Empirical Interpolation Method Research Articles

Reduced basis method for the elastic scattering by multiple shape-parametric open arcs in two dimensions

We consider the elastic scattering problem by multiple disjoint arcs or cracks in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

Efficient Electromagnetic Scattering Signature Restoration for UAV Swarm Targets Based on Equivalent Principle Algorithm and Empirical Interpolation Method

Efficient Electromagnetic Scattering Signature Restoration for UAV Swarm Targets Based on Equivalent Principle Algorithm and Empirical Interpolation Method

An efficient reduced order model for nonlinear transient porous media flow with time-varying injection rates

An intrusive Reduced Order Model (ROM) is developed for nonlinear porous media flow problems with transient and time-discontinuous fluid injection rates. The proposed ROM is significantly more computationally efficient than the Full Order Model (FOM). The training regime is generated using the FOM with constant injection rates during the offline stage. The trained ROM exhibits high accuracy for complex pumping schedules (rate vs time) simulated online. The proposed ROM uses the combination of Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method (POD-DEIM), which is compared with the classical POD-Galerkin. The use of an approximated column-reduced Jacobian is shown to be vital to achieving a substantial speedup of ROM vs FOM run-times. An analysis of the trade-off between accuracy and run-time is conducted for ROMs of different sizes and hyper-parameters. The impact of the training regime on the performance of the presented ROM is assessed. The performance of the ROM is studied in the context of a two-dimensional analysis of time-varying injection into a two-well system in a layered porous media reservoir. The accuracy and efficiency of POD-DEIM motivate its potential use as a surrogate model in the real-time control and monitoring of fluid injection processes.

Continental scale spatial temporal interpolation of near-surface air temperature: do 1 km hourly grids for Australia outperform regional and global reanalysis outputs?

Near-surface air temperature is an essential climate variable for the study of many biophysical phenomena, yet is often only available as a daily mean or extrema (minimum, maximum). While many applications require sub-diurnal dynamics, temporal interpolation methods have substantial limitations and atmospheric reanalyses are complex models that typically have coarse spatial resolution and may only be periodically updated. To overcome these issues, we developed an hourly air temperature product for Australia with spatial interpolation of hourly observations from 621 stations between 1990 and 2019. The model was validated with hourly observations from 28 independent stations, compared against empirical temporal interpolation methods, and both regional (BARRA-R) and global (ERA5-Land) reanalysis outputs. We developed a time-varying (i.e., time-of-day and day-of-year) coastal distance index that corresponds to the known dynamics of sea breeze systems, improving interpolation performance by up to 22.4% during spring and summer in the afternoon and evening hours. Cross-validation and independent validation (n = 24/4 OzFlux/CosmOz field stations) statistics of our hourly output showed performance that was comparable with contemporary Australian interpolations of daily air temperature extrema (climatology/hourly/validation: R2 = 0.99/0.96/0.92, RMSE = 0.75/1.56/1.78 °C, Bias = -0.00/0.00/-0.03 °C). Our analyses demonstrate the limitations of temporal interpolation of daily air temperature extrema, which can be biased due to the inability to represent frontal systems and assumptions regarding rates of temperature change and the timing of minimum and maximum air temperature. Spatially interpolated hourly air temperature compared well against both BARRA-R and ERA5-Land, and performed better than both reanalyses when evaluated against the 28 independent validation stations. Our research demonstrates that spatial interpolation of sub-diurnal meteorological fields, such as air temperature, can mitigate the limitations of alternative data sources for studies of near-surface phenomena and plays an important ongoing role in supporting numerous scientific applications.

On the optimality of voronoi‐based column selection

AbstractWhile there exists a rich array of matrix column subset selection problem (CSSP) algorithms for use with interpolative and CUR‐type decompositions, their use can often become prohibitive as the size of the input matrix increases. In an effort to address these issues, in earlier work we developed a general framework that pairs a column‐partitioning routine with a column‐selection algorithm. Two of the four algorithms presented in that work paired the Centroidal Voronoi Orthogonal Decomposition (CVOD; Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis, 2003, 137–150) and an adaptive variant (adaptCVOD) with the Discrete Empirical Interpolation Method (DEIM; SIAM J. Sci. Computer. 38 (2016), no. 3, A1454–A1482). In this work, we extend this framework and pair the CVOD‐type algorithms with any CSSP algorithm that returns linearly independent columns. Our results include detailed error bounds for the solutions provided by these paired algorithms, as well as expressions that explicitly characterize how the quality of the selected column partition affects the resulting CSSP solution. In addition to examples involving matrix approximation, we test several of our partition‐based constructions on tasks commonly encountered in model order reduction (MOR).

Hyper-reduced-order model for estimating convection heat transfer coefficients of turbine rotors

Forced convection heat transfer occurs inside the steam turbine, and it is crucial for thermal and strength analysis to accurately estimate the convection heat transfer coefficients of the turbine rotor. This paper presents a hyper-reduced-order model for efficiently estimating the forced convection heat transfer coefficients of a steam turbine rotor. Unlike the full-order finite element method, the hyper-reduced-order model uses the discrete empirical interpolation method to approximate the domain integration of transient heat conduction simulations. This approach greatly reduces the degrees of freedom of numerical integration, thus substantially improving the computational efficiency of forward heat conduction problems. The hyper-reduced-order model and Levenberg–Marquardt algorithm are combined to minimize temperature errors between the numerical simulation results and remote sensor measurements and iteratively estimate the heat transfer coefficients of forced convection in the rotor. The accuracy and efficiency of the proposed model are verified through a transient heat conduction simulation. Moreover, the effects of the initial guess values and measurement errors are investigated to demonstrate the universality and robustness of the proposed approach in determining the convection heat transfer coefficient. Compared with the full-order finite element model, the proposed hyper-reduced-order model can increase the efficiency of forward heat conduction simulations of the turbine rotor by more than 10 times, and it only takes 0.87 s computer runtime in single transient step simulation, and then rapidly estimate the forced convection heat transfer coefficients. Furthermore, the performance of the proposed approach is insensitive to the initial guess values, and it exhibits great robustness under measurement error. The mean errors are 4.78 %, 2.67 % and 10.51 % when the initial values are 0.015 mW·mm−3·°C−1, 0.05 mW·mm−3·°C−1 and 0.15 mW·mm−3·°C−1 respectively. When there is no measurement error, the estimated error of the convection heat transfer coefficients is 2.67 %, and even if the measurement error reaches 5 %, and the accuracy loss is only 5.57 %. Consequently, this approach is highly suitable for estimating convection heat transfer coefficients during the dynamic operation of steam turbines, and this research provides reliable guidance for the optimization design and safe operation of steam turbines.

Randomized greedy magic point selection schemes for nonlinear model reduction

An established way to tackle model nonlinearities in projection-based model reduction is via relying on partial information. This idea is shared by the methods of gappy proper orthogonal decomposition (POD), missing point estimation (MPE), masked projection, hyper reduction, and the (discrete) empirical interpolation method (DEIM). The selected indices of the partial information components are often referred to as “magic points.” The original contribution of the work at hand is a novel randomized greedy magic point selection. It is known that the greedy method is associated with minimizing the norm of an oblique projection operator, which, in turn, is associated with solving a sequence of rank-one SVD update problems. We propose simplification measures so that the resulting greedy point selection has the following main features: (1) The inherent rank-one SVD update problem is tackled in a way, such that its dimension does not grow with the number of selected magic points. (2) The approach is online efficient in the sense that the computational costs are independent from the dimension of the full-scale model. To the best of our knowledge, this is the first greedy magic point selection that features this property. We illustrate the findings by means of numerical examples. We find that the computational cost of the proposed method is orders of magnitude lower than that of its deterministic counterpart. Nevertheless, the prediction accuracy is just as good if not better. When compared to a state-of-the-art randomized method based on leverage scores, the randomized greedy method outperforms its competitor.

A DEIM-CUR factorization with iterative SVDs

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of one-round sampling and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify A after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.

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

AbstractThe discrete empirical interpolation method (DEIM) is well established as a means of performing model order reduction in approximating solutions to differential equations, but it has also more recently demonstrated potential in performing data class detection through subset selection. Leveraging the singular value decomposition (SVD) for dimension reduction, DEIM uses interpolatory projection to identify the representative rows and/or columns of a data matrix. This approach has been adapted to develop additional algorithms, including a CUR matrix factorization for performing dimension reduction while preserving the interpretability of the data. DEIM‐oversampling techniques have also been developed expressly for the purpose of index selection in identifying more DEIM representatives than would typically be allowed by the matrix rank. Even with these developments, there is still a relatively large gap in the literature regarding the use of DEIM in performing unsupervised learning tasks to analyze large datasets. Known examples of DEIM's demonstrated applicability include contexts such as physics‐based modeling/monitoring, electrocardiogram data summarization and classification, and document term subset selection. This overview presents a description of DEIM and some DEIM‐related algorithms, discusses existing results from the literature with an emphasis on more statistical‐learning‐based tasks, and identifies areas for further exploration moving forward.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

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.

Entropy approximations for simple fluids.

Liquid state entropy formulas based on configurational probability distributions are examined for Lennard-Jones fluids across a range temperatures and densities. These formulas are based on expansions of the entropy in a series of n-body distribution functions. We focus on two special cases. One, which we term the "perfect gas" series, starts with the entropy of an ideal gas; the other, which we term the "dense liquid" series, removes a many-body contribution from the ideal gas entropy and reallocates it among the subsequent n-body terms. We show that the perfect gas series is most accurate at low density, while the dense liquid series is most accurate at high density. We propose empirical interpolation methods that are capable of connecting the two series and giving consistent predictions in most situations.

Nonlinear reduction using the extended group finite element method

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem’s size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we consider model order reduction and complexity reduction techniques to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For proof of concept, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.

Optimal Sparse Energy Sampling for X-ray Spectro-Microscopy: Reducing the X-ray Dose and Experiment Time Using Model Order Reduction.

The application of X-ray spectro-microscopy to image changes in the chemical state in application areas such as catalysis, environmental science, or biological samples can be limited by factors such as the speed of measurement, the presence of dilute concentrations, radiation damage, and thermal drift during the measurement. We have adapted a reduced-order model approach, known as the discrete empirical interpolation method, which identifies how to optimally subsample the spectroscopic information, accounting for background variations in the signal, to provide an accurate approximation of an equivalent full spectroscopic measurement from the sampled material. This approach uses readily available prior information to guide and significantly reduce the sampling requirements impacting both the total X-ray dose and the acquisition time. The reduced-order model approach can be adapted more broadly to any spectral or spectro-microscopy measurement where a low-rank approximation can be made from prior information on the possible states of a system, and examples of the approach are presented.

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.