Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems

Abstract

Uncertainty quantification and propagation in physical systems appear as a critical path for the improvement of the prediction of their response. Galerkin-type spectral stochastic methods provide a general framework for the numerical simulation of physical models driven by stochastic partial differential equations. The response is searched in a tensor product space, which is the product of deterministic and stochastic approximation spaces. The computation of the approximate solution requires the solution of a very high dimensional problem, whose calculation costs are generally prohibitive. Recently, a model reduction technique, named Generalized Spectral Decomposition method, has been proposed in order to reduce these costs. This method belongs to the family of Proper Generalized Decomposition methods. It takes part of the tensor product structure of the solution function space and allows the a priori construction of a quasi optimal separated representation of the solution, which has quite the same convergence properties as a posteriori Hilbert Karhunen-Loeve decompositions. The associated algorithms only require the solution of a few deterministic problems and a few stochastic problems on deterministic reduced basis (algebraic stochastic equations), these problems being uncoupled. However, this method does not circumvent the “curse of dimensionality” which is associated with the dramatic increase in the dimension of stochastic approximation spaces, when dealing with high stochastic dimension. In this paper, we propose a marriage between the Generalized Spectral Decomposition algorithms and a separated representation methodology, which exploits the tensor product structure of stochastic functions spaces. An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.