Singular Dynamic Mode Decompositions

This manuscript is aimed at addressing several long standing limitations of dynamic mode decompositions in the application of Koopman analysis. Principle among these limitations are the convergence of associated Dynamic Mode Decomposition algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in this analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. The manuscript concludes with the description of a Dynamic Mode Decomposition algorithm that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.