Search Results for « Low-Rank Dynamic Mode Decomposition »

Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative domains, e.g., for climate prediction, to study molecular dynamics or microelectromechanical devices. This low-rank extension takes the form of a non-convex optimization problem. To the best of our knowledge, only sub-optimal algorithms have been proposed in the literature to compute the solution of this problem. In this paper, we prove that there exists a closed-form optimal solution to this problem and design an effective algorithm to compute it based on Singular Value Decomposition (SVD). A toy-example illustrates the gain in performance of the proposed algorithm compared to state-of-the-art techniques.

This technical note reviews sate-of-the-art algorithms for linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). While repeating several parts of our article "low-rank dynamic mode decomposition: an exact and tractable solution", this work provides additional details useful for building a comprehensive picture of state-of-the-art methods.

Abstract Traffic forecasting is an important part in realising intelligent traffic management, which helps traffic controllers and travellers make effective decisions. However, traffic forecasting accuracy is often affected by missing traffic data due to hardware and software failure. Therefore, accurate prediction based on incomplete traffic data is an important problem as well as a challenge. Though many approaches recover the missing values before prediction, the errors from the data‐filling step are likely to cause additional bias to the prediction result. Besides, this tactic is difficult to guarantee the timeliness and may impede real‐time prediction. In this case, a traffic forecasting model is proposed to directly predict traffic data with missing values. This model develops tensor formed dynamic mode decomposition, recording the dynamic information of traffic data into a state transition tensor. In addition, the model takes low rank property of the dynamic tensor and the similarity of temporal variation trend into consideration. In order to verify the effectiveness and the robustness of the proposed model, experiments were performed on two real‐world time series datasets. The results demonstrate that the model achieves better performance on forecasting than other baseline approaches under the impact of missing data.

In this study, we provide an alternative approach for computing the dynamic mode decomposition (DMD) in real-time for streaming datasets. It is a low-storage method that updates the DMD approx­imation of a given dynamic as new data becomes available. Unlike the standard online DMD method, which is applicable only to over­constrained and full-rank datasets, the new method is applicable for both overconstrained and underconstrained datasets. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations and is entirely data-driven. Several numerical examples are presented to demonstrate the performance of the method.

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of “big data.” The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a quickly decaying singular value spectrum.

Koopman theory has become widely used for predictions and analysis of nonlinear systems. Especially, kernel extended dynamic mode decomposition (kEDMD) embeds nonlinear dynamics in a reproducing-kernel Hilbert space. However, its Gram matrix grows with every snapshot, blocking online use when parameters drift. We present a low-rank approximation-based kEDMD algorithm that swaps out arbitrary old snapshots, inserts new ones, and updates the eigendecomposition with rank-two corrections, keeping both memory and cost fixed. The method efficiently tracks evolving systems on the van der Pol oscillator and adapts to unseen data and slow coefficient changes. In addition, it efficiently tracks a 50-point discretized Korteweg-de Vries flow and preserves predictive accuracy.