Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Adaptive rational Krylov subspaces for large-scale dynamical systems

Parametric model order reduction (PMOR) has received a tremendous amount of attention in recent years. Among the first approaches considered, mainly in system and control theory as well as computational electromagnetics and nanoelectronics, are methods based on multi-moment matching. Despite numerous other successful methods, including the reduced-basis method (RBM), other methods based on (rational, matrix, manifold) interpolation, or Kriging techniques, multi-moment matching methods remain a reliable, robust, and flexible method for model reduction of linear parametric systems. Here we propose a numerically stable algorithm for PMOR based on multi-moment matching. Given any number of parameters and any number of moments of the parametric system, the algorithm generates a projection matrix for model reduction by implicit moment matching. The implementation of the method based on a repeated modified Gram-Schmidt-like process renders the method numerically stable. The proposed method is simple yet efficient. Numerical experiments show that the proposed algorithm is very accurate.

https://link.springer.com/content/pdf/10.1007%2F978-3-319-02090-7_6.pdf

A Robust Algorithm for Parametric Model Order Reduction Based on Implicit Moment Matching

In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear state-space systems has been a topic of research for about 50 years at the time of writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the so-called stochastic balanced truncation method to descriptor systems which so far cannot be found in the literature.

Model Order Reduction for Differential-Algebraic Equations: A Survey

Summary
This paper investigates the problem of robust performance analysis for Lurie nonlinear control system with parameter uncertainties and interval time-varying delays. Based on an augmented Lyapunov-Krasovskii functional including multiple integral terms, new delay-dependent robust stability criteria are derived by proposing a novel secondary delay-partition approach. Moreover, to obtain less conservative stability conditions, an optimized integral inequality is developed by introducing an adjustable parameter ς1. Finally, 5 numerical simulation examples are given to illustrate the effectiveness and advantages of the proposed results.

Secondary delay-partition approach on robust performance analysis for uncertain time-varying Lurie nonlinear control system

This paper presents a multiparameter moment-matching-based model order reduction technique for parameterized interconnect networks via a novel two-directional Arnoldi process (TAP). It is referred to as a Parameterized Interconnect Macromodeling via a TAP (PIMTAP) algorithm. PIMTAP inherits the advantages of previous multiparameter moment-matching algorithms and avoids their shortfalls. It is numerically stable and adaptive. PIMTAP model yields the same form of the original state equations and preserves the passivity of parameterized RLC networks like the well-known method passive reduced-order interconnect macromodeling algorithm for nonparameterized RLC networks.

/pdf/model-order-reduction-of-parameterized-interconnect-networks-49i2hdgnko.pdf

Model Order Reduction of Parameterized Interconnect Networks via a Two-Directional Arnoldi Process

A two-directional Arnoldi process and its application to parametric model order reduction

This paper presents a multiparameter moment-matching based model order reduction technique for parameterized interconnect networks via a novel two-directional Arnoldi process. It is referred to as a PIMTAP algorithm, which stands for Parameterized Interconnect Macromodeling algorithm via a Two-directional Arnoldi Process. PIMTAP inherits the advantages of previous multiparameter moment-matching algorithms and avoids their shortfalls. It is numerically stable and adaptive, and preserves the passivity of parameterized RLC networks.

/pdf/parameterized-model-order-reduction-via-a-two-directional-13ycn2uf5b.pdf

Parameterized model order reduction via a two-directional Arnoldi process

https://web.cs.ucdavis.edu/~bai/publications/libailinsu11.pdf

A Structured Quasi-Arnoldi procedure for model order reduction of second-order systems

In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh– Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently, it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally, an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Arnoldi method applied to the linearized quadratic eigenvalue problem. Copyright © 2012 John Wiley & Sons, Ltd.

https://ir.nctu.edu.tw:443/bitstream/11536/21189/1/000314985700009.pdf

Yung-Ta Li

Papers

Model Order Reduction of Parameterized Interconnect Networks via a Two-Directional Arnoldi Process

A two-directional Arnoldi process and its application to parametric model order reduction

Parameterized model order reduction via a two-directional Arnoldi process

A Structured Quasi-Arnoldi procedure for model order reduction of second-order systems

A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems