Reduced order models for random functions. Application to stochastic problems

A method is developed for constructing reduced order models for arbitrary random functions. The reduced order models are simple random functions, that is, functions with a finite range ( x 1 , … , x m ) . The construction of the reduced order models involves two steps. First, a range ( x 1 , … , x m ) is selected based on somewhat heuristic arguments. Second, the probabilities ( p 1 , … , p m ) of ( x 1 , … , x m ) are obtained from the solution of an optimization problem. Reduced order models are applied to calculate the distributions of the modal frequencies of a linear dynamic system with random stiffness matrix and statistics of the hydraulic head in a soil deposit with random heterogeneous conductivity. The performance of reduced order models in both applications is remarkable.