Model Reduction Method Research Articles

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

Multiscale identification and NTFA reduction of the elasto-viscoplastic polycrystalline behaviour of uranium dioxide (UO2) for wide range of loading conditions

This numerical study presents a micromechanical modelling of the behaviour of uranium dioxide fuel (UO2), a polycrystalline ceramic used in nuclear power plants, for loading condition typical of a reactivity initiated accident (RIA). Above a specific temperature, UO2 has an elasto-viscoplastic behaviour with strain hardening depending on both temperature and macroscopic strain rate (T,ε¯˙).In a first step, via a full-field approach, an inverse identification of the local single crystal evolution law is performed based on experimental data obtained at the macroscopic scale [4,5]. Beyond the macroscopic behaviour of the fuel pellet, it is sometimes necessary to go down in scale to study local phenomena. A first approach would be to perform a Finite Element Method squared to two (FEM2) resolution [37]. It consists to solve a thermomechanical problem on a first mesh defined at the pellet scale where at each integration point is solved another thermomechanical problem on a Representative Volume Element. The full-field approach guarantees access to the local field at the expense of relatively high computational time. It is not conceivable within industrial codes.In a second step it appeared necessary to replace the second full-field resolution by a model reduction method based on the Nonuniform Transformation Field Analysis (NTFA) approach. This method allows access to macroscopic quantities and local fields while drastically reducing the computational time. The NTFA model [21] with tangent second order (TSO) approximation [24,21] is developed and applied to the problem at hand. Two decompositions of the hardening variable are studied. It is taken as in the reference [21], as uniform per grain or taken as decomposed, like modes associated with the viscoplastic strain tensor. Then, work is done to integrate the specific loading conditions, i.e. temperatures and strain rates of the RIA, into the NTFA model while keeping the number of internal variables as low as possible. Finally, the results from the experimental, full-field and NTFA TSO models are compared. It is checked that the numerical results are not deteriorated either macroscopically or locally at the macroscopic and local scales for uniaxial and triaxial loading.

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

A non‐linear non‐intrusive reduced order model of fluid flow by auto‐encoder and self‐attention deep learning methods

AbstractThis paper presents a new nonlinear non‐intrusive reduced‐order model (NL‐NIROM) that outperforms traditional proper orthogonal decomposition (POD)‐based reduced order model (ROM). This improvement is achieved through the use of auto‐encoder (AE) and self‐attention based deep learning methods. The novelty of this work is that it uses stacked auto‐encoder (SAE) network to project the original high‐dimensional dynamical systems onto a low dimensional nonlinear subspace and predict fluid dynamics using an self‐attention based deep learning method. This paper introduces a new model reduction neural network architecture for fluid flow problem, as well as, a linear non‐intrusive reduced order model (L‐NIROM) based on POD and self‐attention mechanism. In the NL‐NIROM, the SAE network compresses high‐dimensional physical information into several much smaller sized representations in a reduced latent space. These representations are expressed by a number of codes in the middle layer of SAE neural network. Then, those codes at different time levels are trained to construct a set of hyper‐surfaces using self‐attention based deep learning methods. The inputs of the self‐attention based network are previous time levels' codes and the outputs of the network are current time levels' codes. The codes at current time level are then projected back to the original full space by the decoder layers in the SAE network. The capability of the new model, NL‐NIROM, is demonstrated through two test cases: flow past a cylinder, and a lock exchange. The results show that the NL‐NIROM is more accurate than the popular model reduction method namely POD based L‐NIROM.

Reduced order modeling for parametrized generalized Newtonian fluid flows

Non-Newtonian fluids are present in most manufacturing industry processes. Computational models used to describe the rheological behavior of such materials can be costly due to the non-linear behavior of the viscosity. This work presents a parametrized projection-based model reduction approach to address time-dependent generalized Newtonian fluids in a computational framework. We develop our discrete formulation using three ingredients: an offline-online setting for the model reduction based on a proper orthogonal and Tucker decompositions, the finite element method for the discretization of the spatial domain and finite differences for the temporal integration, and the variational multiscale approach as a stabilization technique for both the full and reduced order models. We also evaluate the reduced method with some numerical tests, where the first part involves testing the accuracy of the model reduction method for time as a single parameter of the dynamic reduced problem. The second part involves the solution of parametrized time-dependent reduced problems with the Reynolds number and the power-law index of the fluid as the varying parameters.

Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method

This paper studies frequency-limited model reduction for linear positive systems. Specifically, the objective is to develop a reduced-order model for a high-order positive system that preserves the positivity, while minimizing the approximation error within a given H∞ upper bound over a limited frequency interval. To characterize the finite-frequency H∞ specification and stability, we first present the analysis conditions in the form of bilinear matrix inequalities. By leveraging these conditions, we derive convex surrogate constraints by means of an inner-approximation strategy. Based on this, we construct a novel iterative algorithm for calculating and optimizing the reduced-order model. Finally, the effectiveness of the proposed model reduction method is illustrated with a numerical example.

Machine learning a surrogate model for transmission loss on a Pekeris waveguide

The connection between ocean environmental parameters and transmission loss (TL) can be modeled using sound propagation models. We can analyze and improve these models using information theoretic tools such as the Fisher information matrix (FIM) and methods for model reduction, where we identify parameters that can be removed from a model without sacrificing accuracy. Critical to these methods for model reduction is the ability to evaluate derivatives of the model predictions of TL with respect to model parameters. Calculating derivatives of TL models using finite difference methods with sufficient accuracy is challenging due to the complexity of sound fields in an ocean environment; instead, we train a surrogate machine learning model to which we can apply automatic differentiation (AD). We propose a general method for model reduction in which a ML model is used as a surrogate model for physical sound propagation ocean models, which is then used to find potential simplifications of the original model. We demonstrate this method using the Pekeris waveguide model of the ocean, which calculates underwater acoustic TL in an ocean environment due to seafloor characteristics. [Work supported by Office of Naval Research]

Manifold Approximations via Transported Subspaces: Model Reduction for Transport-Dominated Problems

This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.

Order Reduction of Large-Scale Linear Dynamic Systems Using Balanced Truncation with Modified Cauer Continued Fraction

This article discusses a new technique to reduce the order of high-dimensional, continuous, linear time-invariant dynamic systems. The proposed approach utilizes the merits of the balanced truncation method, and modified Cauer continued fraction expansion. The reduced model denominator was retrieved via the concept of Lyapunov balancing, and state truncation and numerator coefficients were obtained by using a modified Cauer continued fraction. The proposed technique circumvents the steady-state gain discrepancy and steady-state error mismatch of the conventional balanced truncation method, and the instability issue of continued fraction expansion. The proposed method assures steady-state gain approximation, few moments, Markov parameters, stability, and other essential control features of a high-order system. The efficacy and applicability of the proposed technique are demonstrated using some typical test systems adopted from the literature. This method gives improved performance indices like minimum H∞ norm, H2 norm, a minimum ISE (integral squared error), IAE (integral absolute error), and retains approximately the full IRE (impulse response energy) of the existing system in the reduced model. The simulation findings of the proposed technique are compared with other current model reduction methods in recent literature using MATLAB.

Design of an efficient energy management system for renewables based wireless electric vehicle charging station

AbstractThe radical decline in climatic conditions coerces the world to take steps progressive toward implementing green energy technologies in all facets. The electric vehicle is one such huge leap toward providing a pollution‐free environment to the future generation, but the charging techniques in most cases utilize the power that is obtained using thermal power stations, and so on. This concludes it to be not completely green energy, therefore in this paper, considered two far‐reaching sources viz. photovoltaic and wind energy storage systems. The sources are also supported with the presence of an additional storage unit of battery energy storage system (BESS) and the overall system is managed by energy management system (EMS) that is designed using a simple state flow algorithm instead of complex machine learning methods. The electric vehicle (EV) is charged using wireless power transfer technology. The model is built using MATLAB—Simulink software and the simulation results are validated against the physical calculation carried out over the equivalent circuit model reduction method. The values were taken from the TATA Nexon EV for consideration and the output values are 400 V and 15 A to be provided for the charging of the EV at the receiving end of the wireless charging.

Stability analysis of rock slopes using the interface contact model and strength reduction method

The assessment of rock slope stability is usually controlled by the presence of discontinuities. The block theory is an established method in practical engineering to predict the stability of rock slopes. A maximum of two discontinuity planes are considered in the application of the block theory. It would lead to inaccurate prediction of slopes with multiple discontinuity planes. A novel method for estimating the safety margin of rock slopes is proposed, which is capable of considering the contribution of all discontinuities to the stability problem. The discontinuity planes are simulated by an interface contact model within the theoretical framework of the finite difference method. The factor of safety is obtained by the strength reduction method. The failure criteria of rock slopes are also discussed. The proposed model can simulate discontinuous planes in a more realistic manner and thus is more effective in engineering practice. To demonstrate the effectiveness of the proposed model, several numerical examples are presented, which showcase its superiority for predicting the stability of blocks composed of multiple discontinuities. Several numerical examples are analyzed to confirm the effectiveness of the proposed model and its superiority in stability prediction of blocks formed by multiple discontinuities.

A hierarchical autoencoder and temporal convolutional neural network reduced-order model for the turbulent wake of a three-dimensional bluff body

We propose a novel reduced-order model and examine its applicability to the complex three-dimensional turbulent wake of a generic square-backed bluff body called the Ahmed body at the Reynolds number ReH = U∞H/ν = 9.2 × 104 (where U∞ is free-stream velocity, H the height of the body, and ν viscosity). Training datasets are obtained by large eddy simulation. The model reduction method consists of two components—a Visual Geometry Group (VGG)-based hierarchical autoencoder (H-VGG-AE) and a temporal convolutional neural network (TCN). The first step is to map the high-dimensional flow attributes into low-dimensional features, namely latent modes, which are employed as the input for the second step. The TCN is then trained to predict the low-dimensional features in a time series. We compare this method with a TCN based on proper orthogonal decomposition (POD), which utilizes time coefficients as the input in the second part. It turns out that the H-VGG-AE has a lower reconstruction error than POD when the number of latent modes is relatively small in the first part. As the number of latent modes increases, POD exceeds in the performance of model reduction. However, the H-VGG-AE-based TCN is still more effective in terms of spatiotemporal predictions because it has a lower prediction error and costs much less time.

Design of PIλ-PDμ controller for high-order systems based on model order reduction using BB-BC and time moment matching

In this paper, a practical fractional proportional–integral and proportional–derivative (PIλ-PDμ) controller tuning method and a model reduction method are proposed for high-order systems with or without time delay. First, the Big Bang–Big Crunch (BB-BC) algorithm and the time moment matching method are used to reduce the order of the high-order system function, and the approximate low-order system function is obtained. Second, the parameter region of fractional-order PIλ-PDμ controller is obtained according to the D partition method, and the improved weighted geometric center (WGC) method is introduced to select the special points in the stable region as the control parameters. Then, the fractional-order PIλ controller is discretized by Muir method, and the fractional-order PDμ controller is discretized by continued fraction expansion (CFE). Finally, the BB-BC algorithm is used to optimize the control parameters, and the parameters with better control performance are obtained. The simulation results show that the fractional PIλ-PDμ parameter setting method based on BB-BC algorithm can improve the control performance of the system, and the model reduction scheme based on the combination of moment matching method and BB-BC algorithm is effective in both time domain and frequency domain analysis.

Dynamic reanalysis of structures with geometric variability and parametric uncertainties via an adaptive model reduction method

In this paper, a model reduction method is proposed for the dynamic reanalysis of structures with geometric variability and parametric uncertainties. Geometric variability is introduced by distorting the finite element meshes for some substructures via arbitrary shape functions. Parametric uncertainties are also considered to describe local variations of the stiffnesses of the substructures. The proposed approach involves expressing the substructure transformation matrices using interpolated matrices of Craig–Bampton component modes together with matrices of enrichment vectors. These enrichment vectors are parameter-independent and, as such, they only need to be computed once. This, as a result, leads to reduced substructure models which can be quickly updated to reanalyze structures with geometric and parametric changes. The accuracy and numerical efficiency of the proposed approach are highlighted through numerical experiments.

The efficient calculation methods for stochastic nonlinear transient heat conduction problems

This paper proposes an efficient model reduction strategy in conjunction with non-intrusive statistical moment methods for the stochastic nonlinear transient heat conduction problem. In the proposed strategy, the deterministic reduced basis vectors (RBVs) are adopted to approximate the random temperature response, and the thermal conductivity of the random nonlinear heat conduction matrix in the RBVs space is decoupled from the geometric characteristic matrix (GCM). The computational efficiency of solving the reduced random nonlinear heat conduction models can be enhanced by implementing these two technologies. The large coefficient of variation and the small coefficient of variation stochastic heat conduction problems, respectively, are addressed by the two non-intrusive statistical methods, i.e., the quasi-Monte Carlo method (QMCM) based on number theory and the modified stochastic perturbation method (MSPM). These two methods necessitate deterministic analysis and are easily integrated into the model reduction method. In four numerical instances, the selection of RBVs, the influence of the nonlinear term on the random temperature response, and the accuracy and efficiency of the two non-intrusive statistical moment methods are discussed. The numerical results highlight that the model reduction strategy based on the decoupling of thermal conductivity and GCM, in conjunction with the QMCM and the MSPM, enables the analysis of stochastic nonlinear transient heat conduction problems with a large coefficient of variation and a small coefficient of variation, respectively.

A data-driven optimization method for coarse-graining gene regulatory networks

One major challenge in systems biology is to understand how various genes in a gene regulatory network (GRN) collectively perform their functions and control network dynamics. This task becomes extremely hard to tackle in the case of large networks with hundreds of genes and edges, many of which have redundant regulatory roles and functions. The existing methods for model reduction usually require the detailed mathematical description of dynamical systems and their corresponding kinetic parameters, which are often not available. Here, we present a data-driven method for coarse-graining large GRNs, named SacoGraci, using ensemble-based mathematical modeling, dimensionality reduction, and gene circuit optimization by Markov Chain Monte Carlo methods. SacoGraci requires network topology as the only input and is robust against errors in GRNs. We benchmark and demonstrate its usage with synthetic, literature-based, and bioinformatics-derived GRNs. We hope SacoGraci will enhance our ability to model the gene regulation of complex biological systems.

Model reduction of discrete time-delay systems based on Charlier polynomials and high-order Krylov subspaces

In this paper, we present an efficient model reduction method for discrete time-delay systems based on the expansions of systems under Charlier polynomials. Making full use of the properties of Charlier polynomials and the structure of discrete time-delay systems, the projection space built by state variables is embedded in a high-order Krylov subspace. Further, a high-order Krylov subspace method is developed to generate discrete time-delay reduced systems. The proposed method is independent of the choice of inputs since the high-order Krylov subspace sequence does not involve the expansion coefficients of inputs. Besides, theoretical analysis shows that the resulting discrete time-delay reduced system characterizes the property of invariable coefficients. Finally, two numerical examples demonstrate the effectiveness of the proposed method.

A quasi Monte Carlo-based model reduction method for solving Helmholtz equation in random media

A quasi Monte Carlo-based model reduction method for solving Helmholtz equation in random media

Study of Model-Reduction Method of the Connected Vehicle subjected to Sinusoidal Excitation

Study of Model-Reduction Method of the Connected Vehicle subjected to Sinusoidal Excitation