# Adaptive Model Reduction

Adaptive model reduction is the development of numerical solvers able to recognize the dynamics in a complex, heterogoeneous, multiscale problem that require a more accurate modeling in localized regions of the computational domain for the sake of the best trade-off between accuracy and computational efficiency.

Water dynamics after a dam breaking features complexity only in a small region of the basin, as well as electric potential propagation in the heart exhibits a relatively small front where it is important using the most accurate equations. In these cases, an efficient numerical strategy is the coupling of models with different accuracy (and computational costs), following automatically the dynamics of interest.

We have tackled different methods for reducing model complexity (from ad hoc strategies like on/off model, geometrical multiscale, hierarchical model reduction to general approaches like POD) with adaptivity strategies developed for different purposes (e.g., goal oriented approaches). The automatic selection of the reduced model is usually performed through a suitable a posteriori modeling error analysis.

*Keywords :*

hierarchical model (HiMOD) reduction, geometrical multiscale, on/off model reduction, POD, a posteriori modeling error analysis

*AMS classification :*

78M34, 65T40, 65M15, 65M60

**MOX Staff :**

- Simona Perotto
- Stefano Micheletti

*Collaborations :*

- M. Aletti (Inria Paris-Rocquencourt)
- A. Alvarez, P.J. Blanco (Laboratorio Nacional de Computacao Cientifica, Petropolis, Brasil)
- A. Ern (Université Paris-Est, CERMICS, Ecole des Ponts, Paris)
- M. Lupo Pasini, S. Guzzetti, A. Veneziani (Emory University, Atlanta)
- A. Zilio (Laboratoire CAMS, EHESS, Paris)

*Numerical Gallery :*

- Aneurysm in an artery (S. Guzzetti, S. Perotto, A. Veneziani)
- Dam failure (L. Mauri, S. Perotto, A. Veneziani)
- Flow past a cilynder (S. Perotto)
- Gaussian hill (S. Perotto)
- Lateral flow in a rectangular channel (L. Mauri, S. Perotto, A. Veneziani)
- Velocity and pressure for a Navier-Stokes flow in a stenosed artery (M. Aletti, S. Perotto, A. Veneziani)
- Oxygen transport in a blood oxygenator (A. Zilio, S. Perotto)
- Pulsatile flow in an artery (S. Guzzetti, S. Perotto, A. Veneziani)
- Womersley flow in an artery with an aneurysm (S. Guzzetti, S. Perotto, A. Veneziani)
- Womersley flow in an artery with a stenosis (S. Guzzetti, S. Perotto, A. Veneziani)

**Projects :**

- US National Science Foundation project DMS-1419060

“*Hierarchical model reduction techniques for incompressible fluid dynamics and fluid-structure interaction problems*”

(07/01/2014-06/30/2017)

*Awards :*

- Best poster prize at CAE Conference 2014:

“*HiMOD and HiPOD methods for solving direct and inverse problems in internal fluid dynamics*” by Alessandro Barone, Simona Perotto (MOX, Dipartimento di Matematica, Politecnico di Milano), Sofia Guzzetti, Massimiliano Lupo Pasini, Alessandro Veneziani (Department of Mathematics and Computer Science, Emory University), Matteo Aletti (INRIA, Rocquencourt, Paris)

*Minisymposia and Conferences :*

**USNCCM13**“*Model and Solution Reduction Methods for Direct and Inverse Problems in Computational Mechanics*” (organized by S. Perotto, P. Blanco, A. Veneziani)

*Publications :*

*Peer-reviewed Journals*

- E. Miglio, S. Perotto, F. Saleri. Model coupling techniques for free-surface flow problems. Part I. Nonlinear Analysis, 63 (2005), no.5-7, 1885-1896.
- E. Miglio, S. Perotto, F. Saleri. Model coupling techniques for free-surface flow problems. Part II. Nonlinear Analysis, 63 (2005), no.5-7, 1897-1908.
- S. Perotto. Adaptive modeling for free-surface flows. M2AN Math. Model. Numer. Anal., 40 (2006), no.3, 469-499.
- S. Perotto, A. Ern, A. Veneziani. Hierarchical local model reduction for elliptic problems: a domain decomposition approach. Multiscale Model. Simul., 8 (2010), no.4, 1102-1127.
- S. Micheletti, S. Perotto, F. David. Model adaptation enriched with an anisotropic mesh spacing for nonlinear equations: application to environmental and CFD problems. Numer. Math. Theor. Meth. Appl., 6 (2013), no.3, 447-478.
- S. Perotto, A. Veneziani. Coupled model and grid adaptivity in hierarchical reduction of elliptic problems. J. Sci. Comput., 60 (2014), no.3, 505-536.

*Peer-reviewed Proceedings*

- E. Miglio, S. Perotto, F. Saleri. A multiphysics strategy for free surface flows. In Domain Decomposition Methods in Science and Engineering. Series: Lect. Notes Comput. Sci. Eng., Vol. 40, Springer, Berlin, R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Widlund, J. Xu Eds. (2005), 395-402.
- A. Ern, S. Perotto, A. Veneziani. Hierarchical model reduction for advection-diffusion-reaction problems. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, K. Kunisch, G. Of, O. Steinbach Eds. (2008), 703-710.
- L. Mauri, S. Perotto, A. Veneziani. Adaptive geometrical multiscale modeling for hydrodynamic problems. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley, M.V. Tretyakov Eds (2013), 723-730.
- S. Perotto, A. Zilio. Hierarchical model reduction: three different approaches. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley, M.V. Tretyakov Eds (2013), 851-859.
- S. Perotto. Hierarchical model (Hi-Mod) reduction in non-rectilinear domains. To appear in Domain Decomposition Methods in Science and Engineering. Series: Lect. Notes Comput. Sci. Eng., Vol. 98, Springer, J. Erhel, M. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund Eds. (2014).
- M. Aletti, A. Bortolossi, S. Perotto, A. Veneziani. One-dimensional surrogate models for advection-diffusion problems. To appear in Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso Eds. (2014).

*Contributed Books*

- S. Perotto. A survey of hierarchical model (Hi-Mod) reduction methods for elliptic problems. In Numerical Simulations of Coupled Problems in Engineering. Series: Computational Methods in Applied Sciences, Vol. 33, Springer, S.R. Idelsohn Ed. (2014), 217-241.