In this paper we extend the hierarchical model reduction framework based on reduced basis techniques recently introduced in [M. Ohlberger and K. Smetana, SIAM J. Sci. Comput. 36 (2014) A714–A736] for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method [M. Barrault, et al., C. R. Math. Acad. Sci. Paris Series I 339 (2004) 667–672; Y. Maday and O. Mula, A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In Analysis and Numerics of Partial Differential Equations. Vol. 4 of Springer INdAM Series. Springer Milan (2013) 221–235]. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.
Accepté le :
DOI : 10.1051/m2an/2016031
Mots-clés : Dimensional reduction, hierarchical model reduction, reduced basis methods, a posteriori error estimation, nonlinear partial differential equations, empirical interpolation, finite elements
@article{M2AN_2017__51_2_641_0,
author = {Smetana, Kathrin and Ohlberger, Mario},
title = {Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {641--677},
publisher = {EDP-Sciences},
volume = {51},
number = {2},
year = {2017},
doi = {10.1051/m2an/2016031},
mrnumber = {3626414},
zbl = {1362.65129},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016031/}
}
TY - JOUR AU - Smetana, Kathrin AU - Ohlberger, Mario TI - Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 641 EP - 677 VL - 51 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016031/ DO - 10.1051/m2an/2016031 LA - en ID - M2AN_2017__51_2_641_0 ER -
%0 Journal Article %A Smetana, Kathrin %A Ohlberger, Mario %T Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 641-677 %V 51 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016031/ %R 10.1051/m2an/2016031 %G en %F M2AN_2017__51_2_641_0
Smetana, Kathrin; Ohlberger, Mario. Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 641-677. doi : 10.1051/m2an/2016031. http://archive.numdam.org/articles/10.1051/m2an/2016031/
, , and , A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech. 139 (2006) 153–176. | DOI | Zbl
, and , Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 92 (2012) 891–916. | DOI | MR | Zbl
, , and , Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Automat. Control 53 (2008) 2237–2251. | DOI | MR | Zbl
, , and , ‘An empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris Series I 339 (2004) 667–672. | DOI | MR | Zbl
J. Bear, Dynamics of fluids in porous media. Dover Publications, New York (1988). | Zbl
, , and , Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions. Math. Models Methods Appl. Sci. 24 (2014) 901–936. | DOI | MR | Zbl
, and , Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. | DOI | MR | Zbl
, and , Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42 (2004) 1505–1516. | DOI
R.E. Caflisch, Monte Carlo and quasi-Monte Carlo methods. Vol. 7 of Acta Numerica. Cambridge Univ. Press, Cambridge (1998) 1–49. | MR | Zbl
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems. Vol. V of Handbook of numerical analysis. North-Holland, Amsterdam (1997) 487–637. | MR
, and , Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433–2467. | DOI | MR | Zbl
, and , A posteriori error analysis of the reduced basis method for nonaffine parametrized nonlinear PDEs. SIAM J. Numer. Anal. 47 (2009) 2001–2022. | DOI | MR | Zbl
, and , Efficient non linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86 (2011) 155–181. | DOI | MR | Zbl
, , and , The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242 (2013) 623–647. | DOI | MR | Zbl
and , Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comp. 32 (2010) 2737–2764. | DOI | MR | Zbl
and , A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50 (2012) 46–63. | DOI | MR | Zbl
, and , Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch. Comput. Methods Eng. 17 (2010) 327–350. | DOI | MR | Zbl
P.G. Ciarlet, Mathematical Elasticity Volume II: Theory of Plates. Vol. 27 of Studies in Mathematics and Its Applications. Elsevier (1997). | MR
J.E. Dennis, Jr. and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations. Vol. 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996). | MR | Zbl
R. DeVore, Nonlinear approximation. Vol. 7 of Acta Numerica. Cambridge Univ. Press, Cambridge (1998) 51–150. | MR | Zbl
R. DeVore, G. Petrova and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces. Constructive Approximation (2013) 1–12. | MR | Zbl
M. Drohmann, B. Haasdonk and M. Ohlberger, Adaptive reduced basis methods for nonlinear convection–diffusion equations. In Finite Volumes for Complex Applications VI Problems & Perspectives. Springer (2011) 369–377. | MR | Zbl
, and , Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34 (2012) A937–A969. | DOI | MR | Zbl
and , Parameter multi-domain ‘’ empirical interpolation. Int. J. Numer. Methods Engrg. 90 (2012) 412–428. | DOI | MR | Zbl
, and , A posteriori error bounds for the empirical interpolation method. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 575–579. | DOI | MR | Zbl
A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | MR | Zbl
A. Ern, S. Perotto and A. Veneziani, Hierarchical model reduction for advection-diffusion-reaction problems. In Numerical Mathematics and Advanced Applications, edited by Karl Kunisch, Günther Of and Olaf Steinbach. Springer, Berlin, Heidelberg (2008). | MR | Zbl
L.C. Evans, Partial differential equations. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998). | MR | Zbl
M. Evans and T. Swartz, Approximating integrals via Monte Carlo and deterministic methods. Oxford Statistical Science Series. Oxford University Press, Oxford (2000). | MR | Zbl
W. Feller, An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley & Sons, Inc., New York-London-Sydney (1968). | MR | Zbl
, , and , Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Vis. Sci. 2 (1999) 75–83. | DOI | Zbl
, , and , On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engrg. 191 (2001) 561–582. | DOI | MR | Zbl
L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular mathematics. Modeling and simulation of the circulatory system. Springer (2009). | MR | Zbl
, , and , Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575–605. | DOI | Numdam | MR | Zbl
B. Haasdonk and M. Ohlberger, Adaptive basis enrichment for the reduced basis method applied to finite volume schemes. Finite volumes for complex applications V. ISTE, London (2008) 471–478. | MR
, and , A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst. 17 (2011) 423–442. | DOI | MR | Zbl
and , Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discuss. Math. Differ. Incl. Control Optim. 27 (2007) 95–117. | DOI | MR | Zbl
and , Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492–515. | DOI | MR | Zbl
O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and quasilinear elliptic equations. Vol. 46 of Mathematics in Science and Engineering. Academic Press, New York-London, (1968). | MR | Zbl
Y. Maday and O. Mula, A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In Analysis and Numerics of Partial Differential Equations. Vol. 4 of Springer INdAM Series. Springer Milan (2013) 221–235. | MR | Zbl
, , and , A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404. | DOI | MR | Zbl
Y. Maday, O. Mula and G. Turinici, A priori convergence of the generalized empirical interpolation method. In 10th International Conference on Sampling Theory and Applications (SampTA 2013) (2013) 168–171.
, , and , The generalized empirical interpolation method: Stability theory on Hilbert spaces with an application to the Stokes equation. Comput. Methods Appl. Mech. Engrg. 287 (2015) 310–334. | DOI | MR | Zbl
, A finite volume scheme for two-phase immiscible flow in porous media. SIAM J. Numer. Anal. 41 (2003) 1301–1317 (electronic). | DOI | MR | Zbl
M. Ohlberger and K. Smetana, A new hierarchical model reduction-reduced basis technique for advection-diffusion-reaction problems. Proc. of the V International Conference on Adaptive Modeling and Simulation (ADMOS 2011) held in Paris, France, 6-8 June 2011, edited by D. Aubry, P. Díez, B. Tie and N. Parés. Barcelona (2011) 343–354.
and , A dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction. SIAM J. Sci. Comput. 36 (2014) A714–A736. | DOI | MR | Zbl
, A posteriori existence in numerical computations. SIAM J. Numer. Anal. 47 (2009) 2550–2577. | DOI | MR | Zbl
, , and , Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36 (2014) A168–A192. | DOI | MR | Zbl
and , Coupled model and grid adaptivity in hierarchical reduction of elliptic problems. J. Sci. Comput. 60 (2014) 505–536. | DOI | MR | Zbl
, and , Hierarchical local model reduction for elliptic problems: a domain decomposition approach. Multiscale Model. Simul. 8 (2010) 1102–1127. | DOI | MR | Zbl
A. Pinkus, n-widths in approximation theory. Vol. 7. Springer-Verlag, Berlin (1985). | MR | Zbl
, Computer-assisted enclosure methods for elliptic differential equations. Linear Algebra Appl. 324 (2001) 147–187. | DOI | MR | Zbl
and , Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213–231. | DOI | MR | Zbl
K. Smetana, A dimensional reduction approach based on the application of Reduced Basis Methods in the context of Hierarchical Model Reduction. Ph.D. thesis, University of Münster (2013). | Zbl
J.J. Stoker, Water waves. Wiley Classics Library. John Wiley & Sons Inc., New York (1992). | MR | Zbl
G. Talenti, Inequalities in rearrangement-invariant function spaces. Vol. 5 of Nonlinear Analysis, Function Spaces and Applications. Prometheus Publishing House, Prague (1994) 177–230. | MR | Zbl
R. Temam, Navier–Stokes equations. AMS Chelsea Publishing, Providence, RI (2001). | MR | Zbl
T. Tonn, Reduced-Basis Method (RBM) for Non-Affine Elliptic Parametrized PDEs. Ph.D. thesis, Universität Ulm (2011).
K. Urban and B. Wieland, Affine decompositions of parametric stochastic processes for application within reduced basis methods. Proc. of MATHMOD 2012, 7th Vienna International Conference on Mathematical Modelling, held in Vienna, February 14–17, 2012, edited by I. Troch and F. Breitenecker (2012).
and , Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Methods Fluids 47 (2005) 773–788. | DOI | MR | Zbl
and , On a dimensional reduction method. I. The optimal selection of basis functions. Math. Comput. 37 (1981) 31–46. | MR | Zbl
and , On a dimensional reduction method. II. Some approximation-theoretic results. Math. Comput. 37 (1981) 47–68. | MR | Zbl
and , On a dimensional reduction method. III. A posteriori error estimation and an adaptive approach. Math. Comput. 37 (1981) 361–384. | DOI | MR | Zbl
, Implicit partitioning methods for unknown parameter sets. Adv. Comput. Math. 41 (2015) 1159–1186. | DOI | MR | Zbl
, and , A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36 (2014) A311–A338. | DOI | MR | Zbl
Cité par Sources :
