Gappy Proper Orthogonal Decomposition Research Articles

An Application of a Modified Gappy Proper Orthogonal Decomposition on Complexity Reduction of Allen-Cahn Equation

This work considers model reduction techniques that can substantially decrease computational cost in simulating parmetrized Allen–Cahn equation. We first employ the proper orthogonal decomposition (POD) approach to reduce the number of unknowns in the full-order discretized system. Since POD cannot reduce the computational complexity of nonlinearity in Allen–Cahn equation, we also apply discrete empirical interpolation method (DEIM) to approximate the nonlinear term for a substantial reduction in overall simulation time. However, in general, the POD-DEIM approach is less accurate than the POD approach, since it further approximates the nonlinear term. To increase the accuracy of the POD-DEIM approach, this work introduces an extension of the DEIM approximation based on the concept of Gappy POD (GPOD), which is optimal in the least-squares sense. The POD-GPOD approach is tested and compared with the POD and POD-DEIM approaches on Allen–Cahn equation for both cases of fixed parameter value and varying parameter values. The modified GPOD approximation introduced in this work is demonstrated to improve accuracy of DEIM without sacrificing too much efficiency on the computational speedup, e.g., in one of our numerical tests, the POD-GPOD approach provides an approximate solution to the parmetrized Allen–Cahn equation 200 times faster than the full-order system with average error of order O ( 10 − 4 ) . The POD-GPOD approach is therefore shown to be a promising technique that compromises between the accuracy of POD approach and the efficiency of POD-DEIM approach.

Real-Time Reconstruction of Contaminant Dispersion from Sparse Sensor Observations with Gappy POD Method

Real-time estimation of three-dimensional field data for enclosed spaces is critical to HVAC control. This task is challenging, especially for large enclosed spaces with complex geometry, due to the nonuniform distribution and nonlinear variations of many environmental variables. Moreover, constructing and maintaining a network of sensors to fully cover the entire space is very costly, and insufficient sensor data might deteriorate system performance. Facing such a dilemma, gappy proper orthogonal decomposition (POD) offers a solution to provide three-dimensional field data with a limited number of sensor measurements. In this study, a gappy POD method for real-time reconstruction of contaminant distribution in an enclosed space is proposed by combining the POD method with a limited number of sensor measurements. To evaluate the gappy POD method, a computational fluid dynamics (CFD) model is utilized to perform a numerical simulation to validate the effectiveness of the gappy POD method in reconstructing contaminant distributions. In addition, the optimal sensor placement is given based on a quantitative metric to maximize the reconstruction accuracy, and the sensor placement constraints are also considered during the sensor design process. The gappy POD method is found to yield accurate reconstruction results. Further works will include the implementation of real-time control based on the POD method.

Combining CFD-EFD-FFD data via Gappy Proper Orthogonal Decomposition

ABSTRACT This paper demonstrates how, by means of Reduced Order Modelling (ROM), CFD cost can be drastically reduced and, in addition, a more complete investigation of a continuous design space obtained by adding experimental fluid dynamics (EFD) and flight fluid dynamics (FFD) data. The ‘Gappy’ Proper Orthogonal Decomposition (GPOD) method is used to enrich the EFD and FFD data to the size of their companion CFD. All three are then combined in a multidimensional database providing a much finer real-time exploration of a design space. Examples from aerodynamics and in-flight icing are given.

A novel inversion approach for identifying the shape of cavity by combining Gappy POD with direct inversion scheme

A novel inversion method is proposed for the identification of cavity shape by combining Gappy proper orthogonal decomposition (POD) with a direct inversion scheme. Firstly, the virtual boundary is introduced to construct a new computing domain, and the virtual boundary condition formed by polynomial of increasing order one by one is imposed on the virtual boundary, and then the needed snapshot to generate the POD vector is calculated with the boundary element method (BEM). Secondly, the full-field temperature data are reconstructed by Gappy POD using partial measurement data. Finally, the shape of cavity is identified by searching the isothermal. In order to improve the ability of the proposed method to adapt to the inversion of complex geometry problems, we propose an adaptive modified virtual boundary technique. Numerical examples show that the present method even for complex geometry such as pear shape, can not only obtain high precision inversion results, but also has a good tolerance of noise.

Stability of Discrete Empirical Interpolation and Gappy Proper Orthogonal Decomposition with Randomized and Deterministic Sampling Points

Stability of Discrete Empirical Interpolation and Gappy Proper Orthogonal Decomposition with Randomized and Deterministic Sampling Points

Modal decomposition from partial measurements

A data set over space and time is assumed to have a low-rank representation in separated spatial and temporal modes. The problem of evaluating these modes from a temporal series of partial measurements is considered. Each elementary instantaneous measurement captures only a “window” (in space) of the observed data set, but the position of this window varies in time so as to cover the entire region of interest and would allow for a complete measurement would the scene be static. A novel procedure, alternative to the Gappy Proper Orthogonal Decomposition (GPOD) methodology, is introduced. It is a fixed-point iterative procedure where modes are evaluated sequentially. Tested upon very sparse acquisition (1% of measurements being available) and very noisy synthetic data sets (10% noise), the proposed algorithm is shown to outperform two variants of the GPOD algorithm, with much faster convergence, and better reconstruction of the entire data set.

Gappy POD-based reconstruction of the temperature field in Tibet

Meteorological observations in Tibet are poor in quality with a severe amount of missing data; this is mostly caused by extreme climatological conditions and higher maintenance costs. This paper focuses on the imputation of missing data and the reconstruction of the regional temperature field. Due to insufficient observation stations and complicated topography, we employ the weather research and forecasting (WRF) model to produce the proper orthogonal decomposition (POD) basis for the study region. We then develop the gappy POD method for the imputation of missing data. Both methods are compared and tested for various missing data cases, and the results show that the gappy POD method dramatically outperforms the regularized EM algorithm when the amount of missing spatial data is not severe. Furthermore, between the two methods, only the gappy POD method is capable of reconstructing the temperature field at locations where the data are absent. The gappy POD method can also be generalized for data assimilation with the assumption that the data across all model grids have missing values.

Effective sensor placement in a steam reformer using gappy proper orthogonal decomposition

In this study, a gappy proper orthogonal decomposition (POD) method was applied to a steam reformer model with combustion, flow, and catalysis to determine the optimal number and placement of sensors. The dominant POD modes were identified based on a limited number of snapshots obtained from a spatial domain simulation, and the POD modal content was calculated from the corresponding gappy data. This information was used to estimate the differences between the sensor measurements and actual fields. The estimation results were utilized to verify the accuracies of gappy POD projections of 20 snapshots of the positions of six, three, and two sensors, and the sensor arrangements determined by using a proposed objective function were compared to those resulting from applying the conventional method. In addition, reconstructions based on gappy data were evaluated in four validation cases, and the accuracy and robustness of the sensor positions in various situations were confirmed. Consequently, this paper optimized sensor placement for the steam reformer in terms of temperature prediction and proposed modified the objective function which maintains orthogonality of the mask matrix.

Fusing wind-tunnel measurements and CFD data using constrained gappy proper orthogonal decomposition

In this article gappy proper orthogonal decomposition (POD) is used to fuse wind-tunnel measurements and computational fluid dynamics (CFD) data to provide a consistent and more comprehensive output of greater utility. The technique is used to fuse a very limited and ‘gappy’ set of wind-tunnel surface pressure measurements with computational surface pressure and friction data. By integrating the resulting fused surface data over a complete aerodynamic surface, the corresponding aerodynamic coefficients are obtained. A comparison between these integrated aerodynamic coefficients and the wind-tunnel balance measurements show a better agreement than that obtained between CFD and the wind-tunnel experiment. In addition, the original gappy POD algorithm is modified and extended to provide an accurate and consistent match between the integrated coefficients evaluated from the fused data set and the wind-tunnel balance measurements, thus producing a set of data distributed over the entire aerodynamic surface that conforms to experiment. This is achieved by considering the wind-tunnel balance measurements as equality constraints in the gappy POD least-squares problem formulation. The technique is demonstrated for steady flow around an RAE 2822 airfoil and an industrial wing-body aircraft configuration. Also, an important feature of the methodology that is essential for its use in an industrial environment, namely regularization, is discussed.

Data-Driven Time Parallelism via Forecasting

This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the dynamical-system state are available. The method adopts the parareal framework for time parallelism, which is defined by an initialization method, a coarse propagator that advances solutions on a coarse time grid, and a fine propagator that operates on an underlying fine time grid. Rather than employing usual approaches for initialization and coarse propagation, we propose novel data-driven techniques that leverage the available time-evolution bases. The coarse propagator is defined by a forecast (proposed in [K. Carlberg, J. Ray, and B. van Bloemen Waanders, Comput. Methods Appl. Mech. Engrg., 289 (2015), pp. 79--103]) applied locally within each coarse time interval, which comprises the following steps: (1) apply the fine propagator for a small number of time steps, (2) approximate the state over the entire coarse time interval using gappy proper orthogonal decomposition (POD) with the local time-evolution bases, and (3) select the approximation at the end of the time interval as the propagated state. We also propose both local-forecast and global-forecast initialization techniques. The method is particularly well suited for POD-based reduced-order models (ROMs), as the time-evolution bases can be extracted from readily available data, i.e., the right singular vectors arising during POD computation. In addition to performing analyses related to the method's accuracy, speedup, stability, and convergence, we also numerically demonstrate the method's performance. Here, numerical experiments on ROMs for a nonlinear convection--reaction problem demonstrate the method's ability to realize near-ideal speedups.

Reconstruction and Prediction of Flow Field Fluctuation Intensity and Flow-Induced Noise in Impeller Domain of Jet Centrifugal Pump Using Gappy POD Method

To analyze the interrelations among impeller blade geometry, flow field fluctuation intensity and impeller-induced hydrodynamic noise of jet centrifugal pump (JCP), a Gappy proper orthogonal decomposition (POD) method combined with computational fluid dynamics/computational fluid acoustics (CFD/CFA) technique was proposed to reconstruct and predict the unsteady flow field fluctuation intensity and flow-induced noise in impeller region. The snapshot sets were composed of blade profile parameters, flow field fluctuation intensity data and the data of sound pressure level of hydrodynamic noise in the frequency domain. Similar mesh reconstruction and flow field interpolation were carried out to have the same number of flow field data. The snapshot sets were decomposed into a linear combination of orthogonal bases using the singular value decomposition (SVD) method. The orthogonal basis coefficients corresponding to the objective variables were fitted by the least square method. The results show that the proposed method has a good accuracy in predicting the flow field fluctuation intensity and flow-induced noise of the JCP impeller domain. The relative error of pressure fluctuation intensity field is less than 4.0%, relative velocity fluctuation intensity field is less than 3.0%, turbulent kinetic energy fluctuation intensity field is less than 4.5%, and impeller-induced hydrodynamic noise is less than 10%. Taking the method as a surrogate model to predict the flow field fluctuation intensity and the radiation level of hydrodynamic noise in the optimization process of centrifugal pump impeller, it could not only reduce the calculation amount and time significantly and improve optimization speed and efficiency greatly but could also provide a reference for vibration characteristics of the models.

Stabilized Reduced-Order Model of a Non-Linear Eddy Current Problem by a Gappy-POD Approach

In this paper, a stabilized reduced-order model based on the gappy proper orthogonal decomposition (POD) technique is applied to an eddy-current problem with the non-linear material property. A classical magnetodynamic finite element formulation based on the magnetic vector potential is used as reference and starting point to build up the reduced models. The computational complexity of the non-linear problem might be reduced by the discrete empirical interpolation method-POD technique, but the eddy current effects may make this method unstable. In this paper, we propose an approach which is similar to the gappy method to successfully overcome this instability problem, while keeping the computational cost (time and memory) low.

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

A computationally efficient gradient-based optimization approach for inverse material characterization from incomplete system response measurements that can utilize a generally applicable parameterization (e.g., finite element-type parameterization) is presented and evaluated. The key to this inverse characterization algorithm is the use of a direct inversion strategy with Gappy proper orthogonal decomposition (POD) response field estimation to initialize the inverse solution estimate prior to gradient-based optimization. Gappy POD is used to estimate the complete (i.e., all components over the entire spatial domain) system response field from incomplete (e.g., partial spatial distribution) measurements obtained from some type of system testing along with some amount of a priori information regarding the potential distribution of the unknown material property. The estimated complete system response is used within a physics-based direct inversion procedure with a finite element-type parameterization to estimate the spatial distribution of the desired unknown material property with minimal computational expense. Then, this estimated spatial distribution of the unknown material property is used to initialize a gradient-based optimization approach, which uses the adjoint method for computationally efficient gradient calculations, to produce the final estimate of the material property distribution. The three-step [(1) Gappy POD, (2) direct inversion, and (3) gradient-based optimization] inverse characterization approach is evaluated through simulated test problems based on the characterization of elastic modulus distributions with localized variations (e.g., inclusions) within simple structures. Overall, this inverse characterization approach is shown to efficiently and consistently provide accurate inverse characterization estimates for material property distributions from incomplete response field measurements. Moreover, the solution procedure is shown to be capable of extrapolating significantly beyond the initial assumptions regarding the potential nature of the unknown material property distribution.

Repairing Occluded Data for a Mach 0.6 Jet via Data Fusion

Particle image velocimetry and near-field pressure were collected for an axisymmetric, Mach 0.6 jet. Some of the pressure sensors were in between the laser sheet and camera, causing occlusions in the particle image velocimetry data. Using ideas from the data fusion community, these occluded regions could be repaired. In this case, the particle image velocimetry data could be fused with the knowledge that the velocity field was symmetric about the center axis using a new technique called fused proper orthogonal decomposition, which is inspired by gappy proper orthogonal decomposition and image/wavelet fusion. Using this technique, 90% of the missing data could be estimated with 10% error.

Development and evaluation of gappy-POD as a data reconstruction technique for noisy PIV measurements in gas turbine combustors

Low signal-to-noise in particle image velocimetry (PIV) measurements in systems such as high pressure gas turbine combustors can result in significant data gaps that negatively affect subsequent analysis. Here, gappy proper orthogonal decomposition (GPOD) is evaluated as a method of filling such missing data. Four GPOD methods are studied, including a new method that utilizes a median filter (MF) to adaptively select whether a local missing data point is updated after each iteration. These methods also are compared against local Kriging interpolation. The GPOD methods are tested using PIV data without missing vectors that were obtained in atmospheric pressure swirl flames. Parameters studied include the turbulence intensity, amount of missing data, and the amount of noise in the valid data. Two criteria to check for GPOD convergence also were investigated. The MF method filled in the missing data with the lowest error across all parameters tested, with approximately one-third the computational cost of Kriging. Furthermore, the accuracy of MF GPOD was relatively insensitive to the quality of the convergence criterion. Therefore, compared to the three other GPOD methods and Kriging interpolation, the MF GPOD method is an effective method for filling missing data in PIV measurements in the studied gas turbine combustor flows.

Adaptive infill sampling criterion for multi-fidelity optimization based on Gappy-POD

This paper presents a reformulation of the "Gappy Proper Orthogonal Decomposition" (Gappy-POD) multi-fidelity modeling approach and proposes an enrichment criterion associated with an adaptive infill algorithm. The latter is here applied to the study of the flight domain of the RAE-2822 transonic airfoil at two different levels of accuracy to demonstrate its ability to detect areas in a two-dimensional design space where the surrogate model needs improvement to better drive the optimization process.

Dynamic data-driven model reduction: adapting reduced models from incomplete data

This work presents a data-driven online adaptive model reduction approach for systems that undergo dynamic changes. Classical model reduction constructs a reduced model of a large-scale system in an offline phase and then keeps the reduced model unchanged during the evaluations in an online phase; however, if the system changes online, the reduced model may fail to predict the behavior of the changed system. Rebuilding the reduced model from scratch is often too expensive in time-critical and real-time environments. We introduce a dynamic data-driven adaptation approach that adapts the reduced model from incomplete sensor data obtained from the system during the online computations. The updates to the reduced models are derived directly from the incomplete data, without recourse to the full model. Our adaptivity approach approximates the missing values in the incomplete sensor data with gappy proper orthogonal decomposition. These approximate data are then used to derive low-rank updates to the reduced basis and the reduced operators. In our numerical examples, incomplete data with 30–40 % known values are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.

A hybrid RNN-GPOD surrogate model for real-time settlement predictions in mechanised tunnelling

Realistic 3D simulations of the tunnelling process are increasingly required to investigate the interactions between machine-driven tunnel construction and the surrounding soil in order to provide reliable estimates of the expected settlements and associated risks of damage for existing structures, in particular in urban tunnelling projects. To accomplish the step from large-scale computational analysis to real-time predictions of expected settlements during tunnel construction, the focus of this paper is laid on the generation of a numerically efficient hybrid surrogate modelling strategy, combining Gappy proper orthogonal decomposition (GPOD) and recurrent neural networks (RNN). In this hybrid RNN-GPOD surrogate model, the RNN is employed to extrapolate the time variant settlements at several monitoring points within an investigated surface area and GPOD is utilised to predict the whole field of surface settlements based on the RNN predictions and a POD radial basis functions approximation. Both parts of the surrogate model are created based on results of finite element simulations from geotechnical and process parameters varied within the range of intervals given in the design stage of a tunnel project. In the construction stage, the hybrid surrogate model is applied for real-time reliability analyses of the mechanised tunnelling process to support the machine operator in steering the tunnel boring machine.

A computationally efficient approach for inverse material characterization combining Gappy POD with direct inversion

An approach for computationally efficient inverse material characterization from partial-field response measurements that combines the Gappy proper orthogonal decomposition (POD) machine learning technique with a physics-based direct inversion strategy is presented and evaluated. Gappy POD is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties. Then, the Gappy POD technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method. Lastly, a direct inversion strategy is presented that is derived from the equations governing the system response (i.e., physics of the system), which utilizes the full-field response reconstructed by Gappy POD to produce an estimate of the spatial distribution of the unknown material properties. The direct inversion technique is a particularly computationally efficient inversion technique, requiring a cost equivalent to a single numerical analysis. Therefore, the majority of the computational expense of the presented approach is the one-time potential response generation for the Gappy POD technique, which leads to an approach that is substantially computationally efficient overall. Two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial-field displacement response measurements, both static and dynamic. The inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial-field response measurements. The direct inversion with Gappy POD response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full-field response. Lastly, a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with Gappy POD response reconstruction.

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's “Lagrangian ingredients''---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. Results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.