arXiv subjects

Serkan Gugercin

Publications and source records attributed to Serkan Gugercin.

At least 19 recordsLinked to original sources

arXiv2026

Beyond linear subspaces: Nonlinear moment matching meets quadratic manifolds

Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.

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arXiv2026

Contour integral methods and model order reduction for parametric linear control systems

This paper introduces a contour integral method (CIM) for efficiently computing outputs of parametric linear systems in control form over specified time intervals and to a user-prescribed accuracy. The CIM approximates the inverse Laplace transform via a quadrature rule applied along a modified integration contour. For parametric systems, we show how CIM integrates effectively with projection-based model order reduction (MOR) where a greedy algorithm builds the projection spaces following an error estimate we derive for this setting. We additionally demonstrate that the developed projection framework naturally enforces Hermite interpolation conditions. This combination substantially lowers the cost of evaluating the input-output relations across the parameter domain, for a wide range of input functions, and for initial conditions well captured by a low-dimensional subspace.. We demonstrate the accuracy and efficiency of the approach on benchmark non-parametric and parametric control systems, comparing against state-of-the-art projection-based MOR methods.

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arXiv2026

Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions

Bilinear systems with quadratic outputs (BQO) have recently emerged as an important system class, arising naturally in applications where both the dynamics and the quantities of interest depend nonlinearly on the state. Despite the growing interest in this class of systems, a systematic $\mathcal{H}_2$ framework for BQO systems has been lacking. In this paper, we develop such a framework by establishing an $\mathcal{H}_2$ inner product and norm for BQO systems, deriving output bounds in terms of the $\mathcal{H}_2$ norm, and obtaining first-order optimality conditions for $\mathcal{H}_2$ optimal model reduction. Building on these theoretical foundations, we propose an algorithm that computes a reduced BQO system satisfying these optimality conditions, and thus generalizing existing $\mathcal{H}_2$ optimal methods for bilinear and linear quadratic output systems. The effectiveness of the proposed framework is demonstrated on two numerical examples.

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arXiv2026

Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

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arXiv2026

Load Identification in Bistable Spacecraft Booms via Parametric Data-Driven Modeling

Bistable tape spring booms are used on spacecraft for their ability to self-deploy using stored strain energy. However, their uncontrolled deployment can induce mechanical shocks that are variable as a function of material properties and temperature, and may damage sensitive satellite components and disrupt attitude control. Because traditional Finite Element Analysis (FEA) struggles to accurately capture this highly nonlinear behavior, we solve the inverse problem to estimate these loads from dynamic response measurements. Previous data-driven approaches using Vector Fitting required time-consuming retesting for every specific load level due to the boom's load-dependent dynamic behavior. To overcome this limitation, we introduce a parametric data-driven framework where a parametric transfer-function model of a composite tape spring boom is developed using force and velocity measurements. The parametric Adaptive Antoulas-Anderson algorithm (p-AAA) is used to construct a single parametric (multivariate) transfer function capable of capturing the nonlinear response of the boom to load amplitude. To evaluate the proposed framework, the boom is excited at its base at 15 distinct load levels using a single-axis reference input signal. Results demonstrate that the single parametric model outperformed the best discrete non-parametric case, reducing the total relative force estimation error for the reference signal by nearly 38\%. For experimental validation, the boom is subjected to sinusoidal, triangular and square signals. The cross validation results further supported this generalized performance. Collectively, these results show that the proposed parametric model accurately reconstructs input forces from velocity measurements alone, offering a solution for onboard diagnostics in future space missions.

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arXiv2026

From Interpolation to $\mathcal{H}_2$ Optimality: Model Reduction for Infinite-Dimensional Linear Control Systems

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.

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arXiv2026

A refined nonlinear least-squares method for the rational approximation problem

The adaptive Antoulas-Anderson (AAA) algorithm for rational approximation is a widely used method for the efficient construction of highly accurate rational approximations to given data. While AAA can often produce rational approximations accurate to any prescribed tolerance, these approximations may have degrees larger than what is actually required to meet the given tolerance. In this work, we consider the adaptive construction of interpolating rational approximations while aiming for the smallest feasible degree to satisfy a given error tolerance. To this end, we introduce refinement approaches to the linear least-squares step of the classical AAA algorithm that aim to minimize the true nonlinear least-squares error with respect to the given data. Furthermore, we theoretically analyze the derived approaches in terms of the corresponding gradients from the resulting minimization problems and use these insights to propose a new greedy framework that ensures monotonic error convergence. Numerical examples from function approximation and model order reduction verify the effectiveness of the proposed algorithm to construct accurate rational approximations of small degrees.

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arXiv2026

A parametric Keldysh decomposition

Contour integral algorithms seek to compute a small number of eigenvalues located within a bounded region of the complex plane. These methods can be applied to both linear and nonlinear matrix eigenvalue problems. In the latter case, the foundation of these methods comes from the Keldysh decomposition, which breaks the nonlinear matrix-valued function into two parts: a rational function whose poles match the desired eigenvalues, and a remainder term that is analytic within the target region. Under contour integration this analytic part vanishes (via Cauchy's theorem), leaving only the component containing the desired eigenvalues. We propose an extension of the Keldysh decomposition for matrix-valued functions that depend analytically on an additional parameter. We establish key properties of this parametric Keldysh decomposition, and introduce an algorithm for solving parametric nonlinear eigenvalue problems that is based upon it.

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arXiv2025

Multivariate Rational Approximation of Scattered Data Using the p-AAA Algorithm

Many algorithms for approximating data with rational functions are built on interpolation or least-squares approximation. Inspired by the adaptive Antoulas-Anderson (AAA) algorithm for the univariate case, the parametric adaptive Antoulas-Anderson (p-AAA) algorithm extends this idea to the multivariate setting, combining least-squares and interpolation formulations into a single effective approximation procedure. In its original formulation p-AAA operates on grid data, requiring access to function samples at every combination of discrete sampling points in each variable. In this work we extend the p-AAA algorithm to scattered data sets, without requiring uniform/grid sampling. In other words, our proposed p-AAA formulation operates on a set of arbitrary sampling points and is not restricted to a grid structure for the sampled data. Towards this goal, we introduce several formulations for rational least-squares optimization problems that incorporate interpolation conditions via constraints. We analyze the structure of the resulting optimization problems and introduce structured matrices whose singular value decompositions yield closed-form solutions to the underlying least-squares problems. Several examples illustrate computational aspects and the effectiveness of our proposed procedure.

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arXiv2025

Interpolatory Approximations of PMU Data: Dimension Reduction and Pilot Selection

This work investigates the reduction of phasor measurement unit (PMU) data through low-rank matrix approximations. To reconstruct a PMU data matrix from fewer measurements, we propose the framework of interpolatory matrix decompositions (IDs). In contrast to methods relying on principal component analysis or singular value decomposition, IDs recover the complete data matrix using only a few of its rows (PMU datastreams) and/or a few of its columns (snapshots in time). This row-/column-based compression enables real-time monitoring of power transmission systems using measurements from a smaller subset of pilot datastreams, thereby minimizing communication bandwidth. The ID perspective gives a rigorous error bound on the quality of the data compression. We propose selecting the pilot measurements used in an ID via the discrete empirical interpolation method (DEIM), a greedy algorithm that aims to control the error bound. This bound yields a computable estimate of the reconstruction error during online operations. A violation of this estimate suggests a change in the system's operating conditions and thus serves as a tool for fault detection. Following a disturbance, DEIM can be used to localize the event source across all buses with high accuracy. Numerical tests on synthetic PMU data demonstrate DEIM's excellent performance in data compression and validate the proposed DEIM-based fault-detection and localization method.

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arXiv2025

Data-driven balanced truncation for linear systems with quadratic outputs

We develop the framework for a non-intrusive, quadrature-based method for approximate balanced truncation (QuadBT) of linear systems with quadratic outputs, thus extending the applicability of QuadBT, which was originally designed for data-driven balanced truncation of standard linear systems with linear outputs only. The new approach makes use of the time-domain and frequency-domain quadrature-based representation of the system's infinite Gramians, only implicitly. We show that by sampling solely the extended impulse responses of the original system and their derivatives (or the corresponding transfer functions), we construct a reduced-order model that mimics the approximation quality of the intrusive (projection-based) balanced truncation. We validate the proposed framework on a numerical example.

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arXiv2025

Optimal Damping for the 1D Wave Equation Using a Single Damper

Vibrational structures are susceptible to catastrophic failures or structural damages when external forces induce resonances or repeated unwanted oscillations. One common mitigation strategy is to use dampers to suppress these disturbances. This leads to the problem of finding optimal damper viscosities and positions for a given vibrational structure. Although extensive research exists for the case of finite-dimensional systems, optimizing damper positions remains challenging due to its discrete nature. To overcome this, we introduce a novel model for the damped wave equation (at the PDE level) with a damper of viscosity $\mathfrak{g}$ at position $\mathfrak{p}$ and develop a system-theoretic input/output-based analysis in the frequency domain. In this system-theoretic formulation, while we consider average displacement as the output, for input (forcing), we analyze two separate cases, namely, the uniform and boundary forcing. For both cases, explicit formulas are derived for the corresponding transfer functions, parametrized by $\mathfrak{p}$ and $\mathfrak{g}$. This explicit parametrization by $\mathfrak{p}$ and $\mathfrak{g}$ facilitates analyzing the optimal damping problem (at the PDE level) using norms such as the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ norms. We also examine limiting cases, such as when the viscosity is very large or when no external damping is present. To illustrate our approach, we present numerical examples, compare different optimization criteria, and discuss the impact of damping parameters on the damped wave equation.

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arXiv2025

Bilinear Quadratic Output Systems and Balanced Truncation

Dynamical systems with quadratic outputs have recently attracted significant attention. In this paper, we consider bilinear dynamical systems, a special class of weakly nonlinear systems, with a quadratic output. We develop various primal-dual formulations for these systems and define the corresponding system Gramians. Conditions for the existence and uniqueness of these Gramians are established, and the generalized Lyapunov equations they satisfy are derived. Using these Gramians and their truncated versions, which are computationally more efficient, we construct a balanced truncation framework for bilinear systems with quadratic outputs. The proposed approach is demonstrated through two numerical examples.

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arXiv2025

Second-order AAA algorithms for structured data-driven modeling

The data-driven modeling of dynamical systems has become an essential tool for the construction of accurate computational models from real-world data. In this process, the inherent differential structures underlying the considered physical phenomena are often neglected making the reinterpretation of the learned models in a physically meaningful sense very challenging. In this work, we present three data-driven modeling approaches for the construction of dynamical systems with second-order differential structure directly from frequency domain data. Based on the second-order structured barycentric form, we extend the well-known Adaptive Antoulas-Anderson algorithm to the case of second-order systems. Depending on the available computational resources, we propose variations of the proposed method that prioritize either higher computation speed or greater modeling accuracy, and we present a theoretical analysis for the expected accuracy and performance of the proposed methods. Three numerical examples demonstrate the effectiveness of our new structured approaches in comparison to classical unstructured data-driven modeling.

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arXiv2025

$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

This paper addresses the $\mathcal{H}_2$-optimal approximation of linear dynamical systems with quadratic-output functions, also known as linear quadratic-output systems. Our major contributions are threefold. First, we derive interpolatory first-order optimality conditions for the linear quadratic-output $\mathcal{H}_2$ minimization problem. These conditions correspond to the mixed-multipoint tangential interpolation of the full-order linear- and quadratic-output transfer functions, and generalize the Meier-Luenberger optimality framework for the $\mathcal{H}_2$-optimal model reduction of linear time-invariant systems. Second, given the optimal interpolation data, we show how to enforce the interpolatory optimality conditions explicitly by Petrov-Galerkin projection of the full-order model. Third, to find the optimal interpolation data, we build on this projection framework and propose a generalization of the iterative rational Krylov algorithm for the $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems, called LQO-IRKA. Upon convergence, LQO-IRKA produces reduced linear quadratic-output systems that satisfy the interpolatory optimality conditions. The method only requires solving shifted linear systems and matrix-vector products, thus making it suitable for large-scale problems. Numerical examples are included to illustrate the effectiveness of the proposed method.

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arXiv2025

Energy-Based Approximation of Linear Systems with Polynomial Outputs

Controllability and observability energy functions play a fundamental role in model order reduction and are inherently connected to optimal control problems. For linear dynamical systems the energy functions are known to be quadratic polynomials and various low-rank approximation techniques allow for computing them in a large-scale setting. For nonlinear problems computing the energy functions is significantly more challenging. In this paper, we investigate a special class of nonlinear systems that have a linear state and a polynomial output equation. We show that the energy functions of these systems are again polynomials and investigate under which conditions they can effectively be approximated using low-rank tensors. Further, we introduce a new perspective on the well-established balanced truncation method for linear systems which then readily generalizes to the nonlinear systems under consideration. This new perspective yields a novel energy-based model order reduction procedure that accurately captures the input-output behavior of linear systems with polynomial outputs via a low-dimensional reduced order model. We demonstrate the effectiveness of our approach via two numerical experiments.

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arXiv2025

Balanced truncation with conformal maps

We consider the problem of constructing reduced models for large scale systems with poles in general domains in the complex plane (as opposed to, e.g., the open left-half plane or the open unit disk). Our goal is to design a model reduction scheme, building upon theoretically established methodologies, yet encompassing this new class of models. To this aim, we develop a balanced truncation framework through conformal maps to handle poles in general domains. The major difference from classical balanced truncation resides in the formulation of the Gramians. We show that these new Gramians can still be computed by solving modified Lyapunov equations for specific conformal maps. A numerical algorithm to perform balanced truncation with conformal maps is developed and is tested on three numerical examples, namely a heat model, the Schrödinger equation, and the undamped linear wave equation, the latter two having spectra on the imaginary axis.

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arXiv2025

Multivariate Rational Approximation via Low-Rank Tensors and the p-AAA Algorithm

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 the barycentric form of a rational approximant, has established itself as a powerful tool for efficiently computing such approximations. The p-AAA algorithm, an extension of the AAA algorithm specifically designed to address multivariate approximation problems, has been recently introduced. A common challenge in multivariate approximation methods is that multivariate problems with a large number of variables often pose significant memory and computational demands. To tackle this hurdle in the setting of p-AAA, we first introduce barycentric forms that are represented in the terms of separable functions. This then leads to the low-rank p-AAA algorithm which leverages low-rank tensor decompositions in the setting of barycentric rational approximations. We discuss various theoretical and practical aspects of the proposed computational framework and showcase its effectiveness on four numerical examples. We focus specifically on applications in parametric reduced-order modeling for which higher-dimensional data sets can be tackled effectively with our novel procedure.

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