A Posteriori Error and Optimal Reduced Basis for Stochastic Processes Defined by a Finite Set of Realizations

The use of reduced basis has spread to many scientific fields for the last fifty years to condense the statistical properties of stochastic processes. Among these basis, the classical Karhunen-Loève basis corresponds to the Hilbertian basis that is constructed as the eigenfunctions of the covariance operator of the stochastic process of interest. The importance of this basis stems from its optimality in the sense that it minimizes the total mean square error. When the available information about this stochastic process is characterized by a limited set of independent realizations, the covariance operator is not perfectly known. In this case, there is no reason for the Karhunen-Loève basis associated with any estimator of the covariance that are not converged to be still optimal. This paper presents therefore an adaptation of the Karhunen-Loève expansion in order to characterize optimal basis for projection of stochastic processes that are only characterized by a relatively small set of independent realizations.