Original articlesA reduced order modeling method based on GNAT-embedded hybrid snapshot simulation
Introduction
In spite of significant advances of computing hardware in the past decades, innovative and efficient algorithms are still desirable to accelerate large-scale scientific and engineering computation on resource-limited platforms. Among them, reduced-order modeling (ROM) is one of the effective methods to reduce model sizes and achieve salient computational efficiency without appreciably compromising model accuracy. It has been demonstrated in a variety of applications, such as, real-time simulation of deformation in which the ROM can be adaptively updated [35]; fluid dynamics with applications in shallow water [64], ground water flow [21], [22], flow in porous media [34], [41], and turbulent flows [69]; aerodynamics with applications in aeroelasticity [3] and steady state aerodynamics [75]; thermal analysis in spacecraft [52]; multi-physics modeling with applications in viscoelasticity [31], electromagnetics [45], magnetohydrodynamics [23]; and engineering design for circuits [55] and MEMS [6], [73], among others.
The ROM technique generally consists of two stages: (1) offline snapshot simulation and ROM construction, in which a projection subspace and spanning basis vectors will be extracted from snapshot data produced by a full order model (FOM); and (2) online ROM simulation. In this stage only the ROM will be simulated with prescribed model parameters in an online fashion. In the traditional snapshot simulation approach, the FOMs are almost always used in order to acquire high-quality snapshot data with salient subspace representation. Basis vectors are obtained by applying low-rank approximations, such as proper orthogonal decomposition (POD) [62] onto the snapshot data, and then utilized to construct ROM. The main drawback of this approach is demanding computational cost [32] associated with snapshot simulation and singular value decomposition (SVD) or eigenvalues decomposition (EVD) on the large snapshot data, as well as demanding memory and storage usage.
To overcome these issues, snapshot data selection methods were proposed [1], [68], [70]. The online adaptive methods for data-driven ROMs with sensor data to update POD basis vectors [50] and POD basis vectors for nonlinear terms [51] have also been developed. Advanced ROM techniques for nonlinear advection-dominant problems with adaptive basis and sampling updating have been proposed and investigated [49], [57]. In [49], an approach was developed that exploits both the spatial and temporal locality to propagate coherent structures of transport-dominated problems and obtain ROMs of low dimensions. Hybrid methods incorporating FOMs and local ROMs in streamlined numerical simulation with model switch criteria were reported [29], [53], [65]. In addition to the snapshot data selection approach, an a priori reduction approach was proposed to refine the basis vectors by an iterative procedure [2]. Recently, the present authors proposed a hybrid snapshot simulation approach that alternates the simulation between FOMs and local ROMs [11]. This new approach accelerated snapshot simulation greatly and generated snapshot data with accurate representation, and the ROM constructed by POD basis vectors of snapshot data demonstrated excellent results in online simulation. A DEIM-based ROM can also be constructed by the hybrid snapshot simulation approach with augmented snapshot data of the nonlinear terms of the local ROM [9]. In this hybrid snapshot simulation approach, POD subspaces storing dominant basis vectors of snapshot data are updated by incremental singular value decomposition (iSVD) [12], [13], [58]. The iSVD updates POD basis vectors based on low-rank approximation and vector rotation, and is more efficient than the traditional POD. The iSVD has also been utilized for adaptive snapshot data selection [48] and PDE models with Galerkin projection [25].
One of the major challenges of nonlinear ROMs is acquisition of representative projection subspace and efficient approximation of the nonlinear terms during simulation. Several techniques have been proposed to combat the challenge, including the trajectory piecewise linear approximation (TPWL), which employs Taylor expansion of nonlinear terms at linearization points [55], [56]; the missing point estimation (MPE), which performs Galerkin projections over a subset of grid points in the spatial domain [7], [77]; the discrete empirical interpolation method (DEIM), which utilizes a set of empirically derived basis vectors to determine a few spatial locations that are used as the interpolation points to approximate the nonlinear terms and reduce computational costs [17], [18], [71]; and the reduced basis method (RBM), which approximates the parametrized problems with the construction of a greedy strategy to reduce the computational cost [19], [42], [47]. The Gauss–Newton with approximated tensors (GNAT) is a technique that includes a function-sampling hyper-reduction scheme at the discrete level along with the Petrov–Galerkin projection of nonlinear ROMs. The hyper-reduction is designed to approximate both the residuals and the column-reduced Jacobian through the gappy POD process and is shown to satisfy consistency, optimality and computational efficiency requirements [14], [15], [16], [34]. These approaches have been proven feasible and effective in approximating nonlinear ROMs.
In this paper, we propose a computational method that embeds the GNAT hyper-reduction into the hybrid snapshot simulation developed in our prior work [10], [11]. In this method, three models are used in the snapshot simulation, including the FOM, the ROM based on Petrov–Galerkin projection (ROM-PG), and the ROM based on hyper-reduction by Gauss–Newton with approximated tensors (ROM-GNAT) to form a hierarchy of model approximation. Then the snapshot simulation is partitioned into multiple intervals, and the simulation in each interval is switched among the FOM, ROM-PG, and ROM-GNAT to produce necessary, high-quality snapshot data in a more economic manner. In other words, the local ROM-PG and the local ROM-GNAT will be used to rapidly traverse regions that contribute to negligible new information to update the POD basis vectors of solution variables and nonlinear terms, leading to accelerated snapshot simulation. As shown later, the proposed method is compatible to fully discretized nonlinear PDE models [4], [5], [20], [59], [76].
The novelties of this work include: (1) to the best of our knowledge, combining the GNAT-ROM with the hybrid snapshot simulation to allow efficient generation of snapshot data using hierarchical model approximation (FOM, ROM-PG, and ROM-GNAT) is new and has not been reported before; (2) several new criteria that evaluate the POD basis vectors and the trajectories of the ROM-PG and ROM-GNAT are proposed to determine model switch on-the-fly during the snapshot simulation; (3) two sets of POD modes used, respectively, for Petrov–Galerkin projection and GNAT hyper-reduction are updated by the iSVD during the snapshot simulation. The local POD modes are adopted to guide model switch; (4) the GNAT-embedded hybrid snapshot simulation is developed and demonstrated with the finite volume discretization of 2D Burgers equation in the regime of weak divergence flow recently proposed by Sheng and Zhang [60]; and (5) a ROM-GNAT will be generated at the end of the proposed GNAT-embedded snapshot simulation pipeline, and is immediately applicable for fast and accurate online simulations.
The rest of this article is organized as follows: in Section 2, we will provide an overview of Petrov–Galerkin projection and GNAT in nonlinear dynamic systems. The methodology of the GNAT-embedded hybrid snapshot simulation is described in Section 3. In Section 4 the FOM of 2D Burgers equation is obtained by the finite volume discretization, and the nonlinear implicit Gauss–Newton solver is described. The ROM with Petrov–Galerkin projection (ROM-PG) and the ROM with Gauss–Newton approximated tensor (ROM-GNAT) are presented. In Section 5 the detailed information of the GNAT-embedded hybrid snapshot simulation for 2D Burgers equation will be discussed, including the POD modes of the Petrov–Galerkin projection and the residuals, snapshot data acquisition, and the model switch criteria. In Section 6, the numerical experiments will be presented and discussed in terms of accuracy and computational efficiency. The paper concludes with technical findings of the proposed method in Section 7.
Section snippets
Petrov–Galerkin projection and Gauss–Newton approximated tensors
In this section we introduce the approaches of Petrov–Galerkin projection and Gauss–Newton approximated tensor for generating ROMs of dynamic systems at the discrete time level. The approaches introduce hierarchical approximations when the PDE is discretized in both time and space. First, the residual of the nonlinear system in a fully implicit framework is given by: where , and represents the number of time steps, and represents the variable of the full
Methodology of GNAT-embedded hybrid snapshot approach
Following the brief introduction to the PG projection and the GNAT, in this section the GNAT-embedded hybrid snapshot approach is presented, which is proposed to accelerate snapshot data generation for constructing the efficient ROM-GNAT for online simulation. As shown in Fig. 1, in this method, the total time span of the snapshot simulation (labeled “1” in Fig. 1) is divided into multiple intervals, and each will be simulated by the FOM, ROM-PG, or ROM-GNAT. Since ROM-GNAT includes additional
The FOM, ROM-PG and ROM-GNAT based on finite volume discretization of 2D Burgers equation
In this paper, 2D Burgers equation based on the finite volume method (FVM) will be used to evaluate the proposed GNAT-embedded snapshot simulation method. Therefore, in this section, the numerical approaches used to generate the FOM, ROM-PG and ROM-GNAT are described. Although the numerical solutions of the system can be solved using a variety of temporal integration techniques, the implicit Gauss–Newton method is employed due to the use of the GNAT in the proposed approach.
Snapshot data generation and model switch criteria in GNAT-embedded hybrid snapshot simulation
The methodology of GNAT-embedded hybrid snapshot simulation is introduced in Section 3. In this section, the details how the snapshot data are collected and what are the criteria to enable on-the-fly model switches are provided in the specific context of the 2D Burgers equation.
Numerical experiments and discussion
Numerical experiments with 2D Burgers equation are conducted and the results are presented in this section to evaluate the performance (including efficiency and accuracy) of the proposed GNAT-embedded snapshot simulation methodology.
The computational domain is a 2D square and the time set is . The exact solution of 2D Burgers equation [8], [26] is used to examine the accuracy by comparing the exact solution (when Re 100) with the proposed FOM in Section 4.2.
Conclusions
This paper proposes a GNAT-embedded hybrid snapshot simulation method to accelerate generation of high-quality snapshot data and accurate GNAT-ROMs. By using the obtained snapshot data, a ROM-GNAT model can be constructed more efficiently while preserving the numerical accuracy. In distinct contrast to our previous research [11], the novelties of the present work include: each of the time intervals is simulated by the models with varying levels of approximation, accuracy, and computational
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
YW acknowledges the faculty startup grant from the University of South Carolina, USA for partial funding of this research. The anonymous reviewers are also sincerely appreciated for their careful reading of the manuscript and many valuable and insightful comments and suggestions.
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