Adaptive algorithm for stochastic Galerkin method

Ivana Pultarová

Applications of Mathematics (2015)

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

Abstract

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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

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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

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