Proper Orthogonal Decomposition Reduced-order Model Research Articles

Comparison of the Atmospheric 200 hPa Jet’s Analyses between Proper Orthogonal Decomposition and Advanced Dynamic Mode Decomposition Method

In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.

A POD reduced-order model for resolving the neutron transport problems of nuclear reactor

Due to the high-dimensional integro-differential properties of the neutron transport equation (NTE) and the large-scale property of the nuclear reactor system, the detailed neutron transport simulation can be very time-consuming, which urges researchers to develop a fast and accurate technique. To improve this condition, this work develops a set of reduced-order models for the NTE (NTEROM) by combining the proper orthogonal decomposition (POD) and the Galerkin projection techniques to efficiently solve the neutron transport problems. Results of some typical benchmarks show that the proposed POD based NTEROM can estimate the neutron transport behaviors with high fidelity and outstanding computing efficiency. Furthermore, the prediction function of the proposed NTEROMs can be extended to approximately calculate the neutron transport problems with the parameters outside of the snapshots’ parameter range. This work can provide new perspectives and ideas on the issue of improving the efficiency of numerical calculations in nuclear reactor engineering.

Reduced-Order Modeling Based on Hybrid Snapshot Simulation

This paper presents a hybrid snapshot simulation methodology to accelerate the generation of high-quality data for proper orthogonal decomposition (POD) and reduced-order model (ROM) development. The entire span of the snapshot simulation is divided into multiple intervals, each simulated by either high-fidelity full-order model (FOM) or fast local ROM. The simulation then alternates between FOM and local ROM to accelerate snapshot data generation while maintaining the data fidelity and representation. Model switch is determined on-the-fly by evaluating several criteria that monitor the dominance of leading POD modes and ROM trajectory. The incremental singular value decomposition (iSVD) is employed to continuously update ROMs for enhanced accuracy and utilization. A global ROM broadly applicable to various online simulation is immediately available at the end of the simulation. The hybrid snapshot simulation demonstrates excellent accuracy ([Formula: see text] error) and 2.09–2.6[Formula: see text]X speedup relative to its traditional counterpart. The constructed ROMs also preserve salient accuracy ([Formula: see text] error). The results prove feasibility of the proposed method for robust and efficient snapshot data generation and ROM development.

A Rigorous Model Order Reduction Framework for Waste Heat Recovery Systems Based on Proper Orthogonal Decomposition and Galerkin Projection

A proper orthogonal decomposition (POD) and Galerkin projection-based model order reduction framework is developed for the evaporator heat exchanger in heavy-duty diesel engine organic Rankine cycle waste heat recovery system. The dynamics in the heat exchanger are first modeled by a finite-volume model, composed of highly nonlinear, coupled, partial differential equations, and then used to generate snapshots from which basis functions are defined. Reduced order models (ROMs) are then derived using the Galerkin projection approach. The accuracy and the execution time of different POD ROMs are evaluated against the high-fidelity finite-volume model. The results show that the POD ROM dimension can be selected based on the specific requirements of accuracy and computation time demanded by the intended model utilization. The proposed ROM framework can be utilized to generate ROMs with various dimensions for different purposes, such as estimator design, model-based control, and benchmark generation.

New Proper Orthogonal Decomposition Approximation Theory for PDE Solution Data

In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852--876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for nonorthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds for POD projections. We consider both the discrete and continuous cases of POD. We also apply our results to several example problems and show how the new results improve on previous work.

A novel POD reduced-order model based on EDFM for steady-state and transient heat transfer in fractured geothermal reservoir

A new proper orthogonal decomposition reduced-order model (POD-ROM) for fluid flow and heat transfer processes in a fractured geothermal reservoir is proposed. First, the primary governing equations are established based on the embedded discrete fracture model. Second, the POD-ROM is developed by projecting the discrete matrix equations instead of the primary governing equations onto the low-dimensional space spanned by the pressure and temperature basis functions. Compared with the typical POD modeling method, the new model reduction technique can easily deal with mass and energy transfer between the matrix and fractures, thus ensuring the mass and energy conservation of the whole system in the projection process. The new model does not require complex mathematical deduction and is easy to implement in practical applications. The accuracy and robustness of the model were carefully studied for three complex heat transfer cases. Calculation results indicated that the computational efficiency can be improved by at least 10–15 times while high reconstruction and prediction accuracy is maintained.

A Machine Learning Based Hybrid Multi-Fidelity Multi-Level Monte Carlo Method for Uncertainty Quantification

This paper focuses on reducing the computational cost of the Monte Carlo method for uncertainty propagation. Recently, Multi-Fidelity Monte Carlo (MFMC) method [46, 48] and Multi-Level Monte Carlo (MLMC) method [44, 29] were intro- duced to reduce the computational cost of Monte Carlo method by making use of low- fidelity models that are cheap to an evaluation in addition to the high-fidelity models. In this paper, we use machine learning techniques to combine the features of both the MFMC method and the MLMC method into a single framework called Multi-Fidelity- Multi-Level Monte Carlo (MFML-MC) method. In MFML-MC method, we use a hierarchy of proper orthogonal decomposition (POD) based approximations of high- fidelity outputs to formulate a MLMC framework. Next, we utilize Gradient Boosted Tree Regressor (GBTR) to evolve the dynamics of POD based reduced order model (ROM) [54] on every level of the MLMC framework. Finally, we incorporate MFMC method in order to exploit the POD ROM as a level specific low-fidelity model in the MFML-MC method. We compare the performance of MFML-MC method with the Monte Carlo method that uses either a high-fidelity model or a single low-fidelity model on two subsurface flow problems with random permeability field. Numerical re- sults suggest that MFML-MC method provides an unbiased estimator with speedups by orders of magnitude in comparison to Monte Carlo method that uses high-fidelity model only.

Numerical investigation of the POD reduced-order model for fast predictions of two-phase flows in porous media

PurposeThis paper aims to present an efficient IMPES algorithm based on a global model order reduction method, proper orthogonal decomposition (POD), to achieve the fast solution and prediction of two-phase flows in porous media.Design/methodology/approachThe key point of the proposed algorithm is to establish an accurate POD reduced-order model (ROM) for two-phase porous flows. To this end, two projection methods including projecting the original governing equations (Method I) and projecting the discrete form of original governing equations (Method II) are respectively applied to construct the POD-ROM, and their distinctions are compared and analyzed in detail. It is found the POD-ROM established by Method I is inapplicable to multiphase porous flows due to its failed introduction of fluid saturation and permeability that locate on the edge of grid cell, which would lead to unphysical results.FindingsBy using Method II, an efficient IMPES algorithm that can substantially speed up the simulation of two-phase porous flows is developed based on the POD-ROM. The computational efficiency and numerical accuracy of the proposed algorithm are validated through three numerical examples, and simulation results illustrate that the proposed algorithm displays satisfactory computational speed-up (one to two orders of magnitude) without sacrificing numerical accuracy obviously when comparing to the standard IMPES algorithm that without any acceleration technique. In addition, the determination of POD modes number, the relative errors of wetting phase pressure and saturation, and the influence of POD modes number on the overall performances of the proposed algorithm, are investigated.Originality/value1. Two projection methods are applied to establish the POD-ROM for two-phase porous flows and their distinctions are analyzed. The reason why POD-ROM is difficult to be applied to multiphase porous flows is clarified firstly in this study. 2. A highly efficient IMPES algorithm based on the POD-ROM is proposed to accelerate the simulation of two-phase porous flows. 3. Satisfactory computational speed-up (one to two orders of magnitude) and prediction accuracy of the proposed algorithm are observed under different conditions.

A constrained reduced-order method for fast prediction of steady hypersonic flows

A constrained reduced order model (ROM) based on proper orthogonal decomposition (POD) is proposed to achieve fast and accurate prediction of steady hypersonic flows. The proposed method addresses the convergence issue of the projection-based POD ROM which violates the boundary conditions by using a constrained Gauss-Newton iterative process. The constraints in the iteration are formulated by satisfying the physical boundary conditions. To achieve this, a weighting matrix constructed by an improved Gauss weighting function is adopted to determine the contribution of each cell to the constraint term. The proposed constrained reduced-order method is accelerated by a parallel algorithm based on message passing interface (MPI) and load balancing, which makes the method practical for prediction of complex flows with large memory cost. Fast predictions of hypersonic flows over the two-dimensional cylindrical blunt body and the three-dimensional reentry vehicle using the constrained reduced-order method show that the error is significantly smaller than that of the interpolation-based POD ROM and the projection-based POD ROM. Computing efficiency is increased by 2 ∼ 3 orders of magnitude compared to CFD.

A POD Coupled Adaptive DEIM (POD-ADEIM) Reduced-Order Model for Incompressible Multiphase Flow in Porous Media

The multiphase fluid flow in porous media is one of the most fundamental phenomena in various physical processes, such as oil/gas flow in reservoir, subsurface contamination dispersion, chemical separation, etc. Due to its importance, the efficient and accurate solution and prediction of multiphase flow in porous media is highly required in engineering applications and mechanism studies, which has been a research hot spot with increasing interest in recent years. However, the strong nonlinearity implicated in the multiphase flow model has brought great challenges for the computation and analysis. In addition, the permeability in Darcy-type pressure equation is always represented as a high contrast coefficient, which further complicates the simulation difficulty especially for high fidelity prediction and decision-making process. In this study, we attempt to apply one popular global model reduction method, namely Proper Orthogonal Decomposition (POD), to accelerate the simulation of incompressible multiphase flow in porous media. The cornerstone of POD is processing information from a sequence of snapshots (or instantaneous solutions) to identify a set of basis functions to construct a low-dimensional space, the most energetic part of these basis functions is then selected to represent the solution of the original system. In our work, one key point is how to establish the accurate POD reduced order model (ROM) for incompressible mul-tiphase porous flow problems. According to Li et al. [1], the traditional Galerkin projection method (directly project the original governing equation into the low-dimensional space spanned by basis functions) is incapable of introducing the fluid saturation and permeability located on the cell face in POD-ROM for incompressible multiphase flow whether in homogeneous or heterogeneous porous media, thus we adopt the matrix operation method (project the discrete governing equation instead of the original governing equation into the low-dimensional space spanned by basis functions) to build the POD-ROM in this study. Based on the proposed POD-ROM framework, we use the implicit method to solve the (non-) wetting-phase pressure and saturation equations. However, in the implicit solution of the pressure and saturation equations, the standard POD-ROM alone sometimes cannot yield an expected acceleration owing to two main reasons: (1) the nonlinearity between pressure and saturation is sensitive, thus the saturation would change largely even with slight pressure change. And the small error of pressure results would lead to a large deviation of saturation, which in turn exerts negative impacts on the computation of pressure; (2) the complexity for computing a projected nonlinear term in POD-ROM still depends on the dimension of the original full-order system, it is unavoidable in conventional POD-ROM. Therefore, more POD modes are needed to be used in order to achieve the prescribed numerical accuracy, which could worsen the speed up efficiency of proposed POD-ROM. To further reduce the computational complexity of POD-ROM to speed up the implicit solution of pressure and saturation equations, another model reduction method called Discrete Empirical Interpolation Method (DEIM) [2] is applied to treat the projected nonlinear term in the proposed incompressible multiphase porous flow POD-ROM. Different from previous studies, we develop an adaptive DEIM (ADEIM) in this work to achieve a more flexible selection of the interpolation points. Combined the POD-ROM with ADEIM, a POD-ADEIM-ROM for incompressible multiphase flow in porous media is established. The computational efficiency and numerical accuracy of the proposed model is validated through several representative numerical examples. Numerical results indicate that an outstanding acceleration (2~3 orders of magnitude faster) without sacrificing numerical accuracy significantly is obtained from the proposed model when comparing to the traditional solution method that without any acceleration technique. In addition, the effects of POD mode number, the distribution of permeability field, and the mesh density on the overall performances of the proposed model are investigated in detail.

BFC-POD-ROM Aided Fast Thermal Scheme Determination for China’s Secondary Dong-Lin Crude Pipeline with Oils Batching Transportation

Since the transportation task of China’s Secondary Dong-Lin crude pipeline has been changed from Shengli oil to both Shengli and Oman oils, its transportation scheme had to be changed to “batch transportation”. To determine the details of batch transportation, large amounts of simulations should be performed, but massive simulation times could be costly (they can take hundreds of days with 10 computers) using the finite volume method (FVM). To reduce the intolerable time consumption, the present paper adopts a “body-fitted coordinate-based proper orthogonal decomposition reduced-order model” (BFC-POD-ROM) to obtain faster simulations. Compared with the FVM, the adopted method reduces the time cost of thermal simulations to 2.2 days from 264 days. Subsequently, the details of batch transportation are determined based on these simulations. The Dong-Lin crude oil pipeline has been safely operating for more than two years using the determined scheme. It is found that the field data are well predicted by the POD reduced-order model with an acceptable error in crude oil engineering.

Development of POD reduced-order model and its closure scheme for 2D Rayleigh–Bénard convection

• It is the first time to investigate Rayleigh–Bénard convection at high Ra flow based on POD-Galerkin method. • The reduced-order model with a closure model is built for Rayleigh–Bénard convection. • It is the first time to study the short-time and long-time prediction of Rayleigh–Bénard convection. Reduced-order model (ROM) based on proper orthogonal decomposition (POD) is a fast computational fluid dynamics (CFD) method and has been widely applied to pure flow or heat conduction problems in the past. In this paper, the typical 2D Rayleigh–Bénard convection (RBC) in a square cavity was set as a research target. Firstly, the POD-ROM of 2D RBC problem was constructed at Ra = 10 7 , Pr = 0.71. Combining with direct numerical simulation (DNS) databases, a closure model (CM) was then proposed to correct the evolution process of POD-ROM. Based on the proposed CM, we realized the prediction of flow evolution for a new flow case under the parameters different from that used to get its POD eigenmodes. It showed that the proposed POD-ROM with CM could be able to predict the dynamics of new flow cases. Moreover, the corresponding method proposed in the present study can be also easily extended to other types of flow-heat coupling problems, such as natural heat convection, etc.

HDG–POD reduced order model of the heat equation

We propose a new hybridizable discontinuous Galerkin (HDG) model order reduction technique based on proper orthogonal decomposition (POD). We consider the heat equation as a test problem and prove error bounds that converge to zero as the number of POD modes increases. We present 2D and 3D numerical results to illustrate the convergence analysis.

A POD reduced-order model for wake steering control

Wake steering by active yawing of upstream wind turbines is a promising wind plant control technique. To enable the development of model-based wind plant control methods, there is a need for models that can marry the contrasting requirements of good fidelity and low computational cost. This paper presents a reduced-order model (ROM) obtained by directly compressing high-fidelity computational fluid dynamics (CFD) simulation data using the proper orthogonal decomposition (POD) method. At first, simulations of wake-interacting wind turbines are obtained for time-varying yaw settings using the lifting-line large-eddy simulation (LES) code SOWFA. Next, a ROM is synthesized from the CFD transient simulations, obtaining a discrete-time state-space model that captures the dominant dynamics of the underlying high-fidelity model with only a reduced number of states. The ROM is optionally augmented with a Kalman filter, which feeds back turbine power measurements from the plant to the model, enhancing its accuracy. Results obtained in realistic turbulent conditions show a good agreement between high-fidelity CFD solutions and the proposed POD-based ROM in terms of wake behavior and power output of waked turbines. Additionally, the ROM presents acceptable results when compared to wind tunnel experiments, including the capability of the model to partially correct for an intentionally built-in model mismatch.

A streamline derivative POD-ROM for advection-diffusion-reaction equations

We introduce a new streamline derivative projection-based closure modeling strategy for the numerical stabilization of Proper Orthogonal Decomposition-Reduced Order Models (POD-ROM). As a first preliminary step, the proposed model is analyzed and tested for advection-dominated advection-diffusion-reaction equations. In this framework, the numerical analysis for the Finite Element (FE) discretization of the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Numerical tests for advection-dominated regime show the effciency of the proposed method, as well the increased accuracy over the standard POD-ROM that discovers its well-known limitations very soon in the numerical settings considered, i.e. for low diffusion coeffcients.

Calibration of reduced-order model for a coupled Burgers equations based on PC-EnKF

The proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) are applied to coupled Burgers equations to develop its reduced-order model (ROM) by the Galerkin projection. A calibrated POD ROM is developed in the current study through adding and multiplying a set of time-dependent random parameters to recover the loss of accuracy due to the truncation of the POD modes. Calibrating the ROM becomes essentially a high-dimensional statistical inverse inference problem. To reduce the computational effort, the polynomial chaos based ensemble Kalman filter (PC-EnKF) is adopted in this work. By using a sparse optimization algorithm, a sparse PC expansion is obtained to facilitate further calculation of statistical moments used in ensemble Kalman filter. We apply the well-defined calibrated POD ROM for the coupled Burgers equations with the Reynolds numberRe= 10 000. The numerical results show that the PC-EnKF method is efficient in reducing the uncertainty included in the initial guess of input parameters and feasible in correcting the behavior of the POD ROM. The study suggests that the PC-EnKF is quite general as a calibration tool for calibrating the POD ROM.

Using proper orthogonal decomposition to solve heat transfer process in a flat tube bank fin heat exchanger

Proper orthogonal decomposition (POD) reduced-order model can save computing time by reducing the dimension of physical problems and reconstructing physical fields. It is especially suitable for large-scale complex problems in engineering, such as ground heat utilization, sea energy development, mineral exploitation, multiphase flow and flow and heat transfer with complex structure. In this paper, the POD reduced-order model was used to calculate the heat transfer in a flat tube bank fin heat exchanger. The calculating results of the finite volume method (FVM) were adopted as the snapshot samples. Singular value decomposition method was used to decompose the samples to obtain a series of bases and corresponding coefficients on sampling conditions. With these coefficients, interpolation method was used to calculate the coefficients on predicting conditions. And the physical field has been reconstructed using the bases and the interpolated coefficients directly. In the calculation of heat transfer unit of flat tube fin heat exchanger, air-side Reynolds number, transverse tube spacing and the fin spacing were chosen as the variables. The results obtained by the POD method are in good agreement with the results calculated by the FVM. Moreover, the POD reduced-order model presented in this paper is more advantageous in comparison with the FVM in terms of accuracy, suitability, and computational speed. Cited as : Wang, Y., Xia, X., Wang, Y., et al. Using proper orthogonal decomposition to solve heat transfer process in a flat tube bank fin heat exchanger. Advances in Geo-Energy Research, 2017, 1(3): 158-170, doi: 10.26804/ager.2017.03.03.

Time-resolved flow reconstruction with indirect measurements using regression models and Kalman-filtered POD ROM

This article is devoted to the estimation of time-resolved particle image velocimetry (TR-PIV) flow fields using a time-resolved point measurements of a voltage signal obtained by hot-film anemometry. A multiple linear regression model is first defined to map the TR-PIV flow fields onto the voltage signal. Due to the high temporal resolution of the signal acquired by the hot-film sensor, the estimates of the TR-PIV flow fields are obtained with a multiple linear regression method called orthonormalized partial least squares regression (OPLSR). Subsequently, this model is incorporated as the observation equation in an ensemble Kalman filter (EnKF) applied on a proper orthogonal decomposition reduced-order model to stabilize it while reducing the effects of the hot-film sensor noise. This method is assessed for the reconstruction of the flow around a NACA0012 airfoil at a Reynolds number of 1000 and an angle of attack of $${20}^{\circ }$$ . Comparisons with multi-time delay-modified linear stochastic estimation show that both the OPLSR and EnKF combined with OPLSR are more accurate as they produce a much lower relative estimation error, and provide a faithful reconstruction of the time evolution of the velocity flow fields.

Open-loop control of cavity noise using Proper Orthogonal Decomposition reduced-order model

Flow over open cavities is mainly governed by a feedback mechanism due to the interaction of shear layer instabilities and acoustic forcing propagating upstream in the cavity. This phenomenon is known to lead to resonant tones that can reach 180 dB in the far-field and may cause structural fatigue issues and annoying noise emission. This paper concerns the use of optimal control theory for reducing the noise level emitted by the cavity. Boundary control is introduced at the cavity upstream corner as a normal velocity component. Model-based optimal control of cavity noise involves multiple simulations of the compressible Navier–Stokes equations and its adjoint, which makes it a computationally expensive optimization approach. To reduce the computational costs, we propose to use a reduced-order model (ROM) based on Proper Orthogonal Decomposition (POD) as a surrogate model of the forward simulation. For that, a control input separation method is first used to introduce explicitly the control effect in the model. Then, an accurate and robust POD ROM is derived by using an optimization-based identification procedure and generalized POD modes, respectively. Since the POD modes describe only velocities and speed of sound, we minimize a noise-related cost functional characteristic of the total enthalpy unsteadiness. After optimizing the control function with the reduced-order model, we verify the optimality of the solution using the original, high-fidelity model. A maximum noise reduction of 4.7 dB is reached in the cavity and up to 16 dB at the far-field.

POD-Galerkin reduced-order model for viscoelastic turbulent channel flow

ABSTRACTIn this work, with elasticity governed by the Giesekus constitutive equation, a proper orthogonal decomposition (POD) reduced-order model of viscoelastic turbulent channel flow is established for the first time. The established reduced-order model is based on small sets of basis functions from the POD of the sampling data obtained by direct numerical simulation (DNS) for the studied flow. The POD reduced-order model is tested on cases with that are different from the samplings for viscoelastic turbulent channel flow. The results show that the errors for root-mean-square (rms) velocity fluctuations are significant at the top and bottom walls. It is found that each basis function plays an important role in describing the studied turbulence which makes it unmanageable to obtain accurate velocity field (including mean velocity and velocity fluctuations) through solving the reduced-order model. It is of necessity to take all the basis functions into consideration to depict the flows more accurately. However, the mean velocity obtained from the reduced-order model is of high precision, which states that the POD-based reduced-order model is a potential approach to obtain an accurate mean velocity field for viscoelastic turbulent flow, which has great significance in academic study as well as engineering. The calculation speed of the established reduced-order model is much faster than that of DNS, which indicates that the POD is a highly efficient way of obtaining the statistic characteristics, such as mean velocity in turbulent channel flow.