Reduced Basis Methods: From Low-Rank Matrices To Low-Rank Tensors

We propose a novel combination of the reduced basis method with low-rank tensor techniques for the efficient solution of parameter-dependent linear systems in the case of several parameters. This combination, called rbTensor, consists of three ingredients. First, the underlying parameter-dependent operator is approximated by an explicit affine representation in a low-rank tensor format. Second, a standard greedy strategy is used to construct a problem-dependent reduced basis. Third, the associated reduced parametric system is solved for all parameter values on a tensor grid simultaneously via a low-rank approach. This allows us to explicitly represent and store an approximate solution for all parameter values at a time. Once this approximation is available, the computation of output functionals and the evaluation of statistics of the solution become a cheap online task, without requiring the solution of a linear system.