Approximation of nD systems using tensor decompositions

This paper considers reduced order modeling of systems with multiple independent variables. The method of proper orthogonal decompositions (POD) is a data-based method that is suitable for the reduction of large-scale distributed systems. In this paper we propose a generalization of the POD method so as to take the nD nature of the distributed model into account. Suitable projection spaces can be computed by associating a tensor with the measurement data and computing a suitable decomposition of this tensor. We demonstrate how prior knowledge about the structure of the model reduction problem can be used to improve the quality of approximations. The tensor decomposition techniques are demonstrated on a data approximation example and then the model reduction process is illustrated using a heat diffusion problem.