Structure-preserving Model Research Articles

Structure-preserving model reduction based on multiorder Arnoldi for discrete-time second-order systems

This paper investigates the new structure-preserving model reduction methods based on multiorder Arnoldi for discrete-time second-order systems by using discrete Laguerre polynomials. The expansion coefficients of discrete-time second-order system and its equivalent first-order system satisfy the implicit recursive expressions in space spanned by discrete Laguerre polynomials, and then multiorder Arnoldi algorithm is applied to the implicit recursive expressions to produce the column-orthogonal matrices, where the orthogonal projection matrix for the equivalent first-order system is constructed by partitioning the resulting column-orthogonal matrix appropriately. The reduced-order systems can preserve the special structures of the original systems as well as match a desired number of expansion coefficients of the original outputs. The error estimate of output variables is also given for discrete-time second-order system. Two benchmark examples are simulated to illustrate the effectiveness of the proposed methods.

Incomplete multi-view partial multi-label classification via deep semantic structure preservation

Recent advances in multi-view multi-label learning are often hampered by the prevalent challenges of incomplete views and missing labels, common in real-world data due to uncertainties in data collection and manual annotation. These challenges restrict the capacity of the model to fully utilize the diverse semantic information of each sample, posing significant barriers to effective learning. Despite substantial scholarly efforts, many existing methods inadequately capture the depth of semantic information, focusing primarily on shallow feature extractions that fail to maintain semantic consistency. To address these shortcomings, we propose a novel Deep semantic structure-preserving (SSP) model that effectively tackles both incomplete views and missing labels. SSP innovatively incorporates a graph constraint learning (GCL) scheme to ensure the preservation of semantic structure throughout the feature extraction process across different views. Additionally, the SSP integrates a pseudo-labeling self-paced learning (PSL) strategy to address the often-overlooked issue of missing labels, enhancing the classification accuracy while preserving the distribution structure of data. The SSP model creates a unified framework that synergistically employs GCL and PSL to maintain the integrity of semantic structural information during both feature extraction and classification phases. Extensive evaluations across five real datasets demonstrate that the SSP method outperforms existing approaches, including lrMMC, MVL-IV, MvEL, iMSF, iMvWL, NAIML, and DD-IMvMLC-net. It effectively mitigates the impacts of data incompleteness and enhances semantic representation fidelity.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems

Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.

Neural Autoencoder-Based Structure-Preserving Model Order Reduction and Control Design for High-Dimensional Physical Systems

Neural Autoencoder-Based Structure-Preserving Model Order Reduction and Control Design for High-Dimensional Physical Systems

STRUCTURE-PRESERVING MODEL ORDER REDUCTION OF RANDOM PARAMETRIC LINEAR SYSTEMS VIA REGRESSION

We investigate model order reduction (MOR) of random parametric linear systems via the regression method. By sampling the random parameters contained in the coefficient matrices of the systems, the iterative rational Krylov algorithm (IRKA) is used to generate sample reduced models corresponding to the sample data.We assemble the resulting reduced models by interpolating the coefficient matrices of reduced sample models with the regression technique, where the generalized polynomial chaos (gPC) is adopted to characterize the random dependence coming from the original systems. Noting the invariance of the transfer function with respect to restricted equivalence transformations, the regression method is conducted based on the controllable canonical form of reduced sample models in such a way to improve the accuracy of reduced models greatly.We also provide a posteriori error bound for the projection reduction method in the stochastic setting. We showcase the efficiency of the proposed approach by two large-scale systems along with random parameters: a synthetic model and a mass-spring-damper system.

Novel Gramian-Based Structure-Preserving Model Order Reduction for Power Systems With High Penetration of Power Converters

Renewable energy and high voltage direct current connections increase the number of power electronic devices in the grid, changing how power systems operate and introducing new dynamic behavior. For this reason, it is fundamental to understand what modeling details are needed to represent the system in this new context. This paper proposes a novel model reduction approach using frequency-limited controllability and observability Gramians that preserves the link between the physical structure of the system and the state variables. This method allows understanding and analytically determining how vital each state is to the input-output behavior in different frequency ranges. The primary contribution of this paper is an algorithm that can provide intuitive results to help power system experts understand the simulation tools and minimum modeling details needed to perform different stability studies. Two case studies supported the applicability of the developed method, and a tool for MATLAB/Simulink was developed based on the findings, allowing visualization and immediate identification of the model components that are most relevant for the system dynamics under study. This tool can analyze dynamic behaviors where the cause is poorly understood and provide clarity around modeling detail requirements in converter-rich power systems.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

We investigate linear dynamical systems of second order. Uncertainty quantification is applied, where physical parameters are substituted by random variables. A stochastic Galerkin method yields a linear dynamical system of second order with high dimensionality. A structure-preserving model order reduction (MOR) produces a small linear dynamical system of second order again. We arrange an associated port-Hamiltonian (pH) formulation of first order for the second-order systems. Each pH system implies a Hamiltonian function describing an internal energy. We examine the properties of the Hamiltonian function for the stochastic Galerkin systems. We show numerical results using a test example, where both the stochastic Galerkin method and structure-preserving MOR are applied.

Canonical and noncanonical Hamiltonian operator inference

A method for the nonintrusive and structure-preserving model reduction of canonical and noncanonical Hamiltonian systems is presented. Based on the idea of operator inference, this technique is provably convergent and reduces to a straightforward linear solve given snapshot data and gray-box knowledge of the system Hamiltonian. Examples involving several hyperbolic partial differential equations show that the proposed method yields reduced models which, in addition to being accurate and stable with respect to the addition of basis modes, preserve conserved quantities well outside the range of their training data.

A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

ABSTRACT We present a new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs). Our method exploits the structural properties of the Rosenbrock system matrix for this system class and utilizes condensed forms which often arise in applications and reveal the solution behaviour of a system. Provided that the original system has such a form, our method produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model’s transfer function and are guaranteed to be again in pH-DAE form. This allows the ROM to be safely coupled with other dynamical systems when modelling large system networks, which is useful, for instance, in electric circuit simulation.

Structure-preserving model reduction for port-Hamiltonian systems based on separable nonlinear approximation ansatzes

We discuss structure-preserving model order reduction for port-Hamiltonian systems based on a nonlinear approximation ansatz which is linear with respect to a part of the state variables of the reduced-order model. In recent years, such nonlinear approximation ansatzes have gained more and more attention especially due to their effectiveness in the context of model reduction for transport-dominated systems which are challenging for classical linear model reduction techniques. We demonstrate that port-Hamiltonian reduced-order models can often be obtained by a residual minimization approach where a suitable weighted norm is used for the residual. Moreover, we discuss sufficient conditions for the resulting reduced-order models to be stable. Finally, the methodology is illustrated by means of two transport-dominated numerical test cases, where the ansatz functions are determined based on snapshot data of the full-order state.

MORpH: Model reduction of linear port-Hamiltonian systems in MATLAB

Abstract We present a novel software toolbox MORpH for the efficient storage, analysis, interconnection and structure-preserving model order reduction (MOR) of linear port-Hamiltonian differential-algebraic equation systems (pH-DAEs). The model class of pH-DAEs enables energy-based modeling and a flexible coupling of models across different physical domains. This makes them particularly suited for the simulation and control of complex technical systems. To promote the use of recent theoretical findings in engineering practice, efficient software solutions are required. In this work, we illustrate how possibly large-scale pH-DAEs can be efficiently stored and interconnected in MATLAB in an object-oriented way. We discuss three structure-preserving MOR strategies that are supported by MORpH and demonstrate the application and performance of selected MOR algorithms by means of two benchmark examples.

Port-Hamiltonian fluid–structure interaction modelling and structure-preserving model order reduction of a classical guitar

ABSTRACT A fluid–structure interaction model in a port-Hamiltonian representation is derived for a classical guitar. After discretization, we combine the laws of continuum mechanics for solids and fluids within a unified port-Hamiltonian (pH) modelling approach by adapting the equations through an appropriate coordinate transformation on the second-order level. The high-dimensionality of the resulting system is reduced by model order reduction. The article focuses on pH-systems in different state transformations, a variety of basis generation techniques as well as structure-preserving model order reduction approaches that are independent from the projection basis. As main contribution, a thorough comparison of these method combinations is conducted. In contrast to typical frequency-based simulations in acoustics, transient time simulations of the system are presented. The approach is embedded into a straightforward workflow of sophisticated commercial software modelling and flexible in-house software for multi-physics coupling and model order reduction.

Structure-preserving model order reduction for K-power bilinear systems via Laguerre functions

ABSTRACT This paper presents a series of structure-preserving model order reduction algorithms for K-power bilinear systems via Laguerre functions. The method first aims to rewritten the K-power bilinear system as a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by combining the idea of Laguerre functions expansion of the matrix exponential function. After that, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. In order to achieve the purpose of model order reduction, the states with smaller singular values are then truncated, so as to further obtain the reduced order model. For this approach, there is a disadvantage that unstable systems may be generated although the original one is stable. To alleviate the inadequacies of this approach, we have improved the model reduction procedure, which is based upon the dominant subspace projection method. In addition, we also carried out a correlation analysis on the stability of improved algorithms. Finally, numerical experiments are employed to substantiate the effectiveness of the presented algorithms.

Extended Balancing of Continuous LTI Systems: A Structure-Preserving Approach

In this article, we treat extended balancing for continuous-time linear time-invariant systems. We take a dissipativity perspective, thus, resulting in a characterization in terms of linear matrix inequalities. This perspective is useful for determining a priori error bounds. In addition, we address the problem of structure-preserving model reduction of the subclass of port-Hamiltonian systems. We establish sufficient conditions to ensure that the reduced-order model preserves a port-Hamiltonian structure. Moreover, we show that the use of extended Gramians can be exploited to get a small error bound and, possibly, to preserve a physical interpretation for the reduced-order model. We illustrate the results with a large-scale mechanical system example. Furthermore, we show how to interpret a reduced-order model of an electrical circuit again as a lower dimensional electrical circuit.

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

Comparing the effectiveness of structure-preserving model order reduction methods for vibro-acoustic problems

For the analysis of the vibration and dissipation behavior of structures, the evaluation of the systems' frequency response is essential. However, evaluating a frequency sweep on a high-fidelity model can be computationally very expensive. A remedy to this are model order reduction methods, which allow the computation of cheap-to-evaluate surrogate models. Additionally to the reduction of the internal system dimensions, model order reduction methods that preserve the matrix structure of the transfer function are preferred as the results may allow for physical reinterpretation and the use of similar analysis tools as for the original systems. In this work, we compare system-theoretic, structure-preserving model order reduction methods with a specific focus on their applicability to vibro-acoustic systems. We demonstrate their effectiveness in terms of accuracy and computational costs by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic-acoustic coupling. As a result, this comparative study allows us to derive some general recommendations for the choice of suitable model order reduction methods depending on the specific problem statement.

On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks

This paper deals with the systematic development of structure-preserving and robust approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic Euler equations. Our approach is guided throughout by energy-based modeling concepts (port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the nonlinearities by inexact integration. Thus, it can serve as a basis for structure-preserving model reduction. Combined with an energy stable time integration, we numerically demonstrate the applicability and good stability properties of the approach using the Euler equations as an example.

Riemannian optimization approach to structure-preserving model order reduction of integral-differential systems on the product of two Stiefel manifolds

In this paper, we discuss the structure-preserving H2 optimal model order reduction (MOR) problem of integral-differential systems based on the projection technique. Different from the common idea, we take a new way to deal with this minimization problem and for the first time reduce the integral-differential systems on the product of two Stiefel manifolds. The cost function is viewed as a function whose parameters are a pair of projection matrices with different dimensions, and then the H2 optimal MOR problem is reformulated as the minimization problem of the product of two Stiefel manifolds. We extend the Riemannian geometry of the single Stiefel manifold to the product manifold and the Riemannian gradient of the cost function is derived using the orthogonal projection of the product manifold. Further, we design a Riemannian conjugate gradient scheme on the product manifold, and propose a structure-preserving Riemannian conjugate gradient algorithm. The resulting search direction is a descent direction of the cost function. Besides, we show that our algorithm is globally convergent and has the potential application to second-order systems. Finally, numerical examples demonstrate the effectiveness of the proposed algorithm.

Error bounds for model reduction of feedback-controlled linear stochastic dynamics on Hilbert spaces

We analyze structure-preserving model order reduction methods for Ornstein–Uhlenbeck processes and linear S(P)DEs with multiplicative noise based on balanced truncation. For the first time, we include in this study the analysis of non-zero initial conditions. We moreover allow for feedback-controlled dynamics for solving stochastic optimal control problems with reduced-order models and prove novel error bounds for a class of linear quadratic regulator problems. We provide numerical evidence for the bounds and discuss the application of our approach to enhanced sampling methods from non-equilibrium statistical mechanics.