Abstract
A reduced-order strategy based on the reduced basis (RB) method is developed for the efficient numerical solution of statistical inverse problems governed by PDEs in domains of varying shape. Usual discretization techniques are infeasible in this context, due to the prohibitive cost entailed by the repeated evaluation of PDEs and related output quantities of interest. A suitable reduced-order model is introduced to reduce computational costs and complexity. Furthermore, when dealing with inverse identification of shape features, a reduced shape representation allows to tackle the geometrical complexity. We address both challenges by considering a reduced framework built upon the RB method for parametrized PDEs and a parametric radial basis functions approach for shape representation. We present some results dealing with blood flows modelled by Navier-Stokes equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The common carotid artery (CCA) bifurcates in the lower neck into two branches, the internal and the external carotid arteries (ICA and ECA, respectively). Stenoses, that is the narrowing of the inner portion of an artery, manifest quite often in the ICA.
References
Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Applied Mathematical Sciences, vol. 160. Springer, New York (2005)
Kolachalama, V., Bressloff, N., Nair, P.: Mining data from hemodynamic simulations via Bayesian emulation. Biomed. Eng. Online 6(1), 47 (2007)
Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: A reduced computational and geometrical framework for inverse problems in haemodynamics. Technical report MATHICSE 12.2011. 29(7), 741–776 (2013). http://mathicse.epfl.ch/
Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM: Math. Mod. Numer. Anal. 47(4), 1107–1131 (2013). http://dx.doi.org/10.1051/m2an/2012059
Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32(5), 2523–2542 (2010)
Manzoni, A., Quarteroni, A., Rozza, G.: Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Eng. 28(6–7), 604–625 (2012)
Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1, 3 (2011)
Acknowledgements
This work was partially funded by the European Research Council Advanced Grant “Mathcard, Mathematical Modelling and Simulation of the Cardiovascular System” (Project ERC-2008-AdG 227058), and by the Swiss National Science Foundation (Projects 122136 and 135444).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Manzoni, A., Lassila, T., Quarteroni, A., Rozza, G. (2014). A Reduced-Order Strategy for Solving Inverse Bayesian Shape Identification Problems in Physiological Flows. In: Bock, H., Hoang, X., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes - HPSC 2012. Springer, Cham. https://doi.org/10.1007/978-3-319-09063-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-09063-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09062-7
Online ISBN: 978-3-319-09063-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)