On Least Squares Problems with Certain Vandermonde--Khatri--Rao Structure with Applications to DMD

Abstract

This paper proposes a new computational method for solving the structured least squares problem that arises in the process of identification of coherent structures in dynamic processes, such as, e.g., fluid flows. It is deployed in combination with dynamic mode decomposition (DMD), which provides a nonorthogonal set of modes corresponding to particular temporal frequencies. A selection of these is used to represent time snapshots of the underlying dynamics. The coefficients of the representation are determined from a solution of a structured linear least squares problems with the matrix that involves the Khatri--Rao product of a triangular and a Vandermonde matrix. Such a structure allows for a very efficient normal equation based least squares solution, which is used in state-of-the-art computational fluid dynamics (CFD) tools, such as the sparsity promoting DMD (DMDSP). A new numerical analysis of the normal equations approach provides insights about its applicability and its limitations. Relevant condition numbers that determine numerical robustness are identified and discussed. Further, the paper offers a corrected seminormal solution and the QR factorization based algorithms. It shows how to use the Vandermonde--Khatri--Rao structure to efficiently compute the QR factorization of the least squares coefficient matrix, thus providing a new computational tool for the ill-conditioned cases where the normal equations may fail to compute a sufficiently accurate solution. Altogether, the presented material provides a firm numerical linear algebra framework for a class of structured least squares problems arising in a variety of applications besides the DMD, such as, e.g., multistatic antenna array processing.