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

Anthony Nouy (1) , Christian Soize (2)

1 GeM - Institut de Recherche en Génie Civil et Mécanique
2 MSME - Laboratoire de Modélisation et Simulation Multi Echelle

This paper presents new results allowing an unknown non-Gaussian positive-definite matrix-valued random field (NGPDMVRF) to be identified through a stochastic elliptic boundary value problem (SEBVP), solving a statistical inverse problem. A new general class of NGPDMVRF, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analyzed. Using a minimal parametrization of discretized random fields, a complete identification procedure is proposed. New results of the mathematical analyzes of the parameterized SEBVP are presented. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a mathematical analysis is developed. 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 low-rank approximation methods that exploit the tensor structure of the solution which results from the parametrization of the general class of random fields.

Mots clés: random fields; uncertainty quantification; statistical inverse problems; stochastic elliptic boundary value problems

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Statistiques [math.ST]; Théorie [stat.TH]; Mécanique [physics.med-ph]

Anthony Nouy, Christian Soize. Random fields representations for stochastic elliptic boundary value problems and statistical inverse problems. European Journal of Applied Mathematics, 2014, 25 (3), pp.339-373. ⟨10.1017/S0956792514000072⟩. ⟨hal-00945639⟩