmann equation, gradient-based optimization, stabilized finite elements, boundary reconstruction




We reconstruct the unknown shape of a flow domain using partially available internal velocity measurements. This inverse problem is motivated by applications in cardiovascular imaging where motion-sensitive protocols, such as phase-contrast MRI, can be used to recover three-dimensional velocity fields inside blood vessels. In this context, the information about the domain shape serves to quantify the severity of pathological conditions, such as vessel obstructions. We consider a flow modeled by a linear Brinkman problem with a fictitious resistance accounting for the presence of additional boundaries. To reconstruct these boundaries, we employ a multi-step gradient-based variational method to compute a resistance that minimizes the difference between the computed flow velocity and the available data. Afterward, we apply different post-processing steps to reconstruct the shape of the internal boundaries. To limit the overall computational cost, we use a stabilized equal-order finite element method. We prove the stability and the well-posedness of the considered optimization problem. We validate our method on three-dimensional examples based on synthetic velocity data and using realistic geometries obtained from cardiovascular imaging.

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