Reduced‐order modeling methods for parametric bilinear systems via Walsh functions

In electrodynamic field computations the continuous Maxwell equations are typically discretized in the space variables, i. e the continuous space is mapped onto a finite set of discrete elements leading to a system of differential equations constituting the Maxwell grid equations. These dynamical systems can be very large. Due to limited computational, accuracy and storage capabilities, simplified models, obtained by means of model order reduction (MOR) methods, which capture the main features of the original model are then successfully used instead of the original models. Most commonly MOR via projection is used. Variation of model parameters like geometrical or material parameters give rise to multivariate dynamical systems. It is aimed that also the simplified models keep this parameter dependence. In this work, MOR methods are presented for multivariate systems based on the finite integration technique (FIT). The methods are applied to numerical examples with both geometrical and material variations.

In this article, two model order reduction (MOR) methods of Krylov subspace and Laguerre orthogonal polynomials were employed to the numerical simulation of heat transfer characteristics of ground heat exchangers and surrounding ground. The results show that the relative errors between the direct solution and two MOR methods are less than 0.1% under more than certain orders. Considering both the relative errors and time consumptions, for two MOR methods, it is workable to take about 1% of the original system order as the reduced system order. The larger the nodes of space and time, the more obvious the efficiency of two MOR methods.

The Finite Element Method is a widely used discretization method for mechanical systems. Model Order Reduction is often applied to balance the need for accurate simulations with the requirement for acceptable simulation times by reducing the mathematical complexity of the system. One possible projection based Model Order Reduction method for linear systems is moment matching based on Krylov subspaces. In this method, the size of the reduced order model is directly proportional to the number of inputs and outputs of the system. Therefore, tangential directions can be applied to reduce the number of inputs used for Model Order Reduction. In this contribution, we examine the suitability of the Singular Value Decomposition Model Order Reduction (SVDMOR) method for linear elastic bodies in mechanical systems. SVDMOR is commonly used for electrical circuit simulation and reduces the number of inputs and outputs based on a singular value decomposition of the transfer function. The behavior of the singular values, which correspond to the error between the full order model and the model with a reduced number of inputs and outputs, as well as the tangential directions are investigated in the frequency domain on a numerical example.

In this work, the influence of model order reduction (MOR) methods on optical aberrations is analyzed within a dynamical–optical simulation of a high precision optomechanical system. Therefore, an integrated modeling process and new methods have to be introduced for the computation and investigation of the overall dynamical–optical behavior. For instance, this optical system can be a telescope optic or a lithographic objective. In order to derive a simplified mechanical model for transient time simulations with low computational cost, the method of elastic multibody systems in combination with MOR methods can be used. For this, software tools and interfaces are defined and created. Furthermore, mechanical and optical simulation models are derived and implemented. With these, on the one hand, the mechanical sensitivity can be investigated for arbitrary external excitations and on the other hand, the related optical behavior can be predicted. In order to clarify these methods, academic examples are chosen and the influences of the MOR methods and simulation strategies are analyzed. Finally, the systems are investigated with respect to the mechanical–optical frequency responses, and in conclusion, some recommendations for the application of reduction methods are given.

This paper explores a time-domain parallel parametric model order reduction (PMOR) method for parametric systems based on the block discrete Fourier transform (DFT) and Krylov subspace. The proposed method is suitable for parametric systems with non-affine parametric dependence. With Taylor expansion, the expansion coefficients of the state variable are first obtained. Then, we show that the subspace spanned by the expansion coefficients belongs to a Krylov subspace. To speed up the PMOR process, a parallel strategy based on the block DFT and the structured matrices is proposed to compute the matrices involved in the Krylov subspace. This can avoid directly computing the inverse of the large-scale matrix. After that, the reduced parametric systems are constructed with the projection matrix obtained by the Arnoldi algorithm and orthonormalisation. Furthermore, we analyse the invertibility and the error estimations to guarantee the feasibility of the proposed PMOR method. Finally, the numerical experiments are given to demonstrate the effectiveness of the proposed method.

A strong adaptive piecewise model order reduction method for large-scale dynamical systems with viscoelastic damping

Michiels et al. (SIAM J. Matrix Anal. Appl. 32(4):1399–1421, 2011) proposed a Krylov-based model order reduction (MOR) method for time-delay systems. In this paper, we present an efficient process, which requires less memory consumption, to accomplish the model reduction. Memory efficiency is achieved by replacing the classical Arnoldi process in the MOR method with a two-level orthogonalization Arnoldi (TOAR) process. The resulting memory requirement is reduced from quadratic dependency of the reduced order to linear dependency. Besides, this TOAR process can also be applied to reduce the original delay system into a reduced-order delay system. Numerical experiments are given to illustrate the feasibility and effectiveness of our method.

This paper investigates model order reduction (MOR) methods of discrete-time systems via discrete orthogonal polynomials in the time domain. First, this system is expanded under discrete orthogonal polynomials, and the expansion coefficients are computed from a linear equation. Then the reduced-order system is produced by using the orthogonal projection matrix that is defined in terms of the expansion coefficients, which can match the first several expansion coefficients of the original output, and preserve the asymptotic stability and bounded-input/bounded-output (BIBO) stability. We also study the output error between the original system and its reduced-order system. Besides, the MOR method using discrete Walsh functions is proposed for discrete-time systems with inhomogeneous initial conditions. Finally, three numerical examples are given to illustrate the feasibility of the proposed methods.

This thesis presents the result of the study on model order reduction (MOR) methods, that can be applied to coupled systems. The goal of the research was to develop reduction techniques, that preserve special properties of coupled or interconnected system, e.g. block-structure of the underlying matrices. On the other hand, the new techniques should also be able to benefit from the knowledge, that the system they are applied to, consists of two (or more) sub-systems or describes some phenomena in different physical domains. As a result of this study, two main approaches are proposed. Their general description is given in the following paragraphs. First, the Separate Bases Reduction (SBR) algorithm is developed, which is a projection based MOR technique that uses Krylov subspaces as reduction bases. The novelty of this method is that SBR algorithm, unlike standard reduction methods designed for coupled problems, uses an uncoupled formulation of the system. In other words, an appropriate Krylov subspace is built for each of the sub-system constituting the interconnected system. As a result, the computational costs of application of the SBR algorithm, with respect to time and memory storage needed for calculations, is lower than in case of MOR methods that use the coupled formulation of the system. Moreover, the blockdiagonal form of the reduction matrices allows for preservation of the block-structure of the system matrices and keeps the sub-systems (or different physical domains) still recognizable in the reduced-order model. The SBR algorithm was successfully applied to a few test cases, resulting in the reduced systems that approximate the original ones with accuracy comparable to the accuracy of systems reduced by means of other blockstructure preserving MOR methods. The second topic of the research focuses on the couplings between the sub-systems. Here, the off-diagonal blocks of the system matrices that correspond to the couplings, are approximated by matrices of lower rank. As a main tool, generalized singular value decomposition (GSVD) is used, which allows to find the most important components of a coupling block with respect to one of the sub-systems. Although this method does not reduce the dimension of the considered problem, it gives benefits if used before application of a MOR technique. First of all, the use of low-rank approximations of the coupling blocks can decrease the computational costs of the the Krylov subspaces construction needed for reduction. If the couplings can be approximated by sufficiently low-rank blocks, the necessary matrix inverse calculation can be performed cheaper, by application of the Sherman-Morrison formula. Moreover, the undesired growth of the reduction bases, in case of use of the SBR algorithm to sub-systems with many inputs (outputs), can be lowered by use of only dominant components of the input (output) space. The conducted experiments showed, that for some cases, the number of the components used to define the couplings can be significantly reduced.

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

We propose a general --- i.e., independent of the underlying equation ---\nregistration method for parameterized Model Order Reduction. Given the spatial\ndomain $\\Omega \\subset \\mathbb{R}^d$ and a set of snapshots $\\{ u^k\n\\}_{k=1}^{n_{\\rm train}}$ over $\\Omega$ associated with $n_{\\rm train}$ values\nof the model parameters $\\mu^1,\\ldots, \\mu^{n_{\\rm train}} \\in \\mathcal{P}$,\nthe algorithm returns a parameter-dependent bijective mapping\n$\\boldsymbol{\\Phi}: \\Omega \\times \\mathcal{P} \\to \\mathbb{R}^d$: the mapping is\ndesigned to make the mapped manifold $\\{ u_{\\mu} \\circ \\boldsymbol{\\Phi}_{\\mu}:\n\\, \\mu \\in \\mathcal{P} \\}$ more suited for linear compression methods. We apply\nthe registration procedure, in combination with a linear compression method, to\ndevise low-dimensional representations of solution manifolds with\nslowly-decaying Kolmogorov $N$-widths; we also consider the application to\nproblems in parameterized geometries. We present a theoretical result to show\nthe mathematical rigor of the registration procedure. We further present\nnumerical results for several two-dimensional problems, to empirically\ndemonstrate the effectivity of our proposal.\n

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

Fast Simulation Methods for Dynamic Heat Transfer through Building Envelope Based on Model-order-Reduction

Machine-Learning based model order reduction of a biomechanical model of the human tongue