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Combined Parameter and Model Reduction of Cardiovascular Problems by Means of Active Subspaces and POD-Galerkin Methods

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Mathematical and Numerical Modeling of the Cardiovascular System and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 16))

Abstract

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

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Notes

  1. 1.

    In order to simplify the exposition we will report the FE formulation on the deformed domain Ω(μ). However, it should be noted that only the mesh Ω δ of the reference domain Ω is generated, and deformed meshes Ω δ(μ) are obtained through the mapping \(\mathcal {M}(\boldsymbol {\cdot }; {\boldsymbol {\mu }})\).

  2. 2.

    The non-homogeneous right-hand side accounts for boundary conditions via a lifting.

  3. 3.

    In this section we will omit the dependence on μ. It should be understood that f = f(μ), ρ = ρ(μ), etc.

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Acknowledgements

This work was partially supported by the INDAM-GNCS 2017 project “Advanced numerical methods combined with computational reduction techniques for parameterised PDEs and applications”, and by European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” P.I. Gianluigi Rozza.

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Authors and Affiliations

  1. Mathematics Area, mathLab, SISSA, Trieste, Italy

    Marco Tezzele, Francesco Ballarin & Gianluigi Rozza

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  1. Marco Tezzele
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  2. Francesco Ballarin
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  3. Gianluigi Rozza
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Correspondence to Gianluigi Rozza .

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Editors and Affiliations

  1. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Daniele Boffi

  2. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Luca F. Pavarino

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studies, Trieste, Italy

    Gianluigi Rozza

  4. Department of Mathematics, University of Milan, Milano, Italy

    Simone Scacchi

  5. Department of Mathematics, Politecnico di Milano, Milano, Italy

    Christian Vergara

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Tezzele, M., Ballarin, F., Rozza, G. (2018). Combined Parameter and Model Reduction of Cardiovascular Problems by Means of Active Subspaces and POD-Galerkin Methods. In: Boffi, D., Pavarino, L., Rozza, G., Scacchi, S., Vergara, C. (eds) Mathematical and Numerical Modeling of the Cardiovascular System and Applications. SEMA SIMAI Springer Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-96649-6_8

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