Prediction Algorithm of Mobile User Channel Capacity Based on Dynamic Mode Decomposition

Stable Dynamic Mode Decomposition Algorithm for Noisy Pressure-Sensitive-Paint Measurement Data

Fast simulation algorithms based on reduced-order modeling have been developed in order to facilitate large-scale and complex computationally intensive reservoir simulation and optimization. Methods like proper orthogonal decomposition (POD) and Dynamic Mode Decomposition (DMD) have been successfully used to efficiently capture and predict the behavior of reservoir fluid flow. Non-intrusive techniques (e.g., DMD), are especially attractive as it is a data-driven approach that do not require code modifications (equation free). In this paper, we will further enhance the application of the DMD, by investigating sparse approximations of the snapshots. This is particularly useful when there is a limited number of sparse measurements as in the case of reservoir simulation. The approach taken here is the snapshot-based model reduction, whereby one computes a sequence of reservoir simulation solutions (e.g., pressures and water saturations in the case of two-phase flow model) forming a big data matrix – we call this the offline step - that is used to compute basis for representing the states of the system for different input parameters – the online step. The selection of these few basis is the core of the model reduction methods. DMD selects the basis and apply the reduction without knowledge of the inner works of the reservoir simulator, as opposed to the POD methods. Sparse DMD has been introduced recently to determine the subset of the DMD models that has the most profound influence on the quality of the approximation of the snapshot sequence. Two model reduction process are involved. One is offline process, which does not require running the simulator but rather predicting future behavior with linear combination of DMD modes. The other online process incorporates sparsity DMD modes in numerical simulator to release the burden of linear matrix solver. We first show the methodology applied to a 3-D single phase flow problem. Here we show the DMD modes and its physical interpretations, and then move to two phase flow for 2-D heterogeneous reservoir using the SPE-10 benchmark. Both online and offline process will be used for evaluation. We observe that with a few DMD modes we can capture the behavior of the reservoir models. Sparse DMD leads to the optimal selection of the few DMD modes. We also assess the trade-offs between problem size and computational time for each reservoir model. The novelty of our method is the application of sparse DMD, which is a data-driven technique and the ability to select few optimal basis for the case of reservoir simulation.

The computational simulation of fluid flows over structures still is a major research area in fluid mechanics. Because of the multiscale nature of many of the application models and consequent large amount of data, these simulations are computationally expensive, but can benefit from modern Reduced Order Models and data-driven methods. In this work, Dynamic Mode Decomposition (DMD) and its recent variation, Piecewise DMD (pDMD), are presented and compared in terms of dynamic modes extracted from the data, accuracy in reconstructing an approximation for the original dataset as a reduced order model and, most importantly, computational cost. The pDMD method is shown to be a variation of the traditional DMD that aims to improve and overcome some of the caveats of the standard version. This variation consists in decomposing the entire data into smaller datasets and applying a linear mapping independently on each one of these subsets instead of calculating a global linear fitting. Basically, it is an application of multiple DMD on small subsets instead one DMD over the whole data. Piecewise DMD is a simple and elegant idea that is based on the "divide and conquer" approach well known in the numerical analysis literature. Even though pDMD is new and should be carefully and extensively tested, it can be considered as a promising improvement over the standard DMD. The preliminary results presented in this work show how DMD can capture the dynamics and accurately reconstruct the simulation data and how pDMD can provide more accurate results when traditional DMD reaches its limitations, capture the specific dynamics of different stages of transient flows, and reduce the computational cost by 90% for a two-dimensional flow over a cylinder when compared to standard DMD. Future applications of Reduced Order Models using both DMD and pDMD include future state predictions, computationally cheap parametric simulations, and qualitative dynamic analysis of fluid flows.

Four dynamic mode decomposition (DMD) methods are used to analyze a simulation of the phonation onset carried out by in-house solver based on the nite element method. The dataset consists of several last periods of the flow-induced vibrations of vocal folds (VFs). The DMD is a data-driven and model-free method typically used for finding a low-rank representation of a high-dimensional system. In general, the DMD decomposes a given dataset to modes with mono-frequency content and associated complex eigenvalues providing the growth/decay rate that allows a favourable physical interpretation and in some cases also a short-term prediction of system behaviour. The disadvantages of the standard DMD are non-orthogonal modes and sensitivity to an increased noise level which are addressed by following DMD variants. The recursive DMD (rDMD) is an iterative DMD decomposition producing orthogonal modes. The total least-square DMD and the higher order DMD (hoDMD) are methods substantially reducing a high DMD sensitivity to noise. All methods identi ed very similar DMD modes as well as frequency spectra. Substantial difference was found in the real part of the spectra. The nal dataset reconstruction is the most accurate in the case of the recursive variant. The higher order DMD method also outperforms the standard DMD. Thus the rDMD and the hoDMD decompositions are promising to be used further for the parametrization of a VF motion.

Dynamic mode decomposition (DMD) is a powerful data-driven modal decomposition technique that extracts spatiotemporal coherent structures: a useful process in flow diagnostics and future state estimation of complex nonlinear flow phenomena. Transonic shock buffet is a complicated phenomenon, and modal decomposition techniques such as DMD provide significant insight into its complicated flow physics; but, often, flowfield data are corrupted because of various sources of noise due to the presence of outliers or the absence of critical data components. Therefore, noise corruption renders the modal decomposition inaccurate, and thereby not useful. In this paper, two sources of noise have been considered: simple white noise, and complex salt-and-pepper-type spurious noise. Various DMD techniques including standard DMD, forward–backward DMD, total-least-squares DMD, higher-order DMD, and robust DMD have been implemented. Their effectiveness and limitations in countering noise corruption have been investigated systematically. In the case of white noise corruption, forward–backward DMD, total-least-squares DMD, and higher-order DMD capture the buffet frequency and growth rate with sufficient accuracy, whereas the latter outperforms the other two when the noise variance level is above 5%. In the case of spurious noise, robust DMD handles noise corruption efficiently, with surprisingly high values of pixel corruption of up to 30%.

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.

Analytical and computational studies of reacting flows are extremely challenging due in part to nonlinearities of the underlying system of equations and long-range coupling mediated by heat and pressure fluctuations. However, many dynamical features of the flow can be inferred through low-order models if the flow constituents (e.g., eddies or vortices) and their symmetries, as well as the interactions among constituents, are established. Modal decompositions of high-frequency, high-resolution imaging, such as measurements of species-concentration fields through planar laser-induced florescence and of velocity fields through particle-image velocimetry, are the first step in the process. A methodology is introduced for deducing the flow constituents and their dynamics following modal decomposition. Proper orthogonal (POD) and dynamic mode (DMD) decompositions of two classes of problems are performed and their strengths compared. The first problem involves a cellular state generated in a flat circular flame front through symmetry breaking. The state contains two rings of cells that rotate clockwise at different rates. Both POD and DMD can be used to deconvolve the state into the two rings. In POD the contribution of each mode to the flow is quantified using the energy. Each DMD mode can be associated with an energy as well as a unique complex growth rate. Dynamic modes with the same spatial symmetry but different growth rates are found to be combined into a single POD mode. Thus, a flow can be approximated by a smaller number of POD modes. On the other hand, DMD provides a more detailed resolution of the dynamics. Two classes of reacting flows behind symmetric bluff bodies are also analyzed. In the first, symmetric pairs of vortices are released periodically from the two ends of the bluff body. The second flow contains von Karman vortices also, with a vortex being shed from one end of the bluff body followed by a second shedding from the opposite end. The way in which DMD can be used to deconvolve the second flow into symmetric and von Karman vortices is demonstrated. The analyses performed illustrate two distinct advantages of DMD: (1) Unlike proper orthogonal modes, each dynamic mode is associated with a unique complex growth rate. By comparing DMD spectra from multiple nominally identical experiments, it is possible to identify "reproducible" modes in a flow. We also find that although most high-energy modes are reproducible, some are not common between experimental realizations; in the examples considered, energy fails to differentiate between reproducible and nonreproducible modes. Consequently, it may not be possible to differentiate reproducible and nonreproducible modes in POD. (2) Time-dependent coefficients of dynamic modes are complex. Even in noisy experimental data, the dynamics of the phase of these coefficients (but not their magnitude) are highly regular. The phase represents the angular position of a rotating ring of cells and quantifies the downstream displacement of vortices in reacting flows. Thus, it is suggested that the dynamical characterizations of complex flows are best made through the phase dynamics of reproducible DMD modes.

Dynamic mode (DM) decomposition decomposes spatiotemporal signals into basic oscillatory components (DMs). DMs can improve the accuracy of neural decoding when used with the nonlinear Grassmann kernel, compared to conventional power features. However, such kernel-based machine learning algorithms have three limitations: large computational time preventing real-time application, incompatibility with non-kernel algorithms, and low interpretability. Here, we propose a mapping function corresponding to the Grassmann kernel that explicitly transforms DMs into spatial DM (sDM) features, which can be used in any machine learning algorithm. Using electrocorticographic signals recorded during various movement and visual perception tasks, the sDM features were shown to improve the decoding accuracy and computational time compared to conventional methods. Furthermore, the components of the sDM features informative for decoding showed similar characteristics to the high-γ power of the signals, but with higher trial-to-trial reproducibility. The proposed sDM features enable fast, accurate, and interpretable neural decoding.

The Strait of Gibraltar is a region characterized by intricate oceanic submesoscale features arising from the interplay of topography, tidal forcing, hydrodynamic instabilities, and strongly nonlinear internal hydraulic processes, all governed by the nonlinear equations of fluid motion. In this study, we aim to uncover the underlying physics of these phenomena as represented in the 3D Massachusetts Institute of Technology (MIT) General Circulation Model simulations, including simulated waves, eddies, and gyres. To achieve this, we employ dynamic mode decomposition (DMD) to break down simulation snapshots into Koopman modes, with distinct exponential growth/decay rates and oscillation frequencies. Our objectives encompass evaluating DMD’s efficacy in capturing known features, unveiling new elements, ranking modes, and exploring order reduction. We also introduce modifications to enhance DMD’s numerical accuracy and the robustness of its eigenvalues. DMD analysis yields a comprehensive understanding of flow patterns, internal wave formation, and the dynamics and meandering behaviors within the Strait of Gibraltar, the formation of the secondary western Alboran Gyre, and the propagation of Kelvin and coastal-trapped waves along the African coast. In doing so, it significantly advances our comprehension of intricate oceanographic phenomena and underscores the immense utility of DMD as an analytical tool for such complex datasets, suggesting that DMD could serve as a valuable addition to the toolkit of oceanographers. Significance Statement A challenge arising in all branches of geophysics is making sense of increasingly large datasets describing complex processes. In physical oceanography, these data usually come from high-resolution models and observations. Reduced-order models such as dynamic mode decomposition (DMD) have the potential to identify key processes and interactions through the synthesis of a data-based model with relatively few degrees of freedom. Because of its connection with Koopman operator theory, DMD is also able, in principle, to deal with data describing nonlinear processes. In this work, we test the ability of DMD to describe the essential physics exhibited in a complicated dataset produced by a model of the ocean circulation in the Strait of Gibraltar and western Mediterranean, a region that contains striking, time-dependent, and often nonlinear features, mostly driven or modulated by tides. We carefully describe the connection to Koopman theory and the step-by-step DMD algorithm, and we present a procedure for singling out the most robust of the DMD modes. We then show how particular modes contain information about specific physical processes including modulation of the two-layer exchange flow in the strait, the generation of internal waves, meandering of the Atlantic jet, and the generation of coastal-trapped waves. Some of these features are already well known, but others such as the jet meandering have received little attention.

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 221411, “Accelerated Calibration and CO2 Plume Tracking at the Illinois Basin Decatur Project: A Dynamic Mode-Decomposition and Data-Assimilation Approach,” by James Omeke, SPE, Kassem Alokla, SPE, and Dimitrios Voulanas, SPE, Texas A&M University, et al. The paper has not been peer reviewed. _ Addressing climate change through carbon capture and storage (CCS) technologies requires advanced computational methodologies for subsurface carbon-dioxide (CO2) storage monitoring. This study focuses on the Illinois Basin Decatur Project (IBDP), a CCS demonstration pilot aimed at CO2 injection into a deep saline reservoir. A novel framework combining dynamic mode decomposition (DMD), a data-driven model-reduction technique, with direct data assimilation is introduced to streamline the calibration of CO2 plume evolution models. This approach enhances rapid tracking and overcomes the computational challenges of traditional high-fidelity numerical reservoir simulations known as the full-order model (FOM). Introduction DMD represents a superior approach for flow in porous media compared with most reduced-order models because of its ability to capture complex flow dynamics effectively. DMD excels in identifying key dynamic features, spatial and temporal scales, and localized regions crucial for flow behavior. It offers a data-driven method that can reproduce flow phenomena accurately with high fidelity, even in multiphase and multiscale heterogeneous porous-media scenarios. Additionally, advancements such as sparse DMD and local DMD enhance the accuracy and efficiency of DMD models. The adaptability and robustness of DMD in capturing intricate flow structures make it a preferred choice for studying flow dynamics in porous media during CO2 storage. The foundation of DMD lies in its ability to approximate the Koopman operator. The DMD technique and the Koopman operator are detailed in the complete paper. Workflow Description The steps of the workflow are detailed in the complete paper, including associated equations. Identifying permeability as the principal uncertainty in multilevel pressure measurements, the authors conducted six simulation cases using various permeability multipliers within the IBDP. These simulations, integral to the history-matching process of the used FOM, were designed to span the expected range of uncertainty. Each simulation was specifically influenced by a chosen permeability multiplier. This subsection of the complete paper details the following steps: - Reconstruction of the reduced-order model (ROM) of each FOM case using DMD - Interpolation of DMD modes and eigenvalues for pressure-trend prediction - Kalman filter optimization of permeability multipliers - History-matching of the FOM using the optimized permeability multiplier and construction of the ROM

Fluid flows in porous media exhibit diverse spatially and temporally complex structures arising from intricate interactions between fluid motion and solid interfaces/flow paths. While numerous experimental studies have aimed to quantify these complex flow structures, recent technological advancements have facilitated the widespread adoption of data-driven analysis approaches. These approaches offer invaluable insights into flow physics and can also be used for low-order modeling. The two primary objectives of this study are to apply Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) to velocity flow field data to predict the flow phenomena, develop low-order models, and apply DMD onflow visualization Planar Laser-Induced Fluorescence (PLIF) data to identify the temporal evolution of flow structures for recirculation region in porous media. An in-house refractive index-matched flow loop system was designed for the experiments, where a gear pump was used for fluid circulation, and a syringe pump was used for dye injection. The Reynolds numbers (Re) and the average porosity for the experiment were 42 and 0.39. Irregularly shaped 3 mm Tetrafluoroethylene hexafluoropropylene vinylidene fluoride (THV) beads were used as the porous media. 2D-PIV measurements were made at the center plane of the models to obtain a velocity flow field, which revealed a recirculation region. Applying POD in the PIV dataset, the energy distribution in the first twenty-five POD modes revealed key insights at Re. First twenty-five POD modes captured 74% of the recirculation region’s total energy. DMD provided dynamic insights into the velocity flow field, including frequency and growth rate associated with the frequencies in the recirculation region. Dominant frequencies were obtained using DMD, which were related to the occurrence of vortex in velocity flow field data. Low-order models were created using twenty-five POD modes and three DMD with modes with mean flow mode. The reconstructed velocity flow field data was compared with PIV velocity flow field data, resulting in an average Root Mean Square Error (RMSE) of 0.06% for POD across all regions and 1.37% for DMD low-order models for the recirculation region. Additionally, PLIF measurements using Rhodamine-B dye were used for flow visualization in the recirculation region. PLIF data offered qualitative information complementary to PIV. DMD results from flow visualization provided information about the flow dynamics of dye at Re = 42 in the recirculation region. The frequencies associated with the growth rate closer to zero in the frequency trend could capture the diffusion of dye, and the remaining frequencies captured the advection of dye. In conclusion, this study highlights the applicability of POD and DMD in unraveling the intricate dynamics of fluid flow in porous media, paving the way for future advancements in understanding and modeling these complex systems.

Correcting noisy dynamic mode decomposition with Kalman filters

On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis

Dynamic mode decomposition (DMD) effectively captures the growth and frequency characteristics of individual modes, enabling the construction of reduced-order models for flow evolution, thereby facilitating the prediction of fluid dynamic behavior. However, DMD's predictive accuracy is inherently constrained by its inability to inherently incorporate physical principles. Therefore, for dense particulate pipe flows with complex flow mechanisms, we introduce a physics-informed dynamic mode decomposition (PIDMD) approach, which augments the purely data-driven DMD framework by incorporating the conservation of mass as a constraint. This ensures that the extracted dynamic modes adhere to known physical principles. Initially, we apply the DMD to reconstruct and predict the velocity field, comparing the results against benchmark computational fluid dynamics-discrete element method (CFD-DEM) simulations. Findings indicate that while DMD can reconstruct the flow field simulated by CFD-DEM and provide predictions of future flow states, its predictive accuracy gradually deteriorates over time. Next, we utilize both PIDMD and DMD to reconstruct and predict particle volume fraction, evaluating both models based on CFD-DEM outcomes. The results indicate that both PIDMD and DMD can predict particle aggregation toward the center, but PIDMD provides more accurate predictions regarding the size of particle aggregations and their distribution near the tube wall. Furthermore, the average prediction error for particle volume fraction using PIDMD is 6.54%, which is lower than the error of 13.49% obtained by DMD. Both qualitative and quantitative comparisons highlight the superior predictive capability of PIDMD. The methodology developed in this study provides valuable insights for high-precision predictions of particulate flows.

A rotating detonation combustor (RDC) is a novel approach to achieving pressure gain combustion. Due to the steady propagation of the detonation wave around the perimeter of the annular combustion chamber, the RDC dynamic behavior is well suited to analysis with reduced-order techniques. For flow fields with such coherent aspects, the dynamic mode decomposition (DMD) has been shown to capture well the dominant oscillatory features corresponding to stable limit-cycle or quasi-periodic behavior within its dynamic modes. Details regarding the application of the technique to RDC—such as the number of frames, the effect of subtracting the temporal mean from the processed dataset, the resulting dynamic mode shapes, and the reconstruction of the dynamics from a reduced set of dynamic modes—are analyzed and interpreted in this study. The DMD analysis is applied to two commonly observed operating conditions of rotating detonation combustion, viz., (1) a single spinning wave with weak counter-rotating waves and (2) a clapping operating mode with two counter-propagating waves at equal speed and strength. We show that care must be taken when applying DMD to RDC datasets due to the presence of standing waves (expressed as either counter-propagating azimuthal waves or longitudinal pulsations). Without accounting for these effects, the reduced-order reconstruction fails using the standard DMD approach. However, successful application of the DMD allows for the reconstruction and separation of specific wave modes, from which models of the stabilization and propagation of the primary and counter-rotating waves can be derived.