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

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Methods of Numerical Integration

An overview of model reduction methods and a comparison of the resulting algorithms is presented These approaches are divided into two broad categories, namely SVD based and moment matching based methods It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better

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A Survey of Model Reduction Methods for Large-Scale Systems

Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.

Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function . The results are illustrated on randomly generated systems and on a problem of power-electronic actuator placement in a model of the European power grid.

On Submodularity and Controllability in Complex Dynamical Networks

We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.

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Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems

This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX+XAT=-BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A.
This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

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Low Rank Solution of Lyapunov Equations

This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX+XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. 
This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

Low-Rank Solution of Lyapunov Equations

In this paper we present a new algorithm for computing reduced-order models of interconnect which utilizes the dominant controllable subspace of the system. The dominant controllable modes are computed via a new iterative Lyapunov equation solver, Vector ADI. This new algorithm is as inexpensive as Krylov subspace-based moment matching methods, and often produces a better approximation over a wide frequency range. A spiral inductor and a transmission line example show this new method can be much more accurate than moment matching via Arnoldi.

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An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect

We evaluate the fractional integral $I^\alpha[f](t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-\tau)^{\alpha-1}\,f(\tau)\,d\tau$, $0<\alpha<1$, at time steps $t=\Delta t,2\Delta t,\dots,N\Delta t$ by making use of the integral representation of the convolution kernel $t^{\alpha-1}=\frac{1}{\Gamma(1-\alpha)}\int_0^{\infty}e^{-\xi\,t}\,\xi^{-\alpha}\,d\xi$. We construct an efficient $Q$-point quadrature of this integral representation and use it as a part of a fast time stepping method. The new method has algorithmic complexity $O(NQ)$ and storage requirement $O(Q)$. The number of quadrature nodes $Q$ is independent of $N$ and grows like $O\bigl(\bigl(-\log\epsilon-\log\Delta t\bigr)^2\bigr)$, where $\epsilon$ is the quadrature error tolerance and $\Delta t$ is the size of the time step. The (possible) singularity of $f$ near $\tau=0$ is taken into account. This new method is particularly well-suited for long time simulations.

Jing-Rebecca Li

Papers

Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems

Low Rank Solution of Lyapunov Equations

Low-Rank Solution of Lyapunov Equations

An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect

A Fast Time Stepping Method for Evaluating Fractional Integrals