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

Published online by Cambridge University Press:  21 March 2014

A. NOUY  and
C. SOIZE
A. NOUY
Affiliation:
Ecole Centrale de Nantes, LUNAM Université, GeM UMR CNRS 6183, 1 rue de la Noe, 44321 Nantes, France email: [email protected]
C. SOIZE
Affiliation:
Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Valle, France email: [email protected]
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Abstract

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.

Keywords

60G6065M3260H3562F9935J70
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 3 , June 2014 , pp. 339 - 373
Copyright
Copyright © Cambridge University Press 2014 

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