SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3250-A3284, January 2020.
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.

中文翻译：

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某些Vandermonde-Khatri-Rao结构的最小二乘问题及其在DMD中的应用

SIAM科学计算杂志，第42卷，第5期，第A3250-A3284页，2020年1月。
本文提出了一种新的计算方法，用于解决在动态过程（例如流体流动）的相干结构识别过程中出现的结构化最小二乘问题。它与动态模式分解（DMD）结合部署，后者提供了与特定时间频率相对应的非正交模式集。这些的选择用于表示基础动态的时间快照。表示的系数由矩阵的结构化线性最小二乘问题的解确定，该问题涉及三角形和Vandermonde矩阵的Khatri-Rao乘积。这种结构允许使用非常有效的基于正态方程的最小二乘解，该解决方案可用于最新的计算流体动力学（CFD）工具，例如稀疏性促进DMD（DMDSP）。正规方程方法的新数值分析提供了有关其适用性和局限性的见解。确定并讨论了确定数值鲁棒性的相关条件编号。此外，本文提供了一种校正的半正规解和基于QR分解的算法。它显示了如何使用Vandermonde-Khatri-Rao结构有效地计算最小二乘系数矩阵的QR分解，从而为病态情况提供了一种新的计算工具，其中正常方程可能无法计算出足够准确的解。总体而言，本文提供的材料为除DMD外的各种应用中出现的一类结构化最小二乘问题提供了牢固的数值线性代数框架。