Numerical simulation of large-scale dynamical systems plays a fundamental role in studying wide range complex physical phenomena; however, the inherent nature models often leads to unmanageable demands on computational resources. Model reduction aims reduce this burden by generating reduced that are faster and cheaper simulate, yet accurately represent original system behavior. linear, nonparametric has reached considerable level maturity, as reflected several survey papers books. However,...

Abstract Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey balancing related methods and their corresponding error norms, also introduce some new results. Five are studied: (1) Lyapunov balancing, (2) stochastic (3) bounded real (4) positive (5) frequency weighted balancing. For multiplicative-type bound. Moreover, for certain subclass systems, modified positive-real scheme with an absolute bound proposed. We develop frequency-weighted...

The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area dynamical systems and simulation. In literature, two independent frameworks have evolved focusing either on solution Lyapunov equations one hand or interpolation transfer functions other, without any apparent connection between approaches. this paper, we develop a new unifying framework for approximation using best properties underlying Hilbert space. This leads to set local optimality conditions taking...

This paper introduces a new framework for constructing the discrete empirical interpolation method (\sf DEIM) projection operator. The node selection procedure is formulated using QR factorization with column pivoting, and it enjoys sharper error bound \sf DEIM error. Furthermore, subspace $\mathcal{U}$ given as range of an orthonormal ${\mathsf U}$, does not change if U}$ replaced by U} \Omega$ arbitrary unitary matrix $\Omega$. In large-scale setting, approach allows modifications that use...

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having structured dependence on parameters that we wish to retain in the reduced-order model. The parameter may be or nonlinear and is retained Moreover, are able give conditions under which gradient Hessian system response with respect matched systematic approach built established $\mathcal{H}_2$ optimal methods will produce models...

This paper presents a structure-preserving model reduction approach applicable to large-scale, nonlinear port-Hamiltonian systems. Structure preservation in the step ensures retention of structure which, turn, stability and passivity reduced model. Our analysis provides priori error bounds for both state variables outputs. Three techniques are considered constructing bases needed reduction: one that utilizes proper orthogonal decompositions, $\mathcal{H}_2/\mathcal{H}_{\infty}$-derived...

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, first illustrate that employing subspace conditions from the standard state space settings to systems generically leads unbounded H2 or H-infinity errors due mismatch polynomial parts full and reduced-order transfer functions. We then develop modified based on deflating subspaces guarantee bounded error. For special cases index-1 index-2 systems, also...

Vector fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer as VF, based on least-squares the measurements by function, using an iterative reallocation poles approximant. We show that one can improve performance VF significantly particular choice sampling points and properly weighting their contribution quadrature rules connect objective with ${\mathcal{H}}_2$ error measure. Our modified...

In this paper, we focus on model reduction of large-scale bilinear systems. The main contributions are threefold. First, introduce a new framework for interpolatory contrast to the existing methods where interpolation is forced some leading subsystem transfer functions, shows how enforce multipoint underlying Volterra series. Then, show that first-order conditions optimal H2 systems require multivariate Hermite in terms series framework; and thus extend interpolation-based necessary...

We investigate the optimal model reduction problem for large-scale quadratic-bilinear (QB) control systems. Our contributions are threefold. First, we discuss variational analysis and Volterra series formulation QB then define $\mathcal H_2$-norm a system based on kernels of underlying propose truncated as well. Next, derive first-order necessary conditions an approximation, where optimality is measured in terms error system. iterative algorithm, which upon convergence yields reduced-order...

This work presents a data-driven nonintrusive model reduction approach for large-scale time-dependent systems with linear state dependence. Traditionally, is performed in an intrusive projection-based framework, where the operators of full are required either explicitly assembled form or implicitly through routine that returns action on vector. Our constructs reduced models directly from trajectories inputs and outputs model, without requiring full-model operators. These generated by running...

Stability of Discrete Empirical Interpolation and Gappy Proper Orthogonal Decomposition with Randomized Deterministic Sampling Points

.The AAA algorithm has become a popular tool for data-driven rational approximation of single-variable functions, such as transfer functions linear dynamical system. In the setting parametric systems appearing in many prominent applications, underlying (transfer) function to be modeled is multivariate function. With this mind, we develop framework approximating where approximant constructed barycentric form. The method data driven, sense that it does not require access full state-space model...

Vector Fitting (VF) is a popular method of constructing rational approximants that provides least squares fit to frequency response measurements. In an earlier work, we provided analysis VF for scalar-valued functions and established connection with optimal $H_2$ approximation. We build on this work extend the previous framework include construction effective approximations matrix-valued functions, problem which presents significant challenges do not appear in scalar case. Transfer...

We present a novel reformulation of balanced truncation, classical model reduction method. The principal innovation that we introduce comes through the use system response data has been either measured or computed, without reference to any prescribed realization original model. Data are represented by sampled values transfer function impulse corresponding discuss parallels our approach bears with Loewner framework, another popular data-driven illustrate numerically in both continuous-time...

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and closely related Lur'e equations. The passivation problem in our framework corresponds to finding perturbation given non-passive system that renders passive while minimizing $\mathcal{H}_2$ or frequency-weighted distance between original resulting system. show this can be formulated as an unconstrained optimization whose objective...

Approximations based on rational functions are widely used in various applications across computational science and engineering. For univariate functions, the adaptive Antoulas-Anderson algorithm (AAA), which uses barycentric form of a approximant, has established itself as powerful tool for efficiently computing such approximations. The p-AAA algorithm, an extension AAA specifically designed to address multivariate approximation problems, been recently introduced. A common challenge methods...