arXiv subjects

Matthias Voigt

Publications and source records attributed to Matthias Voigt.

At least 19 recordsLinked to original sources

arXiv2026

Impulse Response Estimation via Laguerre-Fourier Expansion

The empirical transfer function estimate (ETFE) is a widely used method for system identification of linear time-invariant (LTI) systems in engineering disciplines such as acoustics, audio engineering, seismography, and tomography. However, ETFE suffers from numerical limitations when the excitation signal is band-limited or vanishes at certain frequencies, which is a common physical constraint of the excitation in practice. In such cases, division in the frequency domain becomes heavily ill-conditioned, and small measurement disturbances or numerical inaccuracies can degrade the solution. This paper presents L-ETFE, a generalization of ETFE based on Laguerre-Fourier expansions that addresses these limitations. After a suitable transformation, the method can yield a well-conditioned circulant problem even when the original ETFE system is ill-conditioned. Solving this problem via classical ETFE yields the discrete Laguerre-Fourier coefficients of the system's transfer function. The desired impulse response (IR) of the system-to-be-identified can then be recovered by a subsequent transformation pipeline. We derive novel and efficient algorithms for performing these transformations and analyse the conditioning of the transformed problem, explicitly characterizing its dependence on the input and a parameter used in the Laguerre-Fourier expansion. We evaluate the method on two simulated discrete-time LTI systems of varying complexity. The experiments demonstrate accurate IR recovery for spectral-zero and band-limited excitation, where standard ETFE fails.

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arXiv2026

Adaptive Kernel Methods

Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel Hilbert space (RKHS), determined solely by the chosen kernel and the dataset, whose elements identify the basis elements. Consequently, the projection operator underlying the kernel method depends on the loss function, the dataset, and the choice of ambient RKHS. In this study, we consider kernel methods whose solution spaces also depend on learnable parameters that are independent of the dataset. The resulting methods can be viewed as variable projection operators that depend on the loss function, the dataset, and the new learnable parameters instead of a fixed RKHS. This work has two main contributions. First, we propose an efficient approximation of kernels associated with infinite-dimensional RKHSs, commonly used to reduce the solution-space dimension for large datasets. Second, we construct fixed-dimensional, parameter-dependent solution spaces that enable highly efficient kernel models suitable for large-scale problems without the need to approximate kernels of infinite-dimensional RKHSs. Our novel family of adaptive kernel methods generalizes earlier approaches, including Random Fourier Features, and we demonstrate their effectiveness through several numerical experiments.

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arXiv2025

Trust-Region Methods with Low-Fidelity Objective Models

We introduce two multifidelity trust-region methods based on the Magical Trust Region (MTR) framework. MTR augments the classical trust-region step with a secondary, informative direction. In our approaches, the secondary ``magical'' directions are determined by solving coarse trust-region subproblems based on low-fidelity objective models. The first proposed method, Sketched Trust-Region (STR), constructs this secondary direction using a sketched matrix to reduce the dimensionality of the trust-region subproblem. The second method, SVD Trust-Region (SVDTR), defines the magical direction via a truncated singular value decomposition of the dataset, capturing the leading directions of variability. Several numerical examples illustrate the potential gain in efficiency.

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arXiv2025

Adaptive Multilevel Newton: A Quadratically Convergent Optimization Method

Newton's method may exhibit slower convergence than vanilla Gradient Descent in its initial phase on strongly convex problems. Classical Newton-type multilevel methods mitigate this but, like Gradient Descent, achieve only linear convergence near the minimizer. We introduce an adaptive multilevel Newton-type method with a principled automatic switch to full Newton once its quadratic phase is reached. The local quadratic convergence for strongly convex functions with Lipschitz continuous Hessians and for self-concordant functions is established and confirmed empirically. Although per-iteration cost can exceed that of classical multilevel schemes, the method is efficient and consistently outperforms Newton's method, Gradient Descent, and the multilevel Newton method, indicating that second-order methods can outperform first-order methods even when Newton's method is initially slow. The promising empirical results open new avenues for designing reduced-cost second- and high-order methods with extremely fast convergence rates.

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arXiv2025

KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur'e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$ or frequency-weighted $\mathcal{H}_2$ distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur'e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a non-global local minimum. Numerical examples illustrate the effectiveness of the proposed method.

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arXiv2025

Generalized rational Prony and Bernoulli methods

The generalized operator-based Prony method is an important tool for describing signals which can be written as finite linear combinations of eigenfunctions of certain linear operators. On the other hand, Bernoulli's algorithm and its generalizations can be used to recover the parameters of rational functions belonging to finite-dimensional subspaces of $H_2$ Hardy-Hilbert spaces. In this work, we discuss several results related to these methods. We discuss a rational variant of the generalized operator-based Prony method and show that in fact, any Prony problem can be treated this way. This realization establishes the connection between Prony and Bernoulli methods and allows us to address some well-known numerical pitfalls. Several numerical experiments are provided to showcase the usefulness of the introduced methods. These include problems related to the identification of time-delayed linear systems and parameter recovery problems in reproducing kernel Hilbert spaces.

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arXiv2024

Fast and memory-efficient optimization for large-scale data-driven predictive control

Recently, data-enabled predictive control (DeePC) schemes based on Willems' fundamental lemma have attracted considerable attention. At the core are computations using Hankel-like matrices and their connection to the concept of persistency of excitation. We propose an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems. To this end, we apply factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations. To take advantage of this factorization in an optimal control solver and to reduce the effect of inherent bad conditioning of the Hankel-like matrices, we propose an augmented Lagrangian lBFGS-method. We illustrate the performance of our method by means of a numerical study.

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arXiv2024

Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics.

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arXiv2022

Fixed-Order H-Infinity Controller Design for Port-Hamiltonian Systems

We present a new fixed-order H-infinity controller design method for potentially large-scale port-Hamiltonian (pH) plants. Our method computes controllers that are also pH (and thus passive) such that the resulting closed-loop systems is again passive, which ensures closed-loop stability simply from the structure of the plant and controller matrices. In this way, we can avoid computationally expensive eigenvalue computations that would otherwise be necessary. In combination with a sample-based objective function which allows us to avoid multiple evaluations of the H-infinity norm (which is typically the main computational burden in fixed-order H-infinity controller synthesis), this makes our method well-suited for plants with a high state-space dimension. In our numerical experiments, we show that applying a passivity-enforcing post-processing step after using well-established H-infinity synthesis methods often leads to a deteriorated H-infinity performance. In constrast to that, our method computes pH controllers, that are automatically passive and simultaneously aim to minimize the H-infinity norm of the closed-loop transfer function. Moreover, our experiments show that for large-scale plants, our method is significantly faster than the well-established fixed-order H-infinity controller synthesis methods.

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arXiv2022

Structure-Preserving Model Order Reduction for Index Two Port-Hamiltonian Descriptor Systems

We present a new optimization-based structure-preserving model order reduction (MOR) method for port-Hamiltonian descriptor systems (pH-DAEs) with differentiation index two. Our method is based on a novel parameterization that allows us to represent any linear time-invariant pH-DAE with a minimal number of parameters, which makes it well-suited to model reduction. We propose two algorithms which directly optimize the parameters of a reduced model to approximate a given large-scale model with respect to either the H-infinity or the H-2 norm. This approach has several benefits. Our parameterization ensures that the reduced model is again a pH-DAE system and enables a compact representation of the algebraic part of the large-scale model, which in projection-based methods often requires a more involved treatment. The direct optimization is entirely based on transfer function evaluations of the large-scale model and is therefore independent of the system matrices' structure. Numerical experiments are conducted to illustrate the high accuracy and small reduced model orders in comparison to other structure-preserving MOR methods.

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arXiv2022

Structure-Preserving Model Order Reduction for Index One Port-Hamiltonian Descriptor Systems

We develop optimization-based structure-preserving model order reduction (MOR) methods for port-Hamiltonian (pH) descriptor systems of differentiation index one. Descriptor systems in pH form permit energy-based modeling and intuitive coupling of physical systems across different physical domains, scales, and accuracies. This makes pH models well-suited building-blocks for component-wise modeling of large system networks. In this context, it is often necessary to preserve the pH structure during MOR. We discuss current projection-based and structure-preserving MOR algorithms for pH systems and present a new optimization-based framework for that task. The benefits of our method include a simplified treatment of algebraic constraints and often a higher accuracy of the resulting reduced-order model, which is demonstrated by several numerical examples.

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arXiv2022

SOBMOR: Structured Optimization-Based Model Order Reduction

Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non-structure-preserving counterparts. In this paper, we present a framework for structure-preserving MOR, which allows to compute structured reduced order models (ROMs) with a much higher accuracy. The framework is based on parameter optimization, i.e., the elements of the system matrices of the ROM are iteratively varied to minimize an objective functional that measures the difference between the FOM and the ROM. The structural constraints can be encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems. We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second-order system in a comparison with other structure-preserving MOR algorithms.

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arXiv2022

Balanced Truncation Model Reduction with A Priori Error Bounds for LTI Systems with Nonzero Initial Value

In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a joint projection reduced-order model in which the part of the system depending only on the input and the one depending only on the initial value are reduced at once and we prove an a priori error bound. With this result at hand, we derive a separate projection procedure in which the two parts are reduced separately. This gives the freedom to choose different reduction orders for the different subsystems. Moreover, we discuss how the reduced-order models can be constructed in practice. Since the error bounds are parameter-dependent we show how they can be optimized efficiently. We conclude this paper by comparing our results with the ones from the literature by a series of numerical experiments.

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arXiv2022

Structure-Preserving Model Reduction for Dissipative Mechanical Systems

Suppressing vibrations in mechanical systems, usually described by second-order dynamical models, is a challenging task in mechanical engineering in terms of computational resources even nowadays. One remedy is structure-preserving model order reduction to construct easy-to-evaluate surrogates for the original dynamical system having the same structure. In our work, we present an overview of recently developed structure-preserving model reduction methods for second-order systems. These methods are based on modal and balanced truncation in different variants, as well as on rational interpolation. Numerical examples are used to illustrate the effectiveness of all described methods.

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arXiv2022

Derivative Interpolating Subspace Frameworks for Nonlinear Eigenvalue Problems

We first consider the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational part of the rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided or one-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a reduced transfer function. At every iteration, the projection subspaces are expanded to attain Hermite interpolation conditions at the eigenvalues of the reduced transfer function closest to the target, which in turn leads to a new reduced transfer function. We prove in theory that, when a sequence of eigenvalues of the reduced transfer functions converges to an eigenvalue of the full problem, it converges at least at a quadratic rate. In the second part, we extend the proposed framework to locate the eigenvalues of a general square large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we exploit a representation $\mathcal{R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined in terms of the block components of $T(\cdot)$. The numerical experiments illustrate that the proposed framework is reliable in locating a few eigenvalues closest to the target point, and that, with respect to runtime, it is competitive to established methods for nonlinear eigenvalue problems.

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arXiv2021

Adaptive Sampling for Structure Preserving Model Order Reduction of Port-Hamiltonian Systems

We present an adaptive sampling strategy for the optimization-based structure preserving model order reduction (MOR) algorithm developed in [Schwerdtner, P. and Voigt, M. (2020). Structure preserving model order reduction by parameter optimization, Preprint arXiv:2011.07567]. This strategy reduces the computational demand and the required a priori knowledge about the given full order model, while at the same time retaining a high accuracy compared to other structure preserving but also unstructured MOR algorithms. A numerical study with a port-Hamiltonian benchmark system demonstrates the effectiveness of our method combined with its new adaptive sampling strategy. We also investigate the distribution of the sample points.

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arXiv2020

Balanced truncation model reduction for symmetric second order systems -- A passivity-based approach

We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.

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arXiv2020

Discounted Cost Linear Quadratic Gaussian Control for Descriptor Systems

We consider the linear quadratic Gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon. Based on recent results from the deterministic framework, we characterize the feasibility of this problem using a linear matrix inequality. In particular, conditions for existence and uniqueness of optimal controls are derived, which are weaker compared to the standard approaches in the literature. We further show that also for the stochastic problem, the optimal control is given in terms of the stabilizing solution of the Lur'e equation, which generalizes the algebraic Riccati equation.

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