Empirical Interpolation Method Research Articles

An Efficient Integral Equation Model-Order Reduction Method for Complex Radiation Problem

In this letter, an efficient integral equation model-order reduction method is proposed to analyze complex radiation problem. In the offline stage, the randomized singular value decomposition and discrete empirical interpolation method are used to construct a reduced-order model (ROM) for the platform. In the online stage, the equivalence principle algorithm is used to transfer unknowns on the antenna to the unknowns on the equivalence surface. The modified tap basis scheme is applied to decompose the antenna from the platform and solve the current continuity challenge when the platform is cut off by equivalence surface. When the antenna position is changed in the input region, only the tap region needs to be remeshed and resolved. The ROM of platform, antenna, and equivalence surface are reusable in the online computation, which can reduce the computational complexity significantly. Numerical results demonstrate the accuracy and efficiency of the proposed method.

A space-time certified reduced basis method for quasilinear parabolic partial differential equations

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

Sensor Placement for Field Reconstruction in Rotating Electrical Machines

A method is proposed in this article to place sensors in an electrical machine in order to be able to reconstruct the magnetic field distribution. This method is based on the empirical interpolation method combined with the Maxvol technique. The results applied on a surface-mounted permanent magnet machine at no load show that the field distribution can be accurately reconstructed even when the sensor location is imposed in the airgap of the rotating machine.

Efficient Estimation of Electrical Machine Behavior by Model Order Reduction

The proper orthogonal decomposition (POD) is an efficient model order reduction method, which is frequently coupled with the discrete empirical interpolation method (DEIM) to solve nonlinear electromagnetic problems. A drawback of this method is that instabilities can occur related to the reduction operator of the nonlinear part. In this contribution, different DEIMs and the Gappy POD are employed and analyzed. Consecutively, the methods are employed to efficiently estimate the behavior of a permanent magnet synchronous machine in terms of global quantities, such as torque and iron losses.

Optimal and fast field reconstruction with reduced basis and limited observations: Application to reactor core online monitoring

The fast reconstruction of neutronic field in a nuclear core using reduced modeling and limited observations has attracted considerable attention. In particular, four design parameters are considered for developing efficient and robust field reconstruction in this framework, including the choice of reduced basis, the order of reduced dimension, the number of sensors and their placement. In this work, a systematic study has been brought to show the effect of these design parameters on the performance of field reconstruction by investigating i) five basis selection methods and ii) five different sensor placement methods. From a series of practical engineering applications based on HPR1000 reactor core, it is observed that the POD models and the sensor placement determined by the discrete empirical interpolation method and the oversampled discrete empirical interpolation method provide an optimal reconstruction, where the computational complexity, the accuracy, and the robustness to noise are well-balanced. The result is helpful for the practical implementation of the neutronic field reconstruction methods with reduced basis and limited observations in nuclear reactor cores.

An Extended DEIM Algorithm for Subset Selection and Class Identification.

The discrete empirical interpolation method (DEIM) has been shown to be a viable index-selection technique for identifying representative subsets in data. Having gained some popularity in reducing dimensionality of physical models involving differential equations, its use in subset-/pattern-identification tasks is not yet broadly known within the machine learning community. While it has much to offer as is, the DEIM algorithm is limited in that the number of selected indices cannot exceed the rank of the corresponding data matrix. Although this is not an issue for many data sets, there are cases in which the number of classes represented in a given data set is greater than the rank of the data matrix; in such cases, it is impossible for the standard DEIM algorithm to identify all classes. To overcome this issue, we present a novel extension of DEIM, called E-DEIM. With the proposed algorithm, we also provide some theoretical results for using extensions of DEIM to form the CUR matrix factorization in identifying both rows and columns to approximate the original data matrix. Results from applying variations of E-DEIM to two different data sets indicate that the presented extension can indeed allow for the identification of additional classes along with those selected by standard DEIM. In addition, comparing these results to those of some more familiar methods demonstrates that the proposed deterministic E-DEIM approach including coherence performs comparably to or better than the other evaluated methods and should be considered in future class-identification tasks.

On ab initio-based, free and closed-form expressions for gravitational waves

We introduce a new approach for finding high accuracy, free and closed-form expressions for the gravitational waves emitted by binary black hole collisions from ab initio models. More precisely, our expressions are built from numerical surrogate models based on supercomputer simulations of the Einstein equations, which have been shown to be essentially indistinguishable from each other. Distinct aspects of our approach are that: (i) representations of the gravitational waves can be explicitly written in a few lines, (ii) these representations are free-form yet still fast to search for and validate and (iii) there are no underlying physical approximations in the underlying model. The key strategy is combining techniques from Artificial Intelligence and Reduced Order Modeling for parameterized systems. Namely, symbolic regression through genetic programming combined with sparse representations in parameter space and the time domain using Reduced Basis and the Empirical Interpolation Method enabling fast free-form symbolic searches and large-scale a posteriori validations. As a proof of concept we present our results for the collision of two black holes, initially without spin, and with an initial separation corresponding to 25–31 gravitational wave cycles before merger. The minimum overlap, compared to ground truth solutions, is 99%. That is, 1% difference between our closed-form expressions and supercomputer simulations; this is considered for gravitational (GW) science more than the minimum required due to experimental numerical errors which otherwise dominate. This paper aims to contribute to the field of GWs in particular and Artificial Intelligence in general.

L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations

The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However this decomposition, usually implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.

Model order reduction of a reservoir simulation by SOD-DEIM

The large computational costs associated with reservoir simulations restrict using fine and realistic reservoir models for challenging applications such as history matching and well control optimization. This has led to the progress of reduced-order models in order to decrease computational requirements of subsurface flow simulations, especially, those that are based on proper orthogonal decomposition (POD). In this paper, an alternative method named smooth orthogonal decomposition (SOD) is used instead of POD and combined with the discrete empirical interpolation method (DEIM) for flow simulations in two-phase reservoirs. Two assessments included a two-dimensional case and a three-dimensional case are presented. These assessments comprised the accuracy and total simulation times for POD-DEIM and SOD-DEIM models show more precise results for SOD-DEIM compared with POD-DEIM in the regeneration of the full order models despite much lower simulation costs. In the end, the SOD-DEIM is tested for optimizing the well-control parameters of the three-dimensional case.

DEIM reduced order model constructed by hybrid snapshot simulation

This paper presents a methodology that combines the hybrid snapshot simulation (Bai and Wang in Int J Comput Methods 18:2050029, 2020) and the discrete empirical interpolation method (DEIM) to reduce the computational cost of constructing a DEIM-based nonlinear reduced order model (ROM) while preserving its accuracy. In distinct contrast to its traditional counterpart, the time span of the hybrid snapshot simulation is divided into multiple intervals, and a majority of the intervals are simulated by local ROMs, while a fraction by the full order model (FOM). To tackle the challenge associated with limited FOM data for ROM-DEIM construction, a new approach is proposed to reconstruct and enrich the snapshot data of nonlinear model terms using solution data in ROM intervals. A formulation and procedure based on the cell-centered finite volume method (FVM) scheme are also developed to take into account two types of nonlinearities in ROM-DEIM: the componentwise and the transport-related, non-componentwise. A global ROM-DEIM is produced immediately at the end of the snapshot simulation and can be used to accelerate online simulation for different scenarios. The proposed methodology demonstrates excellent computational performance. Specifically, the computational time of the snapshot simulation is reduced by \(\sim\) 50% without compromising ROM-DEIM accuracy (relative error less than 0.8% compared with FOM). The results prove feasibility of combining hybrid snapshot simulation and DEIM for robust and efficient ROM-DEIM development.

Parametric reduced order modeling-based discrete velocity method for simulation of steady rarefied flows

In this work, a parametric reduced order modeling-based discrete velocity method (PROM-DVM) is developed for simulation of steady rarefied flows. This method aims to reduce the number of discrete velocity points so as to improve the computational efficiency of DVM. For solving similar problems with different initial parameters, the developed method can generate a sought-after reduced discrete velocity space for the target parameter value from some pre-computed cases and solve the Boltzmann equation directly in the reduced discrete velocity space. At first, the singular-value decomposition (SVD) method is used to find the reduced-order bases for the cases of pre-computed parameter values and the reduced-order basis for the target parameter value is then constructed from the pre-computed bases by the interpolation method based on the Grassmann manifold and its tangent space. The reduced discrete velocity space and its connection to the original discrete velocity space are further established by the discrete empirical interpolation method (DEIM) based on the interpolated reduced-order basis. Since most points in the original discrete velocity space which are of negligible importance are removed in the computation of the present method, a marked improvement in computational efficiency with respect to the DVM in the original discrete velocity space is achieved. Numerical results show that the PROM-DVM can reach 13 times speed-up in CPU time for lid-driven cavity flow.

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. In the present study, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

The POD–DEIM reduced-order method for stochastic Allen–Cahn equations with multiplicative noise

In this paper, we propose a reduced-order method (ROM) based on the Monte Carlo finite difference method (FDM) or Monte Carlo finite element method (FEM) for the stochastic Allen–Cahn (SAC) equation with multiplicative noise. The reduction method is a combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). In the theoretical part, the error bounds of full-order method (FOM) and POD–DEIM are provided, as well as the computational complexity of FOM and POD–DEIM. In the numerical experiments section, several two-dimensional SAC examples with multiplicative noise are considered. Using the Monte Carlo FDM and Monte Carlo FEM as the reference methods, the POD–DEIM has obvious advantage in the computational efficiency. And with the encryption in the space or time direction, this advantage is more prominent.

Fast solution of the linearized Poisson\u2013Boltzmann equation with nonaffine parametrized boundary conditions using the reduced basis method

The Poisson–Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations, ranging from several hundred thousand to millions. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, for example, in Brownian dynamics simulations or in the computation of similarity indices for protein interaction analysis, this poses great computational challenges to conventional numerical techniques. To accelerate such onerous computations, we suggest to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the PBE with a special focus on simulations of complex biomolecular systems, which greatly reduces this computational complexity by constructing a reduced order model (ROM) of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid (AGMG) method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to the FOM. DEIM is applied to the Dirichlet boundary conditions which are nonaffine in the parameter (ionic strength), to reduce the complexity of the ROM. From the numerical results, we notice that the RBM reduces the model order from {mathcal {N}} = 2times 10^{6} to N = 6 at an accuracy of {mathcal {O}}(10^{-9}) and reduces the runtime by a factor of approximately 7600. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of 20 for a single iteration of the greedy algorithm.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

An Adaptive Segmented Reduced Basis Method for Fast Interpolating the Wideband Scattering of the Dielectric–Metallic Targets

In this letter, an interpolation method, i.e., the volume surface-integral equation-based adaptive segmented reduced basis method (VSIE-ASRBM), is proposed to fast analyze the wideband electromagnetic (EM) scattering of dielectric–metallic targets. First, by integrating with the hybrid multilevel accelerated Cartesian expansion multilevel fast multipole algorithm (MLACE-MLFMA), the broadband EM scattering of electrically large dielectric–metallic targets can be calculated without repeated meshes. Second, replicative calculations of the near-region impedance element are avoided by the empirical interpolation method. Moreover, different from the iteration process of the VSIE enhanced by MLACE-MLFMA, the current distribution can be obtained mainly by calculating the far-region impedance matrix and the sampling current vector multiplications. Meanwhile, an adaptive segmented strategy is adopted to further improve the interpolation efficiency. The proposed VSIE-ASRBM has three advantages: adaptive segmented interpolation, lower interpolation memory cost, and higher interpolation efficiency. The accuracy and efficiency will be demonstrated in two numerical examples.

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

In this paper, an efficient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible flow in porous media. The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of flow equation in original full-order space, and the discrete empirical interpolation method (DEIM) is applied to speed up the solution of Peng–Robinson equation of state in reduced-order subspace. The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reduced-order subspace, which not only preserves good prediction accuracy of full-order model, but also gains dramatic computational acceleration by multigrid and DEIM. Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method. In addition, the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng–Robinson equation of state is demonstrated and carried out in detail. Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors. The general acceleration is up to 50–60 times faster than that of standard semi-implicit method in two-dimensional simulations, but the average relative errors of numerical results between these two methods only have the order of magnitude 10−4–10−6%.

Efficient reduced-order aerodynamic modeling for fast prediction of transonic flutter boundary

In this study, a reduced-order aerodynamic modeling framework by integration of a novel training data interpolation approach and the Eigensystem realization algorithm (ERA) is presented for fast prediction of transonic flutter boundary over a range of flight parameters. First, aerodynamic impulse responses are computed by using the technique of computational fluid dynamics (CFD) at grid points within parameter space, which are selected to cover the transonic regime of concern. Next, the training data interpolation approach by combining the discrete empirical interpolation method with the Kriging technique is used to generate the aerodynamic impulse response at arbitrary flight condition, without performing unsteady CFD simulations. Finally, the interpolated impulse response is used by ERA to extract the linear state-space aerodynamic model, which is coupled with the structural model for flutter characteristics analysis. To illustrate the proposed approach, a NACA 0012 airfoil of two degrees of freedom at zero mean angle of attack is investigated. The transonic flutter boundaries of the airfoil agree well with those obtained by using CFD-based technique.

POD–DEIM model order reduction for nonlinear heat and moisture transfer in building materials

In this paper, the proper orthogonal decomposition (POD) method and the discrete empirical interpolation method (DEIM) are combined to construct a reduced-order model for a cost-optimal simulation of heat and moisture transfer in building materials. The POD-DEIM performance is assessed via two applications: a relatively simple case study on nonlinear heat conduction and a more complex case study on nonlinear moisture redistribution. In both, the results of the POD-DEIM model are compared to results from POD and finite volume methods simulations, to evaluate the accuracy of the POD-DEIM as a function of the numbers of construction modes and interpolation points. Also, as the number of POD construction modes and DEIM interpolation points does not entirely represent the computational cost of different models, the accuracies of the different models are compared as a function of the respective calculation times, to provide a fair comparison of their computational performances. Further, the use of POD-DEIM to simulate a problem different from the POD training snapshot simulation is investigated. The outcomes show that when a rather highly accurate result is desired, POD-DEIM model is capable of reducing the computational cost relative to POD and FVM.

Fast solution of neutron diffusion problem with movement of control rods

Accurate and fast-running reduced order neutronics models are expected in many challenging applications such as multi-physics coupling analysis and data assimilations of reactors. The emphasis of the present study is to investigate an appropriate order reduction method for treating the movement of control rods. The empirical interpolation method is adopted to perform an affine decomposition to the spatial distribution of the cross sections with respect to the positions of control rods. The method can well recover the abrupt change of the cross sections near the end of control rods. With incorporation of the decomposed components, accurate and very efficient reduced order models can be constructed for the neutron diffusion problem with one-step or two-step movements of control rods.