Orthogonal representations of stochastic processes and their propagation in mechanics

The paper reviews the representation of stochastic processes used to model coefficients in partial differential equations. In particular, methods based on the Karhunen-Loeve and Polynomial Chaos expansions are presented. It is shown that these representations permit both the efficient characterization and management of the uncertainty in the solution of the PDE. Management in this context is construed to mean error estimation and control. Furthermore, generalizations of the Karhunen-Loeve expansion to representing processes in Sobolev spaces is also presented. These processes are essential in applications to mechanics since, in this case, many functions of interest are constrained with regards to their smoothness and differentiability.