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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 a distributed model into account. This results in a novel procedure for model reduction of systems with multiple independent variables. Data in multiple independent variables is associated with the mathematical structure of a tensor. We show how orthonormal decompositions of this tensor can be used to derive suitable projection spaces. These projection spaces prove useful for determining reduced order models by performing Galerkin projections on equation residuals. 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 an application in data compression. The proposed model reduction procedure is illustrated on a heat diffusion problem.

A tensor decomposition approach to data compression and approximation of ND systems