Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced whic...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 119; no. 1; pp. 123 - 161
Main Authors: Barth, Andrea, Schwab, Christoph, Zollinger, Nathaniel
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.09.2011
Springer
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical analysis. Scientific computation
Numerical and Computational Physics
Numerical methods in probability and statistics
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Simulation
Theoretical
65N30
Monte Carlo method
Correlation
Error estimation
Differential equation
Initial value problem
Stochastic method
Partial differential equation
Discretization method
Finite element method
Numerical analysis
Elliptic equation
Boundary value problem
Integral equation
Abstract space
Stochastic equation
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Description
Summary:In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k -point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-011-0377-0