A maximum entropy (MaxEnt) based probabilistic approach is developed to model mechanical systems characterized by symmetric positive-definite matrices bounded from below and above. These matrices are typically encountered in the constitutive modeling of heterogeneous materials, where the bounds are deduced by employing the principles of minimum complementary energy and minimum potential energy. Current random matrix based nonparametric approach is only adapted to the Wishart or matrix-variate Gamma probability model supported over the entire space of the symmetric positive-definite matrices, and therefore, unable to exploit additional information available through the lower and upper bounds when appropriate. Specifically, for a given material, the constitutive matrix is construed as a random matrix. A probability measure that reflects the constraints consistent with the energybased bounds, together with an associated sampling scheme, are constructed to synthesize realizations of this random matrix. An additional constraint of the ensemble mean matrix is also considered, and an appropriate probability model and sampling scheme are developed and illustrated numerically for this case.

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A procedure is presented for characterizing the asymptotic sampling distribution of the estimators of the polynomial chaos (PC) coefficients of physical process modeled as non-stationary, non-Gaussian second-order random process by using a collection of observations. These observations made over a denumerable subset of the indexing set of the process are considered to form a set of realizations of a random vector, Y , representing a finite-dimensional model of the random process. The estimators of the PC coefficients of Y are next deduced by relying on its reduced order representation obtained by employing Karhunen-Loève decomposition and subsequent use of the maximum-entropy principle, Metropolis-Hastings Markov chain Monte carlo algorithm and the Rosenblatt transformation. These estimators are found to be maximum likelihood estimators as well as consistent and asymptotically efficient estimators. The computation of the covariance matrix of the associated asymptotic normal distribution of the estimators of these PC coefficients requires evaluation of Fisher information matrix that is evaluated analytically and also estimated by using a sampling technique for the accompanied numerical illustration.Index Terms-Fisher information matrix, Karhunen-Loève expansion, Markov chain Monte Carlo, maximum-entropy probability density estimation, non-Gaussian and nonstationary/non-homogeneous random process/field, polynomial chaos expansion, Rosenblatt transformation.