Order Reduction Problem Research Articles

Sparse Wasserstein Barycenters and Application to Reduced Order Modeling

We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in P2(Ω) with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an available dictionary of measures, but the approximations only involve a reduced number of atoms. We show that the best reconstruction from the class of sparse barycenters is characterized by a notion of best n-term barycenter which we introduce, and which can be understood as a natural extension of the classical concept of best n-term approximation in Banach spaces. We show that the best n-term barycenter is the minimizer of a highly non-convex, bi-level optimization problem, and we develop algorithmic strategies for practical numerical computation. We next leverage this approximation tool to build interpolation strategies that involve a reduced computational cost, and that can be used for structured prediction, and metamodeling of parametrized families of measures. We illustrate the potential of the method through the specific problem of Model Order Reduction (MOR) of parametrized PDEs. Since our approach is sparse, adaptive and preserves mass by construction, it has potential to overcome known bottlenecks of classical linear methods in hyperbolic conservation laws transporting discontinuities. It also paves the way towards MOR for measure-valued PDE problems such as gradient flows.

A Generalized Single‐Step Multi‐Stage Time Integration Formulation and Novel Designs With Improved Stability and Accuracy

ABSTRACTThis paper focuses upon the single‐step multi‐stage time integration methods for second‐order time‐dependent systems. Firstly, a new and novel generalization of the Runge‐Kutta (RK) and Runge‐Kutta‐Nyström (RKN) methods is proposed, featuring an advanced Butcher table for designing new and optimal algorithms. It encompasses not only the classical multi‐stage methods as subsets, but also introduces novel designs with enhanced accuracy, stability, and numerical dissipation/dispersion properties. Secondly, to sharpen the focus on the present developments, several existing multi‐stage explicit time integration methods (which are of interest and the focus of this paper) are revisited within the proposed unified mathematical framework, such that it highlights the differences, advantages, and disadvantages of various existing methods. Thirdly, the consistency analysis is rigorously demonstrated using both single‐step and multi‐step local truncation errors, addressing the order reduction problem observed in existing methods when applied to nonlinear dynamics problems. Finally, two sets of single‐step, two‐stage, third‐order time‐accurate schemes with controllable numerical dissipation/dispersion at the bifurcation point are presented. In contrast to existing methods, these newly proposed schemes preserve third‐order time accuracy in nonlinear dynamics applications and exhibit improved stability in cases involving physical damping. Numerical examples are demonstrated to verify the theoretical analysis and the superior performance of the proposed schemes compared to existing methods.

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Interconnection-based model order reduction - a survey

In this survey we present in an organic, complete, and accessible style the interconnection framework for model order reduction. While this framework originally started as a revisitation of the moment matching method, in the last 15 years it has expanded to provide solutions to the model order reduction problem for general classes of systems, such as nonlinear, time-delay, stochastic, and hybrid, and to give insight on topics as diverse as circuit analysis and optimal control. The main theme of the paper is the characterization of “interconnection behaviors” as primary properties to be maintained by the reduction process. The survey is enriched by historical notes on the development of the interconnection framework.

Model Reduction for Linear Port-Hamiltonian Systems in the Loewner Framework

The problem of model order reduction with assignment and preservation of port-Hamiltonian structure in the reduced order model is tackled in the Loewner framework. Given a set of right-tangential interpolation data, the (subset of) left-tangential interpolation data that allow for the construction of an interpolant possessing port-Hamiltonian structure is characterized. Conditions under which an interpolant retains the underlying port-Hamiltonian structure of the system generating the data are given by requiring a particular structure of the generalized observability matrix.

Stability preserving data-driven models with latent dynamics

In this paper, we introduce a data-driven modeling approach for dynamics problems with latent variables. The state-space of the proposed model includes artificial latent variables, in addition to observed variables that can be fitted to a given data set. We present a model framework where the stability of the coupled dynamics can be easily enforced. The model is implemented by recurrent cells and trained using backpropagation through time. Numerical examples using benchmark tests from order reduction problems demonstrate the stability of the model and the efficiency of the recurrent cell implementation. As applications, two fluid–structure interaction problems are considered to illustrate the accuracy and predictive capability of the model.

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.

Riemannian optimization model order reduction method for general linear port-Hamiltonian systems

Abstract This paper presents a Riemannian optimal model order reduction method for general linear stable port-Hamiltonian systems based on the Riemannian trust-region method. We consider the $\mathcal{H}_2$ optimal model order reduction problem of the general linear port-Hamiltonian systems. The problem is formulated as an optimization problem on the product manifold, which is composed of the set of skew symmetric matrices, the manifold of the positive definite matrices, the manifold of the positive semidefinite matrices with fixed rank and the Euclidean space. To solve the optimal problem, the Riemannian geometry of the product manifold is given. Moreover, the Riemannian gradient and the Riemannian Hessian of the objective function are derived. Furthermore, we propose the Riemannian trust-region method for the optimization problem and introduce the truncated conjugate gradient method to solve the trust-region subproblem. Finally, the numerical experiments illustrate the efficiency of the proposed method.

Adaptive frequency‐limited ‐model order reduction

AbstractIn this paper, we present an adaptive framework for constructing a pseudo‐optimal reduced model for the frequency‐limited ‐optimal model order reduction problem. We show that the frequency‐limited pseudo‐optimal reduced‐order model has an inherent property of monotonic decay in error if the interpolation points and tangential directions are selected appropriately. We also show that this property can be used to make an automatic selection of the order of the reduced model for an allowable tolerance in error. The proposed algorithm adaptively increases the order of the reduced model such that the frequency‐limited ‐norm error decays monotonically irrespective of the choice of interpolation points and tangential directions. The stability of the reduced‐order model is also guaranteed. Additionally, it also generates the approximations of the frequency‐limited system Gramians that monotonically approach the original solution. Further, we show that the low‐rank alternating direction implicit iteration method for solving large‐scale frequency‐limited Lyapunov equations implicitly performs frequency‐limited pseudo‐optimal model order reduction. We consider two numerical examples to validate the theory presented in the paper.

H∞Model Match Output-Feedback Control by Means of an LMI-Based Algorithm

This letter addresses the problem of model match control for uncertain continuous-time linear systems by means of fixed order dynamic output-feedback. The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm is used as performance metric for the approximation error and the synthesis conditions are formulated in terms of LMIs, which are solved iteratively. The novelty of the approach with respect to previous results is that the matrices of the controller, as well as the matrices of the reference model, appear affinely in the conditions, allowing the immediate treatment of structured controllers and providing extra degrees of freedom for the reference model, in the sense that some of the entries of the matrices do not need to be fixed a priori and, therefore, can be determined by the synthesis conditions. This latter feature allows an extension of the method to address the problem of order reduction of uncertain systems. The results are illustrated by an example based on a model borrowed from the literature.

Computational techniques for [formula omitted] optimal frequency-limited model order reduction of large-scale sparse linear systems

We consider the problem of frequency limited H2 optimal model order reduction for large-scale sparse linear systems. A set of first-order H2 optimality conditions are derived for the frequency limited model order reduction problem. These conditions involve the solution of two frequency limited Sylvester equations that are known to be computationally complex. We discuss a framework for solving these matrix equations efficiently. The idea is also extended to the frequency limited H2 optimal model order reduction of index-1 descriptor systems. Numerical experiments are carried out using Python programming language and the results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed technique.

Model order reduction and dynamic characteristic analysis of the bolted flange structure with free-free boundaries

The problem of model order reduction and dynamic characteristic analysis of the bolted flange structure with free-free boundary conditions is studied by proposing a novel simplified modeling method. The proposed simplified modeling method is able to reduce the finite element model of the linear components through free-interface mode synthesis method and condense the number of degree-of-freedom of the joint interface through coordinate condensation, which can fix the model size not to increase with the number of bolts and greatly reduce the computational cost. Rigid-body mode elimination is also considered to further reduce the order of the integrated dynamic equation and avoid the problem of computational instability in the proposed simplified modeling method. An experimental bolted flange structure with free-free boundary conditions is built and the corresponding modal testing is carried out to verify the accuracy of the proposed simplified modeling method. Comparative results of modal testing and simulation indicate that gravity leads to unequal bending frequencies of the same order in X and Y directions, which also proves that the proposed simplified modeling method is able to analyze the dynamic characteristic of the bolted flange structure with free-free boundary conditions.

Harris hawks optimization for model order reduction of power system

This paper aims to investigate the application of Harris hawks optimization (HHO) optimization for the solution of model order reduction (MOR) problem of power system application. The proposed approach is implemented to determine the reduced order equivalent model of large-scale power system model. The obligations encountered by the higher-scale model like stability, calculative effort and problem into local optima can be enhanced by this propounded methodology. These yielded reduced models have been tried in addition to the existing algorithms and the obtained results are contrasted considering various technical parameters to accomplish its effectiveness, reliability and robustness of the adopted strategy. Thus, superior performance of the proposed method is demonstrated. The validation of the methodology in terms of error index, time and frequency domain outputs, convergence curves with scalability of the approach are outlined in the investigation of the system.

Frequency-weighted ℋ2-optimal model order reduction via oblique projection

In projection-based model order reduction, a reduced-order approximation of the original full-order system is obtained by projecting it onto a reduced subspace that contains its dominant characteristics. The problem of frequency-weighted -optimal model order reduction is to construct a local optimum in terms of the -norm of the weighted error transfer function. In this paper, a projection-based model order reduction algorithm is proposed that constructs a reduced-order model, which nearly satisfies the first-order optimality conditions for the frequency-weighted -optimal model order reduction problem. It is shown that as the order of the reduced model is increased, the deviation in the satisfaction of the optimality conditions reduces further. Numerical methods are also discussed that improve the computational efficiency of the proposed algorithm. Four numerical examples are presented to demonstrate the efficacy of the proposed algorithm.

Model Order Reduction of Positive Real Systems Based on Mixed Gramian Balanced Truncation with Error Bounds

In this paper, we discuss the problem of model order reduction for positive real systems based on balancing methods. The mixed gramian balanced truncation (MGBT) method, which is a modification of the positive real balanced truncation (PRBT) method, focuses on solving one Lyapunov equation and one Riccati equation resulting in less computational effort compared to PRBT requiring solving two Riccati equations. One major disadvantage of MGBT is that it cannot provide an error bound in contrast to PRBT. To overcome this issue, we have developed some novel modifications to MGBT which not only work with one Lyapunov and one Riccati equations but also provide error bounds. Thus, we can say that the presented methods take the key features of both MGBT and PRBT. These algorithms are presented with the aid of the new gramians which are extracted from new Lyapunov equations. The second algorithm is the frequency weighted version of the first algorithm. Additionally, it is also observed that the proposed methods can provide better error bounds compared to PRBT. Finally, comprehensive numerical examples are included to figure out the effectiveness of the presented method.

Frequency-weighted ℌ2-pseudo-optimal model order reduction

Abstract The frequency-weighted model order reduction techniques are used to find a lower-order approximation of the high-order system that exhibits high-fidelity within the frequency region emphasized by the frequency weights. In this paper, we investigate the frequency-weighted $\mathcal{H}_2$-pseudo-optimal model order reduction problem wherein a subset of the optimality conditions for the local optimum is attempted to be satisfied. We propose two iteration-free algorithms, for the single-sided frequency-weighted case of $\mathcal{H}_2$-model reduction, where a subset of the optimality conditions is ensured by the reduced system. In addition, the reduced systems retain the stability property of the original system. We also present an iterative algorithm for the double-sided frequency-weighted case, which constructs a reduced-order model that tends to satisfy a subset of the first-order optimality conditions for the local optimum. The proposed algorithm is computationally efficient as compared to the existing algorithms. We validate the theory developed in this paper on three numerical examples.

Time- and frequency-limited H 2-optimal model order reduction of bilinear control systems

ABSTRACT In the time- and frequency-limited model order reduction, a reduced-order approximation of the original high-order model is sought to ensure superior accuracy in some desired time and frequency intervals. We first consider the time-limited -optimal model order reduction problem for bilinear control systems and derive first-order optimality conditions that a local optimum reduced-order model should satisfy. We then propose a heuristic algorithm that generates a reduced-order model, which tends to achieve these optimality conditions. The frequency-limited and the time-limited -pseudo-optimal model reduction problems are also considered wherein we restrict our focus on constructing a reduced-order model that satisfies a subset of the respective optimality conditions for the local optimum. Two new algorithms have been proposed that enforce two out of four optimality conditions on the reduced-order model upon convergence. The algorithms are tested on three numerical examples to validate the theoretical results presented in the paper. The numerical results confirm the efficacy of the proposed algorithms.

QALO-MOR: Improved antlion optimizer based on quantum information theory for model order reduction

The field of optimization science is proliferating that has made complex real-world problems easy to solve. Metaheuristics based algorithms inspired by nature or physical phenomena based methods have made its way in providing near-ideal (optimal) solutions to several complex real-world problems. Ant lion Optimization (ALO) has inspired by the hunting behavior of antlions for searching for food. Even with a unique idea, it has some limitations like a slower rate of convergence and sometimes confines itself into local solutions (optima). Therefore, to enhance its performance of classical ALO, quantum information theory is hybridized with classical ALO and named as QALO or quantum theory based ALO. It can escape from the limitations of basic ALO and also produces stability between processes of explorations followed by exploitation. CEC2017 benchmark set is adopted to estimate the performance of QALO compared with state-of-the-art algorithms. Experimental and statistical results demonstrate that the proposed method is superior to the original ALO. The proposed QALO extends further to solve the model order reduction (MOR) problem. The QALO based MOR method performs preferably better than other compared techniques. The results from the simulation study illustrate that the proposed method effectively utilized for global optimization and model order reduction.

Frequency-Limited Reduced Models for Linear and Bilinear Systems on the Riemannian Manifold

In this article, we propose two new iterative algorithms to solve the frequency-limited Riemannian optimization model order reduction problems of linear and bilinear systems. Different from the existing Riemannian optimization methods, we design a new Riemannian conjugate gradient scheme based on the Riemannian geometry notions on a product manifold, and then generate a new search direction. Theoretical analysis shows that the resulting search direction is always descent with depending neither on the line search used, nor on the convexity of the cost function. The proposed algorithms are also suitable for generating reduced systems over a frequency interval in bandpass form. Finally, two numerical examples are simulated to demonstrate the efficiency of our algorithms.