Elliptic equations of higher stochastic order

Sergey V. Lototsky; Boris L. Rozovskii; Xiaoliang Wan

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 1135-1153
  • ISSN: 0764-583X

Abstract

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This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.

How to cite

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Lototsky, Sergey V., Rozovskii, Boris L., and Wan, Xiaoliang. "Elliptic equations of higher stochastic order." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 1135-1153. <http://eudml.org/doc/250757>.

@article{Lototsky2010,
abstract = { This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented. },
author = {Lototsky, Sergey V., Rozovskii, Boris L., Wan, Xiaoliang},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Elliptic PDE; random coefficients; Wiener Chaos; spectral finite elements; elliptic PDE; Wiener chaos; white noise; Wick product; Itô-Skorokhod calculus; Fourier method; numerical experiments},
language = {eng},
month = {8},
number = {5},
pages = {1135-1153},
publisher = {EDP Sciences},
title = {Elliptic equations of higher stochastic order},
url = {http://eudml.org/doc/250757},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Lototsky, Sergey V.
AU - Rozovskii, Boris L.
AU - Wan, Xiaoliang
TI - Elliptic equations of higher stochastic order
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 1135
EP - 1153
AB - This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.
LA - eng
KW - Elliptic PDE; random coefficients; Wiener Chaos; spectral finite elements; elliptic PDE; Wiener chaos; white noise; Wick product; Itô-Skorokhod calculus; Fourier method; numerical experiments
UR - http://eudml.org/doc/250757
ER -

References

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