A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

Pulsed high-intensity focused ultrasound (pHIFU) can induce sparse de novo inertial cavitation without the introduction of exogenous contrast agents, promoting mild mechanical disruption in targeted tissue. Because the bubbles are small and rapidly dissolve after each HIFU pulse, mapping transient bubbles and obtaining real-time quantitative metrics correlated to tissue damage are challenging. Prior work introduced Bubble Doppler, an ultrafast power Doppler imaging method as a sensitive means to map cavitation bubbles. The main limitation of that method was its reliance on conventional wall filters used in Doppler imaging and optimized for imaging blood flow rather than transient scatterers. This study explores Bubble Doppler enhancement using dynamic mode decomposition (DMD) of a matrix created from a Doppler ensemble for mapping and extracting the characteristics of transient cavitation bubbles. DMD was first tested in silico with a numerical dataset mimicking the spatiotemporal characteristics of backscattered signal from tissue and bubbles. The performance of DMD filter was compared to other widely used Doppler wall filters - singular value decomposition (SVD) and infinite impulse response (IIR) highpass filter. DMD was then applied to an ex vivo tissue dataset where each HIFU pulse was immediately followed by a plane wave Doppler ensemble. In silico DMD outperformed SVD and IIR high pass filter and ex vivo provided physically interpretable images of the modes associated with bubbles and their corresponding temporal decay rates. These DMD modes can be trackable over the duration of pHIFU treatment using k-means clustering method, resulting in quantitative indicators of treatment progression.

Dynamic mode decomposition (DMD) is essentially a hybrid algorithm based on mode decomposition and singular value decomposition, and it inevitably inherits the drawbacks of these two algorithms, including the selection strategy of truncated rank order and wanted mode components. A novel denoising and feature extraction algorithm for multi-component coupled noisy mechanical signals is proposed based on the standard DMD algorithm, which provides a new method solving the two intractable problems above. Firstly, a sparse optimization method of non-convex penalty function is adopted to determine the optimal dimensionality reduction space in the process of DMD, obtaining a series of optimal DMD modes. Then, multiscale permutation entropy calculation is performed to calculate the complexity of each DMD mode. Modes corresponding to the noise components are discarded by threshold technology, and we reconstruct the modes whose entropies are smaller than a threshold to recover the signal. By applying the algorithm to rolling bearing simulation signals and comparing with the result of wavelet transform, the effectiveness of the proposed method can be verified. Finally, the proposed method is applied to the experimental rolling bearing signals. Results demonstrated that the proposed approach has a good application prospect in noise reduction and fault feature extraction.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena

Simulating particle deposition in the respiratory tract requires high computational effort due to the intricate airway geometry and complex airflow–particle interactions. To address this challenge, this study introduces the first demonstration of Dynamic Mode Decomposition (DMD) as a reduced-order model to infer the trajectories of inhaled particles during a breathing cycle and to evaluate the applicability of DMD as a fluid field interpolator. The periodic nature of respiration and the predominance of sinusoidal boundary conditions make it well-suited for DMD analysis. Three high-fidelity computational fluid dynamics (CFD) simulations were performed under three different inlet volume airflow conditions for the same realistic adult male anthropomorphic phantom respiratory tract model. Reduced-rank DMD reconstructions were compared to the CFD ground truth, yielding a Mean Relative Error (MRE) of 12% in the velocity field. Additionally, a fourth simulation was conducted at an intermediate point to evaluate the interpolation capability of the parametric DMD framework in complex systems. This interpolation resulted in an MRE of 20%, with the reconstructed flow field capturing dominant fluid modes and overall dynamics, though localized discrepancies reached relative errors up to 70%.While DMD effectively reconstructed fluid fields, preserving mean flow regimes, some deviations were observed in Lagrangian particle tracking, specifically in spatial deposition resolution. However, the method approximated overall particle distribution with an 85% correlation to ground truth and was effective in representing regional deposition patterns across the tracheobronchial tree. These findings support the utility of DMD a computationally efficient approach for fluid field reconstruction and particle transport analysis in respiratory flow simulations.

Operating convergent–divergent nozzles at low pressure ratios can lead to flow separation. Certain nozzle contours are known to produce a discrete acoustical tone under such conditions that are believed to be generated by a phenomenon known as transonic resonance, which involves a standing pressure wave situated between the separation shock and the nozzle exit plane. This paper reports the findings of a dynamic mode decomposition analysis of a perturbed axisymmetric unsteady Reynolds-averaged Navier–Stokes simulation of a nozzle flow known to exhibit this phenomenon. Two cases of different pressure ratios were studied in depth. The results show that the two cases differ in the shape of the standing pressure wave. The lower pressure ratio produces a standing pressure wave, whereas the higher pressure ratio produces a wave. In both cases, dynamic mode decomposition modes that match the standing pressure wave shape were found to be the least damped and the most energetic modes of the modes produced by the dynamic mode decomposition algorithm. The frequency of the mode for the lower pressure ratio matched the experimentally observed transonic tone frequency extraordinarily well, whereas the higher pressure ratio case was in fair agreement. The dynamic mode decomposition algorithm captured the transonic frequency for five additional pressure ratios.

To address the issue of low accuracy in the dynamic modal decomposition (DMD) method used for operational modal analysis (OMA) under noise conditions of aerospace structures, an enhanced identification approach is proposed in this paper, which integrates subspace orthogonal projection with DMD to better determine the modal properties of linear mechanical systems with noisy observations. Subspace orthogonal projection applied to the Hankelized matrix is utilized for denoising observation signals. Compact singular value decomposition (SVD) is employed on the projection matrix in order to acquire the optimal realization of system matrix. Subsequently, DMD is introduced to reduce the dimensionality of the system matrix and extract the dominant modal features. The effectiveness and practicality of the proposed method are confirmed through numerical and experimental examples. The proposed method demonstrates marginally improved identification accuracy in modal frequency and enhanced performance in damping ratios when compared to representative OMA methods under different white noise conditions.

Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.

With the increase in collected data volumes, either from experimental measurements or high fidelity simulations, there is an ever-growing need to develop computationally efficient tools to process, analyze, and interpret these datasets. Modal analysis techniques have gained great interest due to their ability to identify patterns in the data and extract valuable information about the system being considered. Dynamic mode decomposition (DMD) relies on elements of the Koopman approximation theory to compute a set of modes, each associated with a fixed oscillation frequency and a decay/growth rate. Extracting these details from large datasets can be computationally expensive due to the need to implement singular value decomposition of the input data matrix. Sketching algorithms have become popular in numerical linear algebra where statistical theoretic approaches are utilized to reduce the cost of major operations. A sketch of a matrix is another matrix, which is significantly smaller, but still sufficiently approximates the original system. We put forth an efficient DMD framework, SketchyDMD, based on a core sketching algorithm that captures information about the range and corange (their mutual relationship) of input data. The proposed sketching-based framework can accelerate various portions of the DMD routines, compared to classical methods that operate directly on the raw input data. We conduct numerical experiments using the spherical shallow water equations as a prototypical model in the context of geophysical flows. We show that the proposed SketchyDMD is superior to existing randomized DMD methods that are based on capturing only the range of the input data.

A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

Two-dimensional open-cavity flows for free stream Mach number of 0.1 to 1.6 are investigated with direct numerical simulations for cavities of aspect ratio 2 and 6 to characterize the flow stability. We determine the neutral stability curve over a wide range of Mach numbers and Reynolds numbers. In particular, we examine the effects of compressibility on the flow in transonic regime at low Reynolds numbers. Rossiter modes are identified by spectral analysis. The corresponding spatial structures associated with the dominant and subdominant Rossiter modes are extracted with dynamic mode decomposition. The nature of these structures provides us insight to their contribution to instability of the flow. The interaction between the shock waves and the shear layer modes for the transonic flows are investigated.

This study proposes a novel data-driven framework for rapid and accurate prediction of unsteady hemodynamics in a patient-specific cardiovascular model, based on a Deep Learning-based Reduced-Order Model (DL-ROM). High-fidelity simulations of pulsatile blood flow were used to generate a wall shear stress (WSS) dataset. To reduce dimensionality while preserving essential dynamics, Dynamic Mode Decomposition (DMD) was employed, capturing dominant spatiotemporal modes linked to flow instabilities and coherent structures. The extracted DMD time coefficients were then used to train an attention-augmented Convolutional Neural Network with a Long Short-Term Memory layer, enabling the prediction of temporal WSS evolution. The DL-ROM accurately reconstructs high-resolution flow fields at unseen time points, achieving pointwise relative errors below 3% across most of the aneurysm surface. Compared to traditional full-order models, the proposed approach offers over two orders of magnitude reduction in computational cost, supporting near real-time inference. Analysis of DMD mode components further revealed region-specific flow dynamics, with high-frequency oscillations concentrated near the aneurysm dome. This work introduces an efficient and interpretable surrogate model for transient hemodynamic analysis, with promising implications for clinical applications such as rupture risk stratification, fast patient screening, and future integration across multiple aneurysm geometries.

On many occasions, fan or compressor stages have to face azimuthal flow distortion at inlet, which affects their performance and stability. These flow distortions can be caused by external events or by some particular geometrical features. The aim of this work is to propose a joined numerical and experimental analysis of the flow behavior in a single axial compressor stage under flow distortion. The distortions are generated by different grids that are placed upstream to the rotor. Experimentally, the flow analysis is based on the measurements obtained by a series of unsteady pressure sensors flush-mounted at the casing of the machine rotor. URANS computations are conducted using the elsA software. The flow distortion is simulated by a drop of stagnation pressure ratio at the inlet boundary condition. The study is focusing first on the ability of a pressure drop, imposed as an inlet boundary condition in CFD, to reproduce accurately the effect of a flow distortion. The analysis is conducted using singular value decomposition (SVD) and dynamic mode decomposition (DMD). A special attention is then paid, on the experimental level, to the arising of rotating stall, from the onset of the instability up to completely developed stall cells.

ABSTRACTThe purpose of this paper is the identification of a reduced order model (ROM) from numerical code output by non-intrusive techniques (i.e. not requiring projecting of the governing equations onto the reduced basis modes). In this paper, we perform a comparison between two methods of model order reduction based on dynamic mode decomposition (DMD). The first method is a deterministic (classic) DMD technique endowed with a dynamic filtering criterion of selection of modes used in the ROM model. The second method is an adaptive randomised DMD algorithm (ARDMD) based on a randomised singular value decomposition. This produced an accelerating algorithm, which is endowed with a few additional advantages. In addition, the reduced order model is guaranteed to satisfy the boundary conditions of the full model, which is crucial for surrogate modelling. For numerical illustration, we use the shallow water equations model.

The modes in the electrocardiogram (ECG) signal can be divided into stable modes and unstable modes. The unstable modes are of great significance for signal analysis and classification. When the traditional dynamic mode decomposition (DMD) method is directly applied to these signals, it is difficult to effectively extract these key modes due to the rank mismatch problem of the data matrix. In order to better capture and analyze unstable modes, this study introduced the Hankel matrix to expand ECG data and used it as the input of DMD to propose the Hankel dynamic mode decomposition (HDMD) method. Although HDMD has been applied in other physiological signal processing, this study is the first to successfully apply it to multi-lead ECG signal analysis. By optimizing the delay parameters of the Hankel matrix and retaining the modes that account for 90% of the singular value decomposition energy, we significantly improve the effectiveness of feature extraction. Each type of abnormal ECG signal data is classified on the PTB Diagnostic ECG database. The experimental results of the proposed model are compared with the direct use of DMD for modal extraction. The highest classification accuracy of multi-modal ECG signal data extracted by HDMD is more than 10% higher than that using DMD. At the same time, the mean squared error (MSE) of the reconstruction using HDMD is 0.282 lower than that of DMD, indicating a significant improvement in reconstruction accuracy. Further illustrating the proposed method, the HDMD method can better capture and analyze the unstable modes in the ECG signal, thereby significantly improving the accuracy and robustness of signal classification. Future work will focus on further optimizing the parameter selection of the HDMD model and exploring its application potential in real-time heart disease monitoring and early warning systems.