A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data

2013 ◽  
Vol 51 (6) ◽  
pp. 3163-3185 ◽  
Author(s):  
Peng Chen ◽  
Alfio Quarteroni ◽  
Gianluigi Rozza
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Input Data ◽  
Elliptic Partial Differential Equations ◽  
Reduced Basis Method ◽  
Random Input ◽  
Reduced Basis ◽  
Partial Differential ◽  
Basis Method ◽  
Random Input Data

Stochastic finite element methods for partial differential equations with random input data

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 521-650 ◽  
Author(s):  
Max D. Gunzburger ◽  
Clayton G. Webster ◽  
Guannan Zhang
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Element Methods ◽  
Input Data ◽  
Degrees Of Freedom ◽  
Stochastic Finite Element ◽  
Random Input ◽  
Partial Differential ◽  
Random Input Data

The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

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