Nonlinear model reduction using Space Vectors Clustering POD with application to the Burgers´ equation

Proper Orthogonal Decomposition (POD) fails to capture the nonlinear degrees of freedom in large scale highly nonlinear systems because it assumes that data belongs to a linear space. In this paper we develop a new method that makes POD more accurate in reducing the order of large scale nonlinear systems. 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. We show that clustering the spatial domain into the same number of clusters is more efficient. We call it Space Vector Clustering (SVC) POD where a space vector is the solution over all times at a particular space location. This is consistent with the fact that for infinite dimensional systems described by partial differential equations (PDEs), the large number of states comes from the discretization of the spatial domain of the PDE, not the time domain. We apply our method to reduce a nonlinear convective PDE system governed by the Burgers´ equation over 1D and 2D domains and show a significant improvement over conventional POD.