Generalized Reduced Basis Methods and n-width Estimates for the Approximation of the Solution Manifold of Parametric PDEs

Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 1, page 113-135
  • ISSN: 0392-4041

Abstract

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The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold - only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.

How to cite

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Lassila, Toni, et al. "Generalized Reduced Basis Methods and n-width Estimates for the Approximation of the Solution Manifold of Parametric PDEs." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 113-135. <http://eudml.org/doc/294013>.

@article{Lassila2013,
abstract = {The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold - only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.},
author = {Lassila, Toni, Manzoni, Andrea, Quarteroni, Alfio, Rozza, Gianluigi},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {113-135},
publisher = {Unione Matematica Italiana},
title = {Generalized Reduced Basis Methods and n-width Estimates for the Approximation of the Solution Manifold of Parametric PDEs},
url = {http://eudml.org/doc/294013},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Lassila, Toni
AU - Manzoni, Andrea
AU - Quarteroni, Alfio
AU - Rozza, Gianluigi
TI - Generalized Reduced Basis Methods and n-width Estimates for the Approximation of the Solution Manifold of Parametric PDEs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 113
EP - 135
AB - The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold - only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.
LA - eng
UR - http://eudml.org/doc/294013
ER -

References

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  1. ADOMIAN, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers (1994). Zbl0802.65122MR1282283DOI10.1007/978-94-015-8289-6
  2. BABUŠKA, I. - SZABO, B. A. - KATZ, I. N., The p-version of the finite element method, SIAM J. Numer. Anal.18 (3) (1981), 515-545. Zbl0487.65059MR615529DOI10.1137/0718033
  3. BINEV, P. - COHEN, A. - DAHMEN, W. - DEVORE, R. - PETROVA, G. - WOJTASZCZYK, P., Convergence Rates for Greedy Algorithms in Reduced Basis Methods, SIAM J. Math. Anal.43, n. 3 (2011), 1457-1472. Zbl1229.65193MR2821591DOI10.1137/100795772
  4. BUFFA, A. - MADAY, Y. - PATERA, A. T. - PRUD'HOMME, C. - TURINICI, G., A priori convergence of the Greedy algorithm for the parametrized reduced basis method, ESAIM Math. Model. Numer. Anal.46, n. 3 (2012), 595-603. Zbl1272.65084MR2877366DOI10.1051/m2an/2011056
  5. CANUTO, C. - TONN, T. - URBAN, K., A-posteriori error analysis of the reduced basis method for non-affine parameterized nonlinear PDEs, SIAM J. Numer. Anal47, n. 3 (2009), 2001-2022. Zbl1195.65155MR2519592DOI10.1137/080724812
  6. CARLBERG, K. - FARHAT, C., A low-cost, goal-oriented `compact proper orthogonal decomposition' basis for model reduction of static systems, Int. J. Numer. Methods Engrg.86 (3) (2011), 381-402. Zbl1235.74352MR2814387DOI10.1002/nme.3074
  7. COHEN, A. - DEVORE, R. - SCHWAB, C., Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, Preprint, 2010. MR2763359DOI10.1142/S0219530511001728
  8. DEPARIS, S., Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach, SIAM J. Num. Anal.46, n. 4 (2008), 2039-2067. Zbl1177.35148MR2399407DOI10.1137/060674181
  9. EFTANG, J. L. - KNEZEVIC, D. J. - PATERA, A. T., An hp certified reduced basis method for parametrized parabolic partial differential equations, Math. Comput. Modelling Dynam. Systems, 17, n. 4 (2011), 395-422. Zbl1302.65223MR2823470DOI10.1080/13873954.2011.547670
  10. EVANS, J. A. - BAZILEVS, Y. - BABUŠKA, I. - HUGHES, T. J. R., N-widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method, Comput. Methods Appl. Mech. Engrg.198, n. 21-26 (2009), 1726-1741. MR2517942DOI10.1016/j.cma.2009.01.021
  11. GREPL, M. A. - MADAY, Y. - NGUYEN, N. C. - PATERA, A. T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM Math. Modelling Numer. Anal.41 (3) (2007), 575-605. Zbl1142.65078MR2355712DOI10.1051/m2an:2007031
  12. HAY, A. - BORGGAARD, J. - AKHTAR, I. - PELLETIER, D., Reduced-order models for parameter dependent geometries based on shape sensitivity analysis, J. Comp. Phys.229, n. 4 (2010), 1327-1352. Zbl1329.76058MR2576251DOI10.1016/j.jcp.2009.10.033
  13. HAY, A. - BORGGAARD, J. T. - PELLETIER, D., Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech.629 (2009), 41-72. Zbl1181.76045MR2542636DOI10.1017/S0022112009006363
  14. HUYNH, D. B. P. - KNEZEVIC, D. - CHEN, Y. - HESTHAVEN, J. - PATERA, A. T., A natural-norm successive constraint method for inf-sup lower bounds, Comput. Methods Appl. Mech. Engrg.199 (2010), 29-32. Zbl1231.76208MR2654002DOI10.1016/j.cma.2010.02.011
  15. HUYNH, D. B. P. - ROZZA, G. - SEN, S. - PATERA, A. T., A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability costants, C. R. Acad. Sci. Paris. Sér. I Math.345 (2007), 473-478. Zbl1127.65086MR2367928DOI10.1016/j.crma.2007.09.019
  16. ITO, K. - RAVINDRAN, S. S., A reduced order method for simulation and control of fluid flows, J. Comp. Phys.143 (2) (1998). Zbl0936.76031MR1631176DOI10.1006/jcph.1998.5943
  17. LASSILA, T. - MANZONI, A. - ROZZA, G., On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition, ESAIM Math. Model. Numer. Anal.46 (2012), 1555-1576. Zbl1276.65069MR2996340DOI10.1051/m2an/2012016
  18. MADAY, Y., Reduced basis method for the rapid and reliable solution of partial differential equations, Eur. Math. Soc., volume III (Zürich, 2006), 1255-1270. Zbl1100.65079MR2275727
  19. MADAY, Y. - PATERA, A. T. - TURINICI, G., Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations, C. R. Acad. Sci. Paris. Sér. I Math.335 (2002), 289-294. Zbl1009.65066MR1933676DOI10.1016/S1631-073X(02)02466-4
  20. MADAY, Y. - PATERA, A. T. - TURINICI, G., A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations, J. Sci. Comput.17, n. 1-4 (2002), 437-446. Zbl1009.65066MR1910581DOI10.1023/A:1015145924517
  21. MANZONI, A., Reduced models for optimal control, shape optimization and inverse problems in haemodynamics, PhD thesis, N. 5402, École Polytechnique Fédérale de Lausanne (2012). 
  22. MELENK, J. M., On n-widths for elliptic problems, J. Math. Anal. Appl.247 (2000), 272-289. Zbl0963.35047MR1766938DOI10.1006/jmaa.2000.6862
  23. MELKMAN, A. A. - MICCHELLI, C. A., Spline spaces are optimal for L 2 n-width, Illinois J. Math.22, (4) (1978). Zbl0384.41005MR503961
  24. NOOR, A. K., Recent advances in reduction methods for nonlinear problems, Comput. Struct.13, n. 1-3 (1981), 31-44. MR616719DOI10.1016/0045-7949(81)90106-1
  25. PETERSON, J. S., The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Stat. Comput.10 (1989), 777-786. Zbl0672.76034MR1000745DOI10.1137/0910047
  26. PINKUS, A., n-Widths in Approximation Theory, Springer-Verlag, Ergebnisse, 1985. Zbl0551.41001MR774404DOI10.1007/978-3-642-69894-1
  27. QUARTERONI, A. - ROZZA, G. - MANZONI, A., Certified Reduced Basis Approximation for Parametrized PDEs and Applications, J. Math. Ind.3 n. 1 (2011). Zbl1273.65148MR2824231DOI10.1186/2190-5983-1-3
  28. ROZZA, G. - HUYNH, D. B. P. - PATERA, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Engrg.15 (2008), 229-275. Zbl1304.65251MR2430350DOI10.1007/s11831-008-9019-9
  29. TONN, T. - URBAN, K. - VOLKWEIN, S., Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem, Math. Comp. Model. Dyn.17, n. 4 (2011), 355-369. Zbl1302.49045MR2823468DOI10.1080/13873954.2011.547678
  30. VEROY, K. - PATERA, A. T., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Meth. Fluids, 47 (8-9) (2005), 773-788. Zbl1134.76326MR2123791DOI10.1002/fld.867

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