A Tensor-Based Algorithm for the Optimal Model Reduction of High Dimensional Problems

We propose a method for the approximation of the solution of high-dimensional problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of an ideal minimal residual method with a residual norm corresponding to the error in a solution norm of interest. We introduce and analyze an algorithm for the approximation of the best approximation in a given low-rank tensor subset. A weak greedy algorithm based on this ideal minimal residual formulation is introduced and its convergence is proven under some conditions. The robustness of the method is illustrated on numerical examples in uncertainty propagation.