# 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

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topLototsky, 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 -

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