Adaptive Proper Orthogonal Decomposition Research Articles

An augmented subspace based adaptive proper orthogonal decomposition method for time dependent partial differential equations

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models.

A hybrid numerical model for horizontal ground heat exchanger

The horizontal ground heat exchanger (HGHE) possesses a complicated heat transfer mechanism as its extensive spatial scale, long operational duration, and vulnerability to meteorological conditions. Consequently, the long-period simulations of HGHE usually involve significant computational costs, posing challenges for its dynamic optimization implementation. Therefore, this study initially established and validated a conventional numerical (full-order) model for HGHE as the reference model. Subsequently, a hybrid model was developed using the proposed adaptive proper orthogonal decomposition (POD) method. By analyzing the influential characteristics, the study identified the solution strategy and the key parameter values for adaptive POD, followed by the generality tests. The hybrid model proved to successfully mitigate the issue of error accumulation commonly associated with native POD extrapolation. Finally, employing a long-running engineering case study, the accuracy and the solution efficiency of the hybrid model were compared against those of the conventional (full-order) model. The results demonstrated that the hybrid model maintained computational accuracy at a comparable level while exhibiting a computational efficiency 326 % higher than that of the conventional (full-order) model, without requiring additional computational resources. This study can provide efficient modeling support for the dynamic optimization design of HGHE.

A reduced order model based on adaptive proper orthogonal decomposition incorporated with modal coefficient learning for digital twin in process industry

The digital twin (DT) technology provides a viable and promising direction for improving the level of the production status monitoring and the overall product quality in various fields. However, the accuracy of working condition identification, the timeliness of process adjustment, and the stability of product quality are put forward higher requirements in the process industry, which is characterized by nonlinear, large-scale, and dynamic complex systems. Therefore, it still remains a tricky challenge to construct and maintain an effective and accurate DT model in the process industry. A reduced order model (ROM) with the adaptive updating ability is proposed. The adaptive proper orthogonal decomposition (APOD) is adopted to achieve the continuous iteration and the adaptive optimization of the reduced basis set. Correspondingly, an adaptive learning algorithm based on the least squares support vector regression (LS-SVR) is developed to quickly obtain the modal coefficients and effectively circumvent the prohibitively high computational cost. In this way, the physical field of interest is expressed in a low-dimensional approximation with a high accuracy. The effectiveness of the method is verified by a case study in the process industry. Results show that the proposed model displays a high-precision fitting and a significant time saving for the full order model (FOM).

Parametric Reduced Order Models for wave propagation in 1D media containing defects

This paper presents analyses of reduced order models (ROM) for wave propagation in media containing material defects. Parametric ROMs based on POD (Proper Orthogonal Decomposition) are used here to take into account parameters that characterize local defects such as center position, extension and magnitude (localized loss of stiffness). Local and adaptive (based on the Grassmann interpolation) POD strategies are used. Sequential analyses regarding parametric dependence with one, two and three parameters are performed along with accuracy and speed up as a function of model order. Furthermore, the accuracy of ROMs is also investigated for uncertainty quantification analyses. Our results show that the choice of ROM technique is not straightforward and the best solution, which is case dependent, is not always the most complex one.

Inverse Design and Optimization of Low Specific Speed Centrifugal Pump Blade Based on Adaptive POD Hybrid Model

To improve the prediction accuracy of the surrogate model and reduce the calculation cost for hydraulic optimization design of centrifugal pump impeller, an inverse design and optimization method based on adaptive proper orthogonal decomposition (APOD) hybrid model was proposed. Initial samples were designed by perturbing blade control parameters of the original model. The samples were classified using the K-means clustering algorithm, and the adaptive samples were selected according to the category of the objective sample. The snapshot set is composed of blade shape parameters and the CFD flow field data in impeller, which is decomposed into a linear combination of orthogonal bases by the proper orthogonal decomposition (POD) method to predict the objective parameters. According to the objective load distribution, the low specific speed centrifugal pump was inversely designed by using the APOD model, and its initial blade was obtained. And then, the flow field corresponding to disturbed blade shape was predicted using the APOD method, so as to evaluate the gradient of the objective function to design variables. Finally, the initial blade was optimized by the gradient descent method. The results show that the APOD hybrid model method can be employed to accomplish the blade inverse design and the flow field prediction in the optimization design of centrifugal impeller, which significantly reduces the numerical calculation cost and improves the accuracy of the flow field prediction.

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a two-dimensional wake flow and a three-dimensional flow generated by a flapping insect computed with direct numerical simulation.

Adaptive proper orthogonal decomposition for large scale reliable soil moisture estimation

A major challenge in automatic irrigation of extensive agricultural fields is large scale soil moisture monitoring. Proper orthogonal decomposition (POD) is a widespread data-driven dimension reduction technique which can be combined with QR pivoting method for estimation of high-dimensional signals and optimal sensor placement. However, it requires computation of tailored basis functions which should be extracted from known training data. This is feasible for problems with constant features such as face recognition. However, using fixed bases (and probably fixed sensor selection) may not be an appropriate approach for estimation of signals with time-variant features. This paper demonstrates that the POD algorithm can be implemented adaptively in a reliable fashion to address this issue. The probability of the mean estimation error to remain within a predetermined threshold is studied and theoretical results on permitted update intervals are provided. The adaptive tailored basis and sensor selection increases suitability to large scale applications with time-variant features. The performance of the algorithm on a NASA soil moisture dataset is analyzed.

Lyapunov based on-line model reduction and control of semilinear dissipative distributed parameter systems with minimum feedback information

We focus on the Lyapunov-based output feedback control problem for a class of distributed parameter systems with spatiotemporal dynamics described by input-affine linear and semilinear dissipative partial differential equations (DPDEs). The control problem is addressed via model order reduction. Galerkin projection is applied to discretize the DPDE and derive low-dimensional reduced order models (ROMs). The empirical basis functions needed for this discretization are recursively computed using adaptive proper orthogonal decomposition (APOD). To update the basis functions during process operation, APOD needs measurements of the system state’s complete profile (called snapshots) at revision times. This paper’s main objective is to minimize the demand for snapshots from the spatially distributed sensors by the control structure while maintaining closed-loop stability and performance. A control Lyapunov function is defined, and its value is monitored as the system evolves. Only when the value violates a closed-loop stability threshold, snapshots are requested for a brief period by APOD after which the ROM is updated, and the controller is reconfigured. The proposed approach is applied to stabilize the Kuramoto–Sivashinsky equation.

Adaptive POD-Galerkin Technique for Reservoir Simulation and Optimization

In this work, a novel method with an adaptive functional basis for reduced order models (ROM) based on proper orthogonal decomposition (POD) is introduced. The method is intended to be applied in particular to hydrocarbon reservoir simulations, where a range of varying boundary conditions must be explored. The proposed method allows us to update the POD functional basis constructed for a specific problem setting in order to match varying boundary conditions, such as modified well locations and geometry, without the necessity to recalculate each time the whole set of basis functions. Such an adaptive technique allows us to significantly reduce the number of snapshots required to calculate the new basis, and hence reduce the computational cost of the simulations. The proposed method was applied to a two-dimensional immiscible displacement model, and simulations were performed using a high resolution model, a classical POD reduced model, and a reduced model whose POD basis was adapted to varying well location and geometry. Numerical simulations show that the proposed approach allows us to reduce the required number of model snapshots by a few orders of magnitude compared to a classical POD scheme, without noticeable loss of accuracy of calculated fluid production rates. Such an adaptive POD scheme can therefore provide a significant gain in computational efficiency for problems where multiple or iterative simulations with varying boundary conditions are required, such as optimization of well design or production optimization.

The Transient POD Method Based on Minimum Error of Bifurcation Parameter

An invariable order reduction model cannot be obtained by the adaptive proper orthogonal decomposition (POD) method in parametric domain, there exists uniqueness of the model with different conditions. In this paper, the transient POD method based on the minimum error of bifurcation parameter is proposed and the order reduction conditions in the parametric domain are provided. The order reduction model equivalence of optimal sampling length is discussed. The POD method was applied for order reduction of a high-dimensional rotor system supported by sliding bearings in a certain speed range. The effects of speed, initial conditions, sampling length, and mode number on parametric domain order reduction are discussed. The existence of sampling length was verified, and two- and three-degrees-of-freedom (DOF) invariable order reduction models were obtained by proper orthogonal modes (POM) on the basis of optimal sampling length.

Adaptive POD surrogate model method for centrifugal pump impeller flow field reconstruction based on clustering algorithm

To reduce the calculation cost and improve the accuracy of flow field prediction, an adaptive proper orthogonal decomposition (APOD) surrogate model based on K-means clustering algorithm was proposed to reconstruct the flow field of impeller. The experiment samples were designed by introducing the perturbation of the blade control parameters such as blade wrap angle and blade angle of outlet. K-means clustering algorithm was used to classify the sample blade shapes, and find out the cluster of the objective blade. The snapshot set, which consisted of the blade shape and the flow field data of impeller, can be described as a linear combination of orthogonal basis by POD method. The radial basis function (RBF) was used to fit the orthogonal basis coefficients of the objective blade, and then the flow field of objective impeller was reconstructed. The traditional fixed sample POD (FPOD) method and the proposed APOD method were used to reconstruct the flow field in impeller, respectively, and the prediction results of the two methods were compared and analyzed. The results show that the proposed APOD method could quickly and accurately reconstruct the objective flow field. The flow field prediction accuracy of the APOD method is significantly higher than the FPOD method, and the calculation time for the flow field prediction is less than 1/360 of the CFD.

A two-stage surrogate model for Neo-Hookean problems based on adaptive proper orthogonal decomposition and hierarchical tensor approximation

The evaluation of robustness and reliability of realistic structures in the presence of uncertainty is numerically costly. This motivates model order reduction techniques like the proper orthogonal decomposition (POD), which gives an approximate model on the basis of a set of precomputations of a full model. This reduces the computational time. The reduction achieved by POD is usually not sufficient in the uncertainty quantification or optimization context where a large number of evaluations has to be carried out. In this context, it is also common that only a few quantities are of interest and a further reduction is possible. The second reduction may be represented by a mapping from a possibly high-dimensional parameter space onto each quantity of interest (QoI). In general, it is difficult to construct such a mapping from an unreduced model. Hence, in this paper a two-stage surrogate model that combines both reduction approaches is introduced. This surrogate model is tailored and applied to Neo-Hookean model problems with plasticity, hardening and damage. Here, the knowledge of the model allows an adaptive selection of the POD basis on the first stage. This idea is elaborated into a generalized framework called adaptive proper orthogonal decomposition (APOD). The second stage consists of the hierarchical tensor approximation (HTA) which is easily adjusted to the accuracy and computational cost of the first stage such that the two-stage surrogate model becomes an efficient option for uncertainty quantification and optimization for Neo-Hookean problems. Additionally, with both stages at hand, the HTA is utilized for a greedy snapshot search to improve the basis of APOD and POD. This is useful if no expert knowledge is available to guide the snapshot selection in the offline phase.

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

SummaryThis article presents a new adaptive reduced order model for resolving the angular direction of the Boltzmann transport equation, based on proper orthogonal decomposition (POD) and the method of snapshots. It builds upon previous methods of applying POD to the angular dimension, with modifications to increase accuracy and solver stability. Previous methods used continuous global functions spanning the whole sphere. The new approach, discontinuous POD (DPOD), partitions the surface of the sphere into angular regions, each with an independent set of POD basis functions. Combined, these can approximate flux distributions which span the sphere using optimized basis functions for each angular region. In addition, a novel implementation of adaptive angular resolution known as adaptive discontinuous POD (ADPOD) is presented, which allows the number of DPOD basis functions to vary by angular octant and spatial element. DPOD and ADPOD are applied to two problems in order to demonstrate their benefits compared with POD. Both are shown to reduce the number of solver iterations required to find a solution and decrease the error in the angular flux.

Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

An Adaptive Proper Orthogonal Decomposition Method for Evaluating Variability Bounds of Antenna Responses

Parameter uncertainties introduced by the manufacturing process will lead to the fluctuation of an actual antenna response. Efficient methods are, thus, required during the design phase to evaluate the variability bounds. This letter proposes an adaptive proper orthogonal decomposition (APOD) method to estimate the variability bounds of the radiation patterns and reflection coefficients of the antenna with uncertain parameters. In this method, a parameterized reduced order model (PROM), as a function of the uncertain parameters, is automatically constructed using a Voronoi diagrams-based multipoint sampling algorithm. Then, the PROM-based Monte Carlo analysis is applied to obtain the variability bounds of the antenna responses. Numerical simulations and experimental tests are conducted, and the results show that the proposed APOD method can obtain comparable accuracy, but achieve a fast speed enhancement, compared with the existing methods.

Efficient reduced order controller design for dissipative PDE systems with strong convective phenomena

The control of dissipative distributed parameter systems with strong convective phenomena is considered, employing model order reduction. The accuracy of the derived reduced order model (ROM) and the associated observer may decrease as the system behavior switches from diffusion-dominant to convection-dominant. To mitigate the issue, we propose to use adaptive proper orthogonal decomposition (APOD) to capture the changing behavior of the system. A Lyapunov controller is designed based on the ROM to control the system in the presence of disturbance and strong convective phenomena. Galerkin method is used to construct the ROM. The basis functions needed in Galerkin method are updated using APOD. A case study of regulating a tubular reactor at an open-loop unstable steady state is presented, where the performance of the ROM-based observer is evaluated. It's also demonstrated that APOD can improve the accuracy of the observer when an unexpected disturbance occurs.

Revisiting APOD accuracy for nonlinear control of transport reaction processes: A spatially discrete approach

This article addresses the problem of output feedback control of dissipative distributed parameter systems. The reduced order model used for controller and observer synthesis is recursively updated using a revised version of adaptive proper orthogonal decomposition (APOD), based on decomposing spatially discrete solution profiles. This approach eliminates the basis size oscillation resulting from the inaccuracy of estimation of energy in APOD when the sampling speed is too slow. The performance of this method is illustrated by applying it to regulate a diffusion-reaction process and a fluid flow system described by the Kuramoto-Sivashinsky equation.

An adaptive proper orthogonal decomposition method for model order reduction of multi-disc rotor system

The proper orthogonal decomposition (POD) method is a main and efficient tool for order reduction of high-dimensional complex systems in many research fields. However, the robustness problem of this method is always unsolved, although there are some modified POD methods which were proposed to solve this problem. In this paper, a new adaptive POD method called the interpolation Grassmann manifold (IGM) method is proposed to address the weakness of local property of the interpolation tangent-space of Grassmann manifold (ITGM) method in a wider parametric region. This method is demonstrated here by a nonlinear rotor system of 33-degrees of freedom (DOFs) with a pair of liquid-film bearings and a pedestal looseness fault. The motion region of the rotor system is divided into two parts: simple motion region and complex motion region. The adaptive POD method is compared with the ITGM method for the large and small spans of parameter in the two parametric regions to present the advantage of this method and disadvantage of the ITGM method. The comparisons of the responses are applied to verify the accuracy and robustness of the adaptive POD method, as well as the computational efficiency is also analyzed. As a result, the new adaptive POD method has a strong robustness and high computational efficiency and accuracy in a wide scope of parameter.

Adaptive POD model reduction for solute transport in heterogeneous porous media

We study the applicability of a model order reduction technique to the solution of transport of passive scalars in homogeneous and heterogeneous porous media. Transport dynamics are modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected times, termed snapshots. The latter are then employed to achieve the desired model order reduction. We introduce a new technique, termed Snapshot Splitting Technique (SST), which allows enriching the dimension of the POD subspace and damping the temporal increase of the modeling error. Coupling SST with a modeling strategy based on alternating over diverse time scales the solution of the full numerical transport model to its reduced counterpart allows extending the benefit of POD over a prolonged temporal window so that the salient features of the process can be captured at a reduced computational cost. The selection of the time scales across which the solution of the full and reduced model are alternated is linked to the Péclet number (P e), representing the interplay between advective and dispersive processes taking place in the system. Thus, the method is adaptive in space and time across the heterogenous structure of the domain through the combined use of POD and SST and by way of alternating the solution of the full and reduced models. We find that the width of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing P e. This suggests that the effects of local-scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost.

Adaptive POD–DEIM basis construction and its application to a nonlinear population balance system

We propose an adaptive algorithm for constructing reduced‐order models (ROMs) of nonlinear systems based on proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). Using an efficient output error estimation, the reduced basis and the DEIM interpolation basis are adaptively adjusted to derive a small, yet accurate ROM. The adaptive algorithm is further explored for a population balance system of a crystallization process. Simulation results show that much smaller and reliable ROMs can be adaptively obtained using the algorithm with ignorable extra computational load as compared with the standard POD–DEIM method. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3832–3844, 2017