The main result of this paper gives a numerically efficient method to bound the error that is made when approximating the output of a nonlinear problem depending on an unknown parameter (described by a probability distribution). The class of nonlinear problems under consideration includes high-dimensional nonlinear problems with a nonlinear output function. A goal-oriented probabilistic bound is computed by considering two phases. An offline phase dedicated to the computation of a reduced model during which the full nonlinear problem needs to be solved only a small number of times. The second phase is an online phase which approximates the output. This approach is applied to a toy model and to a nonlinear partial differential equation, more precisely the Burgers equation with unknown initial condition given by two probabilistic parameters. The savings in computational cost are evaluated and presented.
Accepté le :
DOI : 10.1051/m2an/2018003
Mots-clés : Goal-oriented, probabilistic error estimation, nonlinear problems, uncertainty quantification
@article{M2AN_2018__52_2_705_0,
author = {Janon, Alexandre and Nodet, Ma\"elle and Prieur, Christophe and Prieur, Cl\'ementine},
title = {Goal-oriented error estimation for parameter-dependent nonlinear problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {705--728},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018003},
zbl = {1401.49040},
mrnumber = {3834440},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018003/}
}
TY - JOUR AU - Janon, Alexandre AU - Nodet, Maëlle AU - Prieur, Christophe AU - Prieur, Clémentine TI - Goal-oriented error estimation for parameter-dependent nonlinear problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 705 EP - 728 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018003/ DO - 10.1051/m2an/2018003 LA - en ID - M2AN_2018__52_2_705_0 ER -
%0 Journal Article %A Janon, Alexandre %A Nodet, Maëlle %A Prieur, Christophe %A Prieur, Clémentine %T Goal-oriented error estimation for parameter-dependent nonlinear problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 705-728 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018003/ %R 10.1051/m2an/2018003 %G en %F M2AN_2018__52_2_705_0
Janon, Alexandre; Nodet, Maëlle; Prieur, Christophe; Prieur, Clémentine. Goal-oriented error estimation for parameter-dependent nonlinear problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 705-728. doi : 10.1051/m2an/2018003. http://archive.numdam.org/articles/10.1051/m2an/2018003/
[1] , , and , An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. | DOI | MR | Zbl
[2] and , An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. (2001) 10 1–102. | DOI | MR | Zbl
[3] , , and , Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput. Phys. 224 (2007) 880–896. | DOI | MR | Zbl
[4] and , Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737–2764. | DOI | MR | Zbl
[5] and , The ROMES method for statistical modeling of reduced-order-model error. SIAM/ASA J. Uncertain. Quantif. 3 (2015) 116–145. | DOI | MR | Zbl
[6] and , Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). | DOI | MR | Zbl
[7] and , Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20 (2008) 034103. | DOI | Zbl
[8] , and , Certified reduced-basis solutions of viscous Burgers equations parametrized by initial and boundary values. ESAIM: M2AN 47 (2013) 317–348. | DOI | Numdam | MR | Zbl
[9] , and , Goal-oriented error estimation for the reduced basis method, with application to sensitivity analysis. J. Sci. Comput. 68 (2016) 21–41. | DOI | MR | Zbl
[10] , , and , Global sensitivity analysis for the boundary control of an open channel. Math. Control Signals Syst. 28 (2016) 1–27. | DOI | MR | Zbl
[11] , Design and Analysis of Simulation Experiments. Springer Publishing Company, Inc. (2007). | MR | Zbl
[12] , Simulation experiments in practice: statistical design and regression analysis. J. Simul. 2 (2008) 19–27. | DOI
[13] , , , and , A multigrid solver for two-dimensional stochastic diffusion equations. Comput. Methods Appl. Mech. Eng. 192 (2003) 4723–4744. | DOI | MR | Zbl
[14] , and , A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear Partial Differential Equations and their Applications Collège de France Seminar Volume XIV, edited by and . Vol. 31 of Studies in Mathematics and Its Applications. Elsevier (2002) 533–569. | DOI | MR | Zbl
[15] , Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26 (1981) 17–31. | DOI | MR | Zbl
[16] , and , Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling. Springer (2005) 1523–1558.
[17] , and , Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo 46 (2009) 157–185. | DOI | MR | Zbl
[18] , Low-Rank Tensor Methods for Model Order Reduction. To appear in: Handbook of Uncertainty Quantification (2016) 1–26. DOI: | DOI | MR
[19] , and , The Design and Analysis of Computer Experiments. Springer-Verlag, New York (2003) 283. | MR | Zbl
[20] and , Nonlinear Hilbert adjoints: properties and applications to Hankel singular value analysis. Nonlinear Anal. 51 (2002) 883–901. | DOI | MR | Zbl
[21] , and , Interpolation of spatial data – a stochastic or a deterministic problem? Eur. J. Appl. Math. 24 (2013) 601–629. | DOI | MR | Zbl
[22] , Turbulence and the dynamics of coherent structures. Part I & II. Quart. Appl. Math. 45 (1987) 561–590. | DOI | MR | Zbl
[23] and , Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26 (2004) 395–410. | DOI | MR | Zbl
[24] , and , Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Math. Acad. Sci. Paris 337 (2003) 619–624. | DOI | MR | Zbl
[25] , Proper Orthogonal Decomposition and Singular Value Decomposition. Spezialforschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht. Nr. 153, Graz (1999).
[26] and , Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (2002) 2323–2330. | DOI
[27] and , A space–time variational approach to hydrodynamic stability theory, in Vol. 469 of Proc. R. Soc. A. The Royal Society (2013) 20130036. | DOI | MR | Zbl
[28] , and , A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation. Math. Model. Methods Appl. Sci. 24 (2014) 1903–1935. | DOI | MR | Zbl
[29] and , Model error estimation employing an ensemble data assimilation approach. Mon. Weather Rev. 134 (2006) 1337–1354. | DOI
Cité par Sources :
