A weighted empirical interpolation method: a priori convergence analysis and applications

Peng Chen; Alfio Quarteroni; Gianluigi Rozza

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 4, page 943-953
  • ISSN: 0764-583X

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 efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667–672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method 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 interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404]. We apply our method to geometric Brownian motion, exponential Karhunen–Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

How to cite

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Chen, Peng, Quarteroni, Alfio, and Rozza, Gianluigi. "A weighted empirical interpolation method: a priori convergence analysis and applications." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 943-953. <http://eudml.org/doc/273279>.

@article{Chen2014,
abstract = {We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667–672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method 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 interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404]. We apply our method to geometric Brownian motion, exponential Karhunen–Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.},
author = {Chen, Peng, Quarteroni, Alfio, Rozza, Gianluigi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {empirical interpolation method; a priori convergence analysis; greedy algorithm; Kolmogorov N-width; geometric brownian motion; Karhunen–Loève expansion; reduced basis method; empirical interpolation methods; geometric Brownian motion; error bounds; Kolmogorov -width; Karhunen-Loève expansion; weighted interpolation; random variables; algorithm},
language = {eng},
number = {4},
pages = {943-953},
publisher = {EDP-Sciences},
title = {A weighted empirical interpolation method: a priori convergence analysis and applications},
url = {http://eudml.org/doc/273279},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Chen, Peng
AU - Quarteroni, Alfio
AU - Rozza, Gianluigi
TI - A weighted empirical interpolation method: a priori convergence analysis and applications
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 943
EP - 953
AB - We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667–672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method 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 interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404]. We apply our method to geometric Brownian motion, exponential Karhunen–Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.
LA - eng
KW - empirical interpolation method; a priori convergence analysis; greedy algorithm; Kolmogorov N-width; geometric brownian motion; Karhunen–Loève expansion; reduced basis method; empirical interpolation methods; geometric Brownian motion; error bounds; Kolmogorov -width; Karhunen-Loève expansion; weighted interpolation; random variables; algorithm
UR - http://eudml.org/doc/273279
ER -

References

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  1. [1] M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Anal. Numér.339 (2004) 667–672. Zbl1061.65118MR2103208
  2. [2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica10 (2001) 1–102. Zbl1105.65349MR2009692
  3. [3] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal.43 (2011) 1457–1472. Zbl1229.65193MR2821591
  4. [4] S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput.32 (2010) 2737–2764. Zbl1217.65169MR2684735
  5. [5] P. Chen and A. Quarteroni, Accurate and efficient evaluation of failure probability for partial differential equations with random input data. Comput. Methods Appl. Mech. Eng.267 (2013) 233–260. Zbl1286.65156MR3127301
  6. [6] P. Chen, A. Quarteroni and G. Rozza, Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput.59 (2014) 187–216. Zbl1301.65007MR3167732
  7. [7] P. Chen, A. Quarteroni and G. Rozza, A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal.51 (2013) 3163–3185. Zbl1288.65007MR3129759
  8. [8] R.A. DeVore and G.G. Lorentz, Constructive Approximation. Springer (1993). Zbl0797.41016MR1261635
  9. [9] M.B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica11 (2002) 145–236. Zbl1105.65350MR2009374
  10. [10] M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575–605. Zbl1142.65078MR2355712
  11. [11] T. Lassila, A. Manzoni and G. Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: M2AN 46 (2012) 1555–1576. Zbl1276.65069MR2996340
  12. [12] T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng.199 (2010) 1583–1592. Zbl1231.76245MR2630164
  13. [13] Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal.8 (2009) 383–404. Zbl1184.65020MR2449115
  14. [14] A. Manzoni, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomedical Eng.28 (2012) 604–625. MR2946552
  15. [15] F. Nobile, R. Tempone and C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal.46 (2008) 2309–2345. Zbl1176.65137MR2421037
  16. [16] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications. Springer (2010). Zbl0567.60055MR1619188
  17. [17] A. Pinkus, N-widths in Approximation Theory. Springer (1985). Zbl0551.41001MR774404
  18. [18] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Industry1 (2011) 1–49. Zbl1273.65148MR2824231
  19. [19] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics. Springer (2007). Zbl1136.65001MR2265914
  20. [20] G. Rozza, Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci.12 (2009) 23–35. MR2489209
  21. [21] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives Comput. Meth. Eng.15 (2008) 229–275. Zbl1304.65251MR2430350
  22. [22] 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). 

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