Structure Preserving Model Order Reduction Research Articles

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.

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.

SOBMOR: Structured Optimization-Based Model Order Reduction

Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non-structure-preserving counterparts. In this paper, we present a framework for structure-preserving MOR, which allows us to compute structured reduced order models (ROMs) with a much higher accuracy. The framework is based on parameter optimization, i.e., the elements of the system matrices of the ROM are iteratively varied to minimize an objective functional that measures the difference between the FOM and the ROM. The structural constraints can be encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems. We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second-order system in a comparison with other structure-preserving MOR algorithms.

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.

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.

Passivity preserving model reduction via spectral factorization

We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman–Yakubovich–Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2-optimal reduced-order models.

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.

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.

Model order reduction for (stochastic-) delay equations with error bounds

<p style='text-indent:20px;'>In this article, we analyze a structure-preserving model order reduction technique for deterministic and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring the framework of [<xref ref-type="bibr" rid="b6">6</xref>], we find error estimates for the difference between the dynamics of the full and reduced model. This analysis also yields new error bounds for bilinear systems and stochastic systems with multiplicative noise and non-zero initial states.</p>

Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems

A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Laguerre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states with small approximate Hankel singular values (HSVs). The above method has a disadvantage that it may unexpectedly result in unstable systems although the original one is stable. Therefore, modified reduction procedure combined with the dominant subspace projection method is presented to alleviate the limitation. Finally, two numerical experiments are provided to demonstrate the effectiveness of the algorithms.

Efficient multi-frequency solutions of FE–BE coupled structural–acoustic problems using Arnoldi-based dimension reduction approach

The frequency sweep analysis is indispensable for the performance prediction and optimization design of structural–acoustic interaction problems. The coupled finite element and boundary element (FE–BE) method has been widely used to perform such simulations. However, the straightforward solution of the resulting hybrid model for a large number of frequencies is computationally prohibitive due to the unfavorable properties of involved matrices, e.g. large-scale, non-symmetric and especially frequency-dependent. In order to ease this challenge, a three-step structure-preserving model order reduction method is presented, which is based on an offline-online computing framework. As a first step, the second-order Arnoldi process is used to speed up calculations of the sparse structural part of the strongly coupled system. In a second step, the low-dimensional approximations of the FE unknown variables are eliminated from the global system of equations and after that the remaining dense BE degrees of freedom are reduced by application of the standard Arnoldi-based Krylov subspace algorithm. A transformation technique based on Taylor’s theorem is incorporated to decouple the frequency from the Green’s function and a column-by-column low-memory projection is further sought to favor the overall storage requirements. In the last step, a robust reduced order model can be quickly retrieved by simple algebraic manipulations, which allows a direct solver without any costly matrix operation to be used for the online sweeps, requiring only a very small fraction of the total CPU runtime. Two numerical cases are investigated to highlight the potential of the proposed approach in multi-frequency applications.

Structure-preserving interpolation for model reduction of parametric bilinear systems

In this paper, we present a projection-based interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.