Uncertainty quantification for algebraic systems of equations

We consider the situation where an unknown n-dimensional vector X has to be determined by solving a system of equations having the form F(X, v) = 0, where F is a mapping from the n-dimensional Euclidean space on itself and v is a random k-dimensional vector. We focus on the numerical determination of the distribution of solution X, which is also a random variable. We propose an expansion of X as a function of a vector v and we apply known approaches such as the collocation, moment matching and variational approximation and, we developed a new approach for the solution based on the adaptation of deterministic iterative numerical methods. These approaches are tested and compared in linear and non-linear situations including a laminated composite plate and a beam under nonlinear behavior. The results showed the effectiveness and the advantages of the new approach over the variational one to solve the uncertainty quantification of systems of nonlinear equations. Also, from the comparison among the methods, it is shown that the collocation is the most effective and robust approach, followed by the adaptation one. Finally, the least robust method is the moment matching approach due to the complexity of the resulting optimization problem.