Adaptive algorithm for stochastic Galerkin method

Ivana Pultarová

Applications of Mathematics (2015)

  • Volume: 60, Issue: 5, page 551-571
  • ISSN: 0862-7940

Abstract

top
We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables.

How to cite

top

Pultarová, Ivana. "Adaptive algorithm for stochastic Galerkin method." Applications of Mathematics 60.5 (2015): 551-571. <http://eudml.org/doc/271576>.

@article{Pultarová2015,
abstract = {We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables.},
author = {Pultarová, Ivana},
journal = {Applications of Mathematics},
keywords = {stochastic Galerkin method; a posteriori error estimate; strengthened Cauchy-Bunyakowski-Schwarz constant; adaptive refinement; stochastic Galerkin method; a posteriori error estimate; strengthened Cauchy-Bunyakowski-Schwarz constant; adaptive refinement},
language = {eng},
number = {5},
pages = {551-571},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Adaptive algorithm for stochastic Galerkin method},
url = {http://eudml.org/doc/271576},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Pultarová, Ivana
TI - Adaptive algorithm for stochastic Galerkin method
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 5
SP - 551
EP - 571
AB - We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables.
LA - eng
KW - stochastic Galerkin method; a posteriori error estimate; strengthened Cauchy-Bunyakowski-Schwarz constant; adaptive refinement; stochastic Galerkin method; a posteriori error estimate; strengthened Cauchy-Bunyakowski-Schwarz constant; adaptive refinement
UR - http://eudml.org/doc/271576
ER -

References

top
  1. Ainsworth, M., Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts Wiley, Chichester (2000). (2000) Zbl1008.65076MR1885308
  2. Axelsson, O., Iterative Solution Methods, Cambridge Univ. Press, Cambridge (1994). (1994) Zbl0813.15021MR1276069
  3. Axelsson, O., Blaheta, R., Neytcheva, M., 10.1137/070679673, SIAM J. Matrix Anal. Appl. 31 (2009), 767-789. (2009) Zbl1194.65047MR2530276DOI10.1137/070679673
  4. Babuška, I., Nobile, F., Tempone, R., 10.1137/100786356, SIAM Rev. 52 (2010), 317-355. (2010) Zbl1226.65004MR2646806DOI10.1137/100786356
  5. Babuška, I., Tempone, R., Zouraris, G. E., 10.1137/S0036142902418680, SIAM J. Numer. Anal. 42 (2004), 800-825. (2004) Zbl1080.65003MR2084236DOI10.1137/S0036142902418680
  6. Bank, R. E., Dupont, T. F., Yserentant, H., 10.1007/BF01462238, Numer. Math. 52 (1988), 427-458. (1988) Zbl0645.65074MR0932709DOI10.1007/BF01462238
  7. Bespalov, A., Powell, C. E., Silvester, D., 10.1137/110854898, SIAM J. Numer. Anal. 50 (2012), 2039-2063. (2012) Zbl1253.35228MR3022209DOI10.1137/110854898
  8. Bosq, D., 10.1007/978-1-4612-1154-9_8, Lecture Notes in Statistics 149 Springer, New York (2000). (2000) Zbl0962.60004MR1783138DOI10.1007/978-1-4612-1154-9_8
  9. Bryant, C. M., Prudhomme, S., Wildey, T., A posteriori error control for partial differential equations with random data, ICES Report 13-08, 2013, https://www.ices.utexas.edu/media/reports/2013/1308.pdf. 
  10. Butler, T., Constantine, P., Wildey, T., 10.1137/110840522, SIAM J. Matrix Anal. Appl. 33 (2012), 195-209. (2012) Zbl1248.65021MR2902678DOI10.1137/110840522
  11. Deb, M. K., Babuška, I. M., Oden, J. T., 10.1016/S0045-7825(01)00237-7, Comput. Methods Appl. Mech. Eng. 190 (2001), 6359-6372. (2001) Zbl1075.65006MR1870425DOI10.1016/S0045-7825(01)00237-7
  12. Eigel, M., Gittelson, C. J., Schwab, C., Zander, E., A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes, Preprint No. 1911. Weierstrass Institute für Angewadte Analysis und Stochastic, Berlin, 2013. 
  13. Eigel, M., Gittelson, C. J., Schwab, C., Zander, E., 10.1016/j.cma.2013.11.015, Comput. Methods Appl. Mech. Eng. 270 (2014), 247-269. (2014) Zbl1296.65157MR3154028DOI10.1016/j.cma.2013.11.015
  14. Ernst, O. G., Ullmann, E., 10.1137/080742282, SIAM J. Matrix Anal. Appl. 31 (2010), 1848-1872. (2010) Zbl1205.65021MR2644740DOI10.1137/080742282
  15. Frauenfelder, P., Schwab, C., Todor, R. A., 10.1016/j.cma.2004.04.008, Comput. Methods Appl. Mech. Eng. 194 (2005), 205-228. (2005) Zbl1143.65392MR2105161DOI10.1016/j.cma.2004.04.008
  16. Maître, O. P. Le, Knio, O. M., Najm, H. N., Ghanem, R. G., 10.1016/j.jcp.2003.11.033, J. Comput. Phys. 197 (2004), 28-57. (2004) Zbl1052.65114MR2061240DOI10.1016/j.jcp.2003.11.033
  17. Powell, C. E., Elman, H. C., 10.1093/imanum/drn014, IMA J. Numer. Anal. 29 (2009), 350-375. (2009) Zbl1169.65007MR2491431DOI10.1093/imanum/drn014
  18. Pultarová, I., Hierarchical preconditioning for the stochastic Galerkin method: upper bounds to the strengthened CBS constants, Submitted. Available in ERC-CZ project LL1202 database, http://more.karlin.mff.cuni.cz. 
  19. Ralston, A., A First Course in Numerical Analysis, McGraw-Hill Book, New York (1965). (1965) Zbl0139.31603MR0191070
  20. Sousedík, B., Ghanem, R. G., 10.1615/Int.J.UncertaintyQuantification.2014007353, Int. J. Uncertain. Quantif. 4 (2014), 333-348. (2014) MR3260480DOI10.1615/Int.J.UncertaintyQuantification.2014007353
  21. Sousedík, B., Ghanem, R. G., Phipps, E. T., 10.1002/nla.1869, Numer. Linear Algebra Appl. 21 (2014), 136-151. (2014) MR3150614DOI10.1002/nla.1869
  22. Ullmann, E., Elman, H. C., Ernst, O. G., 10.1137/110836675, SIAM J. Sci. Comput. 34 (2012), A659--A682. (2012) Zbl1251.35200MR2914299DOI10.1137/110836675
  23. Xiu, D., Numerical Methods for Stochastic Computations. A Spectral Method Approach, Princeton University Press, Princeton (2010). (2010) Zbl1210.65002MR2723020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.