One of the most versatile parametrizations of stochastic processes or random fields for large-scale computations is provided by the Karhunen-Loève expansion. Its computational realization requires (at least the partial) solution of an eigenvalue problem for a covariance operator. We give an overview of the functional analytic background of covariance models and describe our contributions to the numerical solution of the covariance eigenproblem based on Krylov subspace methods, hierarchical matrix techniques and adapted quadrature.
Oliver Ernst received his Diploma degree in Mathematics from the Technical University of Karlsruhe, Germany, in 1989 and earned a PhD in Scientific Computing and Computational Mathematics from Stanford University supervised by Gene Golub in 1994. Since then he has been at the Department of Mathematics and Computer Science at the Technical University Bergakademie Freiberg, Germany, where he is a professor in the numerical analysis group. He has held visiting appointments at the University of Maryland as well as the University of Wuppertal, Germany.
His research interests are in applied numerical linear algebra, particularly fast solution algorithms for large scale linear systems. In recent years he has added an emphasis on the analysis and development of computational methods for probabilistic uncertainty quantification. Further research interests include electromagnetics, time-harmonic scattering and inverse problems.