Abstract
Traditional dynamic mode decomposition (DMD) methods inevitably involve matrix inversion, which often brings in numerical instability and spurious modes. In this paper, a new algorithm is derived to solve DMD as a general eigenvalue problem, which is then computed with a projection to a subspace with minimal errors in terms of least-squares (LS) or total-least-squares (TLS), leading to a more stable DMD algorithm, named DMD-LS or DMD-TLS, respectively. A new residual criterion, along with a typical energy criterion, is then proposed to select the most dynamically relevant DMD eigenvalues and modes. The accuracy and robustness of DMD-LS and DMD-TLS algorithms are demonstrated by application to the direct simulation data of the three-dimensional flow past a fixed long cylinder, which covers the entire evolution from the asymptotic periodic to the completely periodic stages of the flow. The connection between DMD modes and Floquet modes commonly used in stability studies was demonstrated through the DMD analysis of the asymptotic periodic data for the secondary instability of the flow.
Highlights
Dynamic mode decomposition (DMD)1,2 is a data-driven technique to extract dynamic relevant information from time-resolved snapshots
By converting a standard eigenvalue problem to a generalized eigenvalue problems (GEPs), the new framework avoids the numerical operation of matrix inversion and the associated numerical complexity faced in traditional DMD algorithms
To solve the GEP with efficiency and robustness, the solution is approximated through a projection to a well-behaved subspace that is constructed from the original data snapshots
Summary
Dynamic mode decomposition (DMD) is a data-driven technique to extract dynamic relevant information from time-resolved snapshots. Anantharamu and Mahesh discussed the potential risk from the artificial truncation of singular values and proposed a different Arnoldi-based algorithm to avoid it It requires a fairly complex procedure and still faces a risk to lose orthogonality and unbounded growth in storage commonly occurring in Arnoldi-based approaches.. It requires a fairly complex procedure and still faces a risk to lose orthogonality and unbounded growth in storage commonly occurring in Arnoldi-based approaches.31 To address these challenges, the current work proposes a matrix-free framework, which is based on generalized eigenvalue problems (GEPs), to derive DMD modes.
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