Local Proper Orthogonal Decomposition based on space vectors clustering

This paper presents a new method to make the Proper Orthogonal Decomposition (POD) more accurate for reducing the order of nonlinear systems. POD fails to capture the nonlinear degrees of freedom in highly nonlinear systems because it assumes that data belongs to a linear space. In this paper, the solution space is grouped into clusters where the behavior has significantly different features. Although the clustering idea is not new, it has been implemented only on snapshots clustering where a snapshot is the solution over the whole space at a particular time. In this paper, we show that clustering the space domain into the same number of clusters is more efficient. We call it space vectors clustering where a space vector is the solution over all times at a particular space location. This result is consistent with the fact that for infinite dimensional systems described by partial differential equations, the huge number of states comes from the discretization of the space domain of the PDE, not the time domain.