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A weighted empirical interpolation method: a prioriconvergence analysis and applications

Published online by Cambridge University Press:  30 June 2014

Peng Chen,
Alfio Quarteroni and
Gianluigi Rozza
Peng Chen
Affiliation:
Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 Lausanne, Switzerland. peng.chen@epfl.ch; cpempire@gmail.com; alfio.quarteroni@epfl.ch
Alfio Quarteroni
Affiliation:
Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 Lausanne, Switzerland. peng.chen@epfl.ch; cpempire@gmail.com; alfio.quarteroni@epfl.chModellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy
Gianluigi Rozza
Affiliation:
SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy; gianluigi.rozza@sissa.it

Abstract

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We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C.Nguyen and A.T. Patera, An empirical interpolation method: application to efficientreduced-basis discretization of partial differential equations. Compt. Rend. Math.Anal. Num. 339 (2004) 667–672] to a weighted empiricalinterpolation method in order to approximate nonlinear parametric functions with weightedparameters, e.g. random variables obeying various probabilitydistributions. A priori convergence analysis is provided for the proposedmethod and the error bound by Kolmogorov N-width is improved from the recent work [Y.Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolationprocedure: the magic points. Commun. Pure Appl. Anal. 8(2009) 383–404]. We apply our method to geometric Brownian motion, exponentialKarhunen–Loève expansion and reduced basis approximation of non-affine stochastic ellipticequations. We demonstrate its improved accuracy and efficiency over the empiricalinterpolation method, as well as sparse grid stochastic collocation method.

Keywords

Empirical interpolation methoda priori convergence analysisgreedy algorithmKolmogorov N-widthgeometric Brownian motionKarhunen–Loève expansionreduced basis method

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014, pp. 943 - 953
Copyright
© EDP Sciences, SMAI, 2014

References

Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Anal. Numér. 339 (2004) 667–672. Google Scholar
Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10 (2001) 1–102. Google Scholar
Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P., Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. Google Scholar
Chaturantabut, S. and Sorensen, D.C., Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737–2764. Google Scholar
Chen, P. and Quarteroni, A., Accurate and efficient evaluation of failure probability for partial differential equations with random input data. Comput. Methods Appl. Mech. Eng. 267 (2013) 233–260. Google Scholar
Chen, P., Quarteroni, A. and Rozza, G., Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59 (2014) 187–216. Google Scholar
Chen, P., Quarteroni, A. and Rozza, G., A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51 (2013) 3163–3185. Google Scholar
R.A. DeVore and G.G. Lorentz, Constructive Approximation. Springer (1993).
Giles, M.B. and Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145–236. Google Scholar
Grepl, M.A., Maday, Y., Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575–605. Google Scholar
Lassila, T., Manzoni, A. and Rozza, G., On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: M2AN 46 (2012) 1555–1576. Google Scholar
Lassila, T. and Rozza, G., Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583–1592. Google Scholar
Maday, Y., Nguyen, N.C., Patera, A.T. and Pau, G.S.H., A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404. Google Scholar
Manzoni, A., Quarteroni, A. and Rozza, G., Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomedical Eng. 28 (2012) 604–625. Google Scholar
Nobile, F., Tempone, R. and Webster, C.G., A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309–2345. Google Scholar
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications. Springer (2010).
A. Pinkus, N-widths in Approximation Theory. Springer (1985).
Quarteroni, A., Rozza, G. and Manzoni, A., Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Industry 1 (2011) 1–49. Google Scholar
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics. Springer (2007).
Rozza, G., Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12 (2009) 23–35. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives Comput. Meth. Eng. 15 (2008) 229–275. Google Scholar
K. Urban and B. Wieland, Affine decompositions of parametric stochastic processes for application within reduced basis methods. In Proc. MATHMOD, 7th Vienna International Conference on Mathematical Modelling (2012).