2018/11/29 by Zlatko Drmač, Igor Mezić, Drmač, Zlatko +3 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Image and Signal Denoising Methods #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.1811.12562
openalex publication_date 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper proposes a new computational method for solving structured least\nsquares problems that arise in the process of identification of coherent\nstructures in fluid flows. It is deployed in combination with dynamic mode\ndecomposition (DMD) which provides a non-orthogonal set of modes ---\ncorresponding to particular temporal frequencies --- a subset of which is used\nto represent time snapshots of the underlying dynamics. The coefficients of the\nrepresentation are determined from a solution of a structured linear least\nsquares problem with the matrix that involves the Khatri--Rao product of a\ntriangular and a Vandermonde matrix. Such a structure allows a very efficient\nnormal equation based least squares solution, which is used in state of the art\nCFD tools such as the sparsity promoting DMD (DMDSP). A new numerical analysis\nof the normal equations approach provides insights about its applicability and\nits limitations. Relevant condition numbers that determine numerical robustness\nare identified and discussed. Further, the paper offers a corrected semi-normal\nsolution and QR factorization based algorithms. It is shown how to use the\nVandermonde--Khatri--Rao structure to efficiently compute the QR factorization\nof the least squares coefficient matrix, thus providing a new computational\ntool for the ill-conditioned cases where the normal equations may fail to\ncompute a sufficiently accurate solution. Altogether, the presented material\nprovides a firm numerical linear algebra framework for a class of structured\nleast squares problems arising in a variety of applications.\n