Nuclear norm regularised dynamic mode decomposition

Abstract

As a data-driven, equation-free decomposition method, the DMD can characterise dynamic behaviour of a non-linear system by using the DMD modes and eigenvalues. However, all current provable algorithms suffer from a separate procedure for obtaining the DMD modes and determining the number of modes. In this study, the authors propose a nuclear norm regularised DMD (NNR-DMD) algorithm that produces low-dimensional spatio-temporal modes. A nuclear norm regularisation term is added to the optimisation problem of the standard DMD algorithm for prompting the sparsity of the projected DMD modes. Split Bregman method is applied to solve the regularised convex, but non-smooth optimisation problem. Several numerical examples demonstrate the potential of the proposed NNR-DMD algorithm: (i) it can identify the low-dimensional spatio-temporal DMD modes in which each of them possesses a single temporal frequency; (ii) the reconstruction errors based on the sparse DMD modes can be reduced when it compares with the sparsity-promoting DMD algorithm penalising the l 1 -norm of the vector of DMD amplitudes; and (iii) it can obtain low-dimensional coherent structures when the NNR-DMD algorithm is applied to coherency identification of generators in an interconnected power system.