On least squares problems with certain Vandermonde--Khatri--Rao structure with applications to DMD

Abstract: This paper proposes a new computational method for solving structured least squares problems that arise in the process of identification of coherent structures in fluid flows. It is deployed in combination with dynamic mode decomposition (DMD) which provides a non-orthogonal set of modes --- corresponding to particular temporal frequencies --- a subset of which 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 problem with the matrix that involves the Khatri--Rao product of a triangular and a Vandermonde matrix. Such a structure allows a very efficient normal equation based least squares solution, which is used in state of the art 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 semi-normal solution and QR factorization based algorithms. It is shown 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.