Discrete Empirical Interpolation Method Research Articles

An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.

The discrete empirical interpolation method in class identification and data summarization

The discrete empirical interpolation method in class identification and data summarization

Fast parametric analysis of trimmed multi-patch isogeometric Kirchhoff-Love shells using a local reduced basis method

AbstractThis contribution presents a model order reduction framework for real-time efficient solution of trimmed, multi-patch isogeometric Kirchhoff-Love shells. In several scenarios, such as design and shape optimization, multiple simulations need to be performed for a given set of physical or geometrical parameters. This step can be computationally expensive in particular for real world, practical applications. We are interested in geometrical parameters and take advantage of the flexibility of splines in representing complex geometries. In this case, the operators are geometry-dependent and generally depend on the parameters in a non-affine way. Moreover, the solutions obtained from trimmed domains may vary highly with respect to different values of the parameters. Therefore, we employ a local reduced basis method based on clustering techniques and the Discrete Empirical Interpolation Method to construct affine approximations and efficient reduced order models. In addition, we discuss the application of the reduction strategy to parametric shape optimization. Finally, we demonstrate the performance of the proposed framework to parameterized Kirchhoff-Love shells through benchmark tests on trimmed, multi-patch meshes including a complex geometry. The proposed approach is accurate and achieves a significant reduction of the online computational cost in comparison to the standard reduced basis method.

A Sensor-Based Simulation Method for Spatiotemporal Event Detection

Human movements in urban areas are essential to understand human–environment interactions. However, activities and associated movements are full of uncertainties due to the complexity of a city. In this paper, we propose a novel sensor-based approach for spatiotemporal event detection based on the Discrete Empirical Interpolation Method. Specifically, we first identify the key locations, defined as “sensors”, which have the strongest correlation with the whole dataset. We then simulate a regular uneventful scenario with the observation data points from those key locations. By comparing the simulated and observation scenarios, events are extracted both spatially and temporally. We apply this method in New York City with taxi trip record data. Results show that this method is effective in detecting when and where events occur.

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

A reduced-order configuration approach for the real-time calculation of three-dimensional flow behavior in a pipe network

The real-time computation of a three-dimensional pipe network flow is crucial for both pipe design and operational maintenance. This study devises a novel reduced-order configuration approach that combines the advantages of the acceleration characteristics of the reduced-order model and the structural applicability of the configuration model. First, a configuration model is established by categorizing sub-pipes extracted from a pipe network into sets based on the sub-pipes' type. Subsequently, reduced-order configurations are realized by a reduced-order model established for each type of configuration, enabling real-time computation of individual sub-pipes. Thus, the concatenation of sub-pipes allows the computation of an entire pipe network. A complex boundary–deep learning–reduced-order configuration model and a complex boundary–deep learning–reduced-order configuration–multi-source data–reduced-order configuration model integrated with a local multi-physical–discrete empirical interpolation method and a multi-source data fusion model are devised. These models were employed for the real-time computation and prediction of a three-dimensional velocity field for 300 snapshots composed of one to four sub-pipes extrapolated from a dataset of 294 pipe network snapshots composed of one to three sub-pipes. The maximum relative errors for snapshots from the dataset were similar to the limit precision of the proper orthogonal decomposition, with more precise accuracy than the relevant studies, indicating the excellent performance of our reduced-order configuration approach.

A DEIM Tucker tensor cross algorithm and its application to dynamical low-rank approximation

We introduce a Tucker tensor cross approximation method that constructs a low-rank representation of a d-dimensional tensor by sparsely sampling its fibers. These fibers are selected using the discrete empirical interpolation method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling (DEIM-FS). For a rank-r approximation of an O(Nd) tensor, DEIM-FS requires access to only dNrd−1 tensor entries, a requirement that scales linearly with the tensor size along each mode. We demonstrate that DEIM-FS achieves an approximation accuracy close to the Tucker-tensor approximation obtained via higher-order singular value decomposition at a significantly reduced cost. We also present DEIM-FS (iterative) that does not require access to singular vectors of the target tensor unfolding and can be viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the computational cost associated with solving nonlinear tensor differential equations (TDEs) using dynamical low-rank approximation (DLRA). The computational cost of solving DLRA equations can become prohibitive when the exact rank of the right-hand side tensor is large. This issue arises in many TDEs, especially in cases involving non-polynomial nonlinearities, where the right-hand side tensor has full rank. This necessitates the storage and computation of tensors of size O(Nd). We show that DEIM-FS results in significant computational savings for DLRA by constructing a low-rank Tucker approximation of the right-hand side tensor on the fly. Another advantage of using DEIM-FS is to significantly simplify the implementation of DLRA equations, irrespective of the type of TDEs. We demonstrate the efficiency of the algorithm through several examples including solving high-dimensional partial differential equations.

CUR and Generalized CUR Decompositions of Quaternion Matrices and their Applications

Low-rank approximations of quaternion matrices have garnered interest across various applications, such as color images and signal processing. In this paper, we propose the CUR and generalized CUR decompositions of quaternion matrices and utilize the CUR decomposition of the quaternion matrix to solve the robust quaternion principal component analysis (RQPCA) problem. Through the discrete empirical interpolation method (DEIM) for the subset selection, we present the error analysis of the approximation of the CUR and the generalized CUR decompositions of quaternion matrices. The error bound depends on the conditioning of sampling submatrices. Next, we employ the alternating projection and adopt the CUR decomposition of the quaternion matrix to tackle the RQPCA. The non-convex algorithm runs fast and significantly reduces computational complexity. The performance advantages of our algorithms have been experimentally verified on artificial datasets. The RQCUR significantly affects color video background subtraction in the experiment.

Advancing wave equation analysis in dual-continuum systems: A partial learning approach with discrete empirical interpolation and deep neural networks

In this work, we propose a partial learning approach using partially explicit discretization for solving the wave equations. The considered mathematical model involves a dual-continuum system of hyperbolic second-order partial differential equations. We employ a finite element method for spatial approximation, while for temporal approximation, we utilize a partially explicit scheme. Specifically, we adopt the explicit scheme for the low-conductive continuum and the implicit scheme for the high-conductive continuum.Our proposed method’s key idea lies in employing a partial learning approach to remove the necessity of finding the difficult-to-compute (implicit) part of the solution. Instead of attempting a full training of the numerical solution, we focus on training only a portion of it that requires the most computational cost. We use a proper orthogonal decomposition with a discrete empirical interpolation method to facilitate machine learning. By employing these methods, we can train a deep neural network to predict the values of the implicit part of the solution at specific observed locations and then restore the field.We consider a two-dimensional model problem in heterogeneous media to test the effectiveness of the proposed approach. The numerical results demonstrate that our partial learning approach yields a satisfactory approximation. By combining the reduced-order modeling techniques, the partially explicit time scheme, and the deep neural network, we removed the necessity of finding the difficult-to-compute part of the solution while maintaining high accuracy.

Data-driven reduced-order modeling for nonlinear aerodynamics using an autoencoder neural network

The design of commercial air transportation vehicles heavily relies on understanding and modeling fluid flows, which pose computational challenges due to their complexity and high degrees of freedom. To overcome these challenges, we propose a novel approach based on machine learning (ML) to construct reduced-order models (ROMs) using an autoencoder neural network coupled with a discrete empirical interpolation method (DEIM). This methodology combines the interpolation of nonlinear functions identified based on selected interpolation points using DEIM with an ML-based clustering algorithm that provides accurate predictions by spanning a low-dimensional subspace at a significantly lower computational cost. In this study, we demonstrate the effectiveness of our approach by the calculation of transonic flows over the National Advisory Committee of Aeronautics 0012 airfoil and the National Aeronautics and Space Administration Common Research Model wing. All the results confirm that the ROM captures high-dimensional parameter variations efficiently and accurately in transonic regimes, in which the nonlinearities are induced by shock waves, demonstrating the feasibility of the ROM for nonlinear aerodynamics problems with varying flow conditions.

Energy-conserving hyper-reduction and temporal localization for reduced order models of the incompressible Navier-Stokes equations

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform energy- and momentum-conserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new energy- and momentum-conserving hyper-reduction methods and the energy- and momentum-conserving temporal localization method is analysed using three convection-dominated test cases; a shear-layer roll-up, two-dimensional homogeneous isotropic turbulence and a time-periodic inviscid flow consisting of a vortex in a uniform background flow. Our main finding is that energy conservation in combination with oversampling or regularization leads to a robust method with excellent long time stability properties. When any of these two ingredients is missing, accuracy and/or stability is significantly impaired.

FastESN: Fast Echo State Network.

Echo state networks (ESNs) are reservoir computing-based recurrent neural networks widely used in pattern analysis and machine intelligence applications. In order to achieve high accuracy with large model capacity, ESNs usually contain a large-sized internal layer (reservoir), making the evaluation process too slow for some applications. In this work, we speed up the evaluation of ESN by building a reduced network called the fast ESN (fastESN) and achieve an ESN evaluation complexity independent of the original ESN size for the first time. FastESN is generated using three techniques. First, the high-dimensional state of the original ESN is approximated by a low-dimensional state through proper orthogonal decomposition (POD)-based projection. Second, the activation function evaluation number is reduced through the discrete empirical interpolation method (DEIM). Third, we show the directly generated fastESN has instability problems and provide a stabilization scheme as a solution. Through experiments on four popular benchmarks, we show that fastESN is able to accelerate the sparse storage-based ESN evaluation with a high parameter compression ratio and a fast evaluation speed.

Reduced order modeling of structural problems with damage and plasticity

AbstractModel order reduction (MOR) techniques are used across the engineering sciences to reduce the computational complexity of high‐fidelity simulations. MOR methods reduce the computation time by representing the problem using a lower number of degrees of freedom (DOF). The use of reduced order models (ROM) in the analysis of structural problems with damage and plasticity has the potential to significantly reduce computational time and increase efficiency. Of course, the approximation of a problem in a lower dimensional space introduces an approximation error that needs to be kept small enough so that the results of the ROM maintain their validity. One well‐known reduced order modeling approach is the proper orthogonal decomposition (POD). POD is used to extract the dominant modes of the structure, which are then used to solve a problem in the smaller dimensional subspace. To overcome the limitations of the POD regarding nonlinear problems, the discrete empirical interpolation method (DEIM) is employed. An exemplary uncertainty quantification application is used to investigate the methodology. The investigation shows that the POD‐based DEIM can significantly reduce the computational effort of highly nonlinear structural simulations incorporating damage while maintaining a high approximation accuracy.

Hyper‐reduced order models for accelerating parametric analyses on reinforced concrete structures subjected to earthquakes

AbstractThis paper combines a hyper‐reduction procedure, the proper orthogonal decomposition unassembled discrete empirical interpolation method, with a non‐iterative α‐operator splitting (α‐OS) time‐integration scheme for accelerating parametric analyses on damageable civil engineering structures subjected to earthquakes. Applications on a two‐story reinforced concrete frame building modeled by multi‐fiber beams provide guidelines to define hyper‐reduced order models (HROMs) for real‐life applications that use seismic databases or variable loading assumptions. The α‐OS HROMs proved their applicability in approximating high‐fidelity dynamic responses with speed‐up factors higher than 60 and errors lower than 0.6% for the present case study.

Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.

Separated representation of the finite element solution of nonlinear magnetostatic problem based on non-intrusive Proper Generalized Decomposition

The Proper Generalized Decomposition has shown its efficiency to solve parameterized problems in nonlinear system events when it is combined with the Discrete Empirical Interpolation Method. Nevertheless, the solution of finite element model with the Proper Generalized Decomposition framework requires to have access to matrices and vectors of the discretized problem, which makes the method highly intrusive. In this context, based on a set of finite element solutions for a set of input parameters, a surrogate model can be developed applying a non-intrusive Proper Generalized Decomposition approach. The proposed non-intrusive approach is based on a canonical decomposition of the finite element solutions combined with an interpolation method. We then obtain a surrogate model approximating the finite element solutions for a wide range of parameters. The surrogate model, given its evaluation speed, can be used for real-time applications. In this paper, the proposed non-intrusive Proper Generalized Decomposition approach is employed to approximate a nonlinear magnetostatic problem and is applied to a single phase standard transformer and to a three-phase inductance.

Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases

Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.

Model order reduction methods applied to neural network training

AbstractNeural networks have emerged as powerful and versatile tools in the field of deep learning. As the complexity of the task increases, so do size and architectural complexity of the network, causing compression techniques to become a focus of current research. Parameter truncation can provide a significant reduction in memory and computational complexity. Originating from a model order reduction framework, the Discrete Empirical Interpolation Method is applied to the gradient descent training of neural networks and analyze for important parameters. The approach for various state‐of‐the‐art neural networks is compared to established truncation methods. Further metrics like L2 and Cross‐Entropy Loss, as well as accuracy and compression rate are reported.

A physics-based reduced order model for urban air pollution prediction

This article presents an innovative approach for developing an efficient reduced-order model to study the dispersion of urban air pollutants. The need for real-time air quality monitoring has become increasingly important, given the rise in pollutant emissions due to urbanization and its adverse effects on human health. The proposed methodology involves solving the linear advection–diffusion problem, where the solution of the Reynolds-averaged Navier–Stokes (RANS) equations gives the convective field. At the same time, the source term consists of an empirical time series.However, the computational requirements of this approach, including microscale spatial resolution, repeated evaluation, and low time scale, necessitate the use of high-performance computing facilities, which can be a bottleneck for real-time monitoring. To address this challenge, a problem-specific methodology was developed that leverages a data-driven approach based on Proper Orthogonal Decomposition with regression (POD-R) coupled with Galerkin projection (POD-G) endorsed with the discrete empirical interpolation method (DEIM). The proposed method employs a feedforward Neural Network (NN) to non-intrusively retrieve the reduced-order convective operator required for online evaluation. The numerical framework was validated on synthetic emissions and real wind measurements.The results demonstrate that the proposed approach significantly reduces the computational burden of the traditional approach and is suitable for real-time air quality monitoring. Overall, the study advances the field of reduced order modeling and highlights the potential of data-driven approaches in environmental modeling and large-scale simulations.

Bayesian decision making using partial data for fractured poroelastic media

Partial data is a common challenge in many applied problems, where we can measure only the part of the solution. This paper considers the poroelasticity problem in fractured media, where only partial measurements of fracture pressures are available. The poroelasticity models have applications in many engineering and scientific fields, such as geomechanics, hydrology, and reservoir simulation. This paper aims to develop a Bayesian decision making approach using partial data and consider its application to the poroelasticity problem in fractured media. The methodology of the proposed approach combines Proper Orthogonal Decomposition, Discrete Empirical Interpolation Method, Bayesian Inversion Approach, and Bayesian Decision Theory. It allows us to compute posterior probabilities and decide according to the optimal decision rule based on the partial data. Moreover, we apply partially explicit temporal discretization to solve forward problems efficiently. For testing the proposed approach, we consider a two-dimensional model problem. We simulate the cases when we need to make a single decision about the reservoir or decisions about the elements of the set of reservoirs. The results show that the proposed approach can efficiently compute the posterior probabilities and make a decision in the presence of only partial data for both cases.