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We use a Monte Carlo method to assemble finite element matrices forpolynomial Chaos approximations of elliptic equations with random coefficients.In this approach, all required expectations are approximated by a Monte Carlomethod. The resulting methodology requires dealing with sparse block-diagonalmatrices instead of block-full matrices. This leads to the solution of acoupled system of elliptic equations where the coupling is given by a Kroneckerproduct matrix involving polynomial evaluation matrices. This generalizes theClassical Monte Carlo approximation and Collocation method for approximatingfunctionals of solutions of these equations.

A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations