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 (x1,…,xm)(x1,…,xm). The construction of the reduced order models involves two steps. First, a range (x1,…,xm)(x1,…,xm) is selected based on somewhat heuristic arguments. Second, the probabilities (p1,…,pm)(p1,…,pm) of (x1,…,xm)(x1,…,xm) 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.