Approximate Solutions of Lagrange Multipliers for Information-Theoretic Random Field Models

Abstract

This work is concerned with the construction of approximate solutions for the Lagrange multipliers involved in information-theoretic non-Gaussian random field models. Specifically, representations of physical fields with invariance properties under some orthogonal transformations are considered. A methodology for solving the optimization problems raised by entropy maximization (for the family of first-order marginal probability distributions) is first presented and exemplified in the case of elasticity fields exhibiting fluctuations in a given symmetry class. Results for all classes ranging from isotropy to orthotropy are provided and discussed. The derivations are subsequently used for proving a few properties that are required in order to sample the above models by solving a family of stochastic differential equations---along the lines of the algorithm constructed in [J. Guilleminot and C. Soize, Multiscale Model. Simul., 11 (2013), pp. 840--870]. The results thus allow for forward simulations of the probabilistic models in stochastic boundary value problems, as well as for a reduction of the computational cost associated with model calibration through statistical inverse problems.

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