TY - JOUR
T1 - Stochastic finite element methods for partial differential equations with random input data
AU - Gunzburger, Max D.
AU - Webster, Clayton G.
AU - Zhang, Guannan
PY - 2014/5
Y1 - 2014/5
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84900467213&partnerID=8YFLogxK
U2 - 10.1017/S0962492914000075
DO - 10.1017/S0962492914000075
M3 - Article
AN - SCOPUS:84900467213
SN - 0962-4929
VL - 23
SP - 521
EP - 650
JO - Acta Numerica
JF - Acta Numerica
ER -