A multi-dimensional residual functional for obtaining the Proper Orthogonal Decomposition coefficients in model reduction

This paper presents a multi-dimensional residual functional for deriving the POD (Proper Orthogonal Decomposition) coefficients of systems described with partial differential equations of one variable in a bidimensional spatial domain, when a POD approach is used for deriving a reduced order model. Model reduction with a POD approach is a technique that uses the signal spectral decomposition and the Galerkin projection for deriving reduced order models. Recently there has been a growing interest in the POD community in the subject of tensor decomposition, since it can address some of the outcomes observed in the current based matrix technique. The main purpose of using tensor decomposition is to derive multidimensional basis functions, which are essential for obtaining a spectral decomposition of the system solution. However, multidimensional basis functions do not match with the residual functional developed in the traditional matrix technique, therefore a generalization of this residual function is needed in order to obtain the POD coefficients.