A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations
For researchers in uncertainty quantification, this offers a computationally cheaper alternative to existing polynomial chaos methods by reducing matrix density.
The paper presents a Monte Carlo method for assembling finite element matrices in polynomial chaos approximations of elliptic equations with random coefficients, resulting in sparse block-diagonal matrices instead of block-full ones. This approach generalizes classical Monte Carlo and collocation methods.
We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.