scholarly journals Statistical inverse problems for non-Gaussian vector valued random fields with a set of experimental realizations

2014 ◽  
pp. 2811-2816
Author(s):  
G Perrin ◽  
C Soize ◽  
D Duhamel ◽  
C Funfschilling
Keyword(s):  
Inverse Problems ◽  
Random Fields ◽  
Gaussian Vector ◽  
Statistical Inverse Problems ◽  
Non Gaussian ◽  
Vector Valued

scholarly journals Random field representations for stochastic elliptic boundary value problems and statistical inverse problems

2014 ◽  
Vol 25 (3) ◽  
pp. 339-373 ◽  
Author(s):  
A. NOUY ◽  
C. SOIZE
Keyword(s):  
Random Fields ◽  
General Class ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Approximation Methods ◽  
Elliptic Boundary Value Problem ◽  
Identification Procedure ◽  
Elliptic Boundary Value ◽  
Statistical Inverse Problems ◽  
Non Gaussian

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.

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