arXiv subjects

Serkan Gugercin

Publications and source records attributed to Serkan Gugercin.

At least 55 records · Page 3Linked to original sources

arXiv2022

A unifying framework for tangential interpolation of structured bilinear control systems

In this paper, we consider the structure-preserving model order reduction problem for multi-input/multi-output bilinear control systems by tangential interpolation. We propose a new type of tangential interpolation problem for structured bilinear systems, for which we develop a new structure-preserving interpolation framework. This new framework extends and generalizes different formulations of tangential interpolation for bilinear systems from the literature and also provides a unifying framework. We then derive explicit conditions on the projection spaces to enforce tangential interpolation in different settings, including conditions for tangential Hermite interpolation. The analysis is illustrated by means of three numerical examples.

math.NA↗

arXiv2022

Structure-Preserving Model Reduction for Nonlinear Power Grid Network

We develop a structure-preserving system-theoretic model reduction framework for nonlinear power grid networks. First, via a lifting transformation, we convert the original nonlinear system with trigonometric nonlinearities to an equivalent quadratic nonlinear model. This equivalent representation allows us to employ the $\mathcal{H}_2$-based model reduction approach, Quadratic Iterative Rational Krylov Algorithm (Q-IRKA), as an intermediate model reduction step. Exploiting the structure of the underlying power network model, we show that the model reduction bases resulting from Q-IRKA have a special subspace structure, which allows us to effectively construct the final model reduction basis. This final basis is applied on the original nonlinear structure to yield a reduced model that preserves the physically meaningful (second-order) structure of the original model. The effectiveness of our proposed framework is illustrated via two numerical examples.

eess.SY↗

arXiv2021

Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

eess.SY↗

arXiv2021

Structure-preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems

We examine interpolatory model reduction methods that are well-suited for treating large scale port-Hamiltonian differential-algebraic systems in a way that is able to preserve and indeed, take advantage of the underlying structural features of the system. We introduce approaches that incorporate regularization together with prudent selection of interpolation data. We focus on linear time-invariant systems and present a systematic treatment of a variety of model classes that include combinations of index-$1$ and index-$2$ systems, describing in particular how constraints may be represented in the transfer function and then preserved with interpolatory methods. We propose an algorithm to generate effective interpolation data and illustrate its effectiveness via two numerical examples.

math.NA↗

arXiv2021

A wavelet-based dynamic mode decomposition for modeling mechanical systems from partial observations

Dynamic mode decomposition (DMD) has emerged as a popular data-driven modeling approach to identifying spatio-temporal coherent structures in dynamical systems, owing to its strong relation with the Koopman operator. For dynamical systems with external forcing, the identified model should not only be suitable for a specific forcing function but should generally approximate the input-output behavior of the underlying dynamics. A novel methodology for modeling those classes of dynamical systems is proposed in the present work, using wavelets in conjunction with the input-output dynamic mode decomposition (ioDMD). Our non-intrusive approach constructs numerical models directly from trajectories of the full model's inputs and outputs, without requiring the full-model operators. These trajectories are generated by running a simulation of the full model or observing the original dynamical systems' response to inputs in an experimental framework. Hence, the present methodology is applicable for dynamical systems whose internal state vector measurements are not available. Instead, data from only a few output locations are only accessible, as often the case in practice. The present methodology's applicability is explained by modeling the input-output response of a finite element beam model. The WDMD provides a linear state-space representation of the dynamical system using the response measurements and the corresponding input forcing functions. The developed state-space model is then used to simulate the beam's response towards different types of forcing functions. The method is further validated on an experimental data set using modal analysis on a simple free-free beam, demonstrating the efficacy of the proposed methodology as an appropriate candidate for modeling practical dynamical systems despite having no access to internal state measurements and treating the full model as a black-box.

eess.SP↗

arXiv2021

Data-driven balancing of linear dynamical systems

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

math.NA↗

arXiv2021

Structured vector fitting framework for mechanical systems

In this paper, we develop a structure-preserving formulation of the data-driven vector fitting algorithm for the case of modally damped mechanical systems. Using the structured pole-residue form of the transfer function of modally damped second-order systems, we propose two possible structured extensions of the barycentric formula of system transfer functions. Integrating these new forms within the classical vector fitting algorithm leads to the formulation of two new algorithms that allow the computation of modally damped mechanical systems from data in a least squares fashion. Thus, the learned model is guaranteed to have the desired structure. We test the proposed algorithms on two benchmark models.

math.NA↗

arXiv2021

Data-driven modeling of power networks

We develop a non-intrusive data-driven modeling framework for power network dynamics using the Lift and Learn approach of \cite{QianWillcox2020}. A lifting map is applied to the snapshot data obtained from the original nonlinear swing equations describing the underlying power network such that the lifted-data corresponds to quadratic nonlinearity. The lifted data is then projected onto a lower dimensional basis and the reduced quadratic matrices are fit to this reduced lifted data using a least-squares measure. The effectiveness of the proposed approach is investigated by two power network models.

eess.SY↗

arXiv2021

Parametric model reduction via rational interpolation along parameters

We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.

math.NA↗

arXiv2021

Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework

We extend the AAA (Adaptive-Antoulas-Anderson) algorithm to develop a data-driven modeling framework for linear systems with quadratic output (LQO). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of LQO systems. This analysis leads to the proposed AAA-LQO algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven LQO models. Two numerical test cases illustrate the efficiency of the proposed method.

math.NA↗

arXiv2021

Structure-preserving Model Reduction of Parametric Power Networks

We develop a structure-preserving parametric model reduction approach for linearized swing equations where parametrization corresponds to variations in operating conditions. We employ a global basis approach to develop the parametric reduced model in which we concatenate the local bases obtained via $\mathcal{H}_2$-based interpolatory model reduction. The residue of the underlying dynamics corresponding to the simple pole at zero varies with the parameters. Therefore, to have bounded $\mathcal{H}_2$ and $\mathcal{H}_\infty$ errors, the reduced model residue for the pole at zero should match the original one over the entire parameter domain. Our framework achieves this goal by enriching the global basis based on a residue analysis. The effectiveness of the proposed method is illustrated through two numerical examples.

eess.SY↗

arXiv2021

Estimating Experimental Dispersion Curves from Steady-State Frequency Response Measurements

Dispersion curves characterize the frequency dependence of the phase and the group velocities of propagating elastic waves. Many analytical and numerical techniques produce dispersion curves from physics-based models. However, it is often challenging to accurately model engineering structures with intricate geometric features and inhomogeneous material properties. For such cases, this paper proposes a novel method to estimate group velocities from experimental data-driven models. Experimental frequency response functions (FRFs) are used to develop data-driven models, {which are then used to estimate dispersion curves}. The advantages of this approach over other traditionally used transient techniques stem from the need to conduct only steady-state experiments. In comparison, transient experiments often need a higher-sampling rate for wave-propagation applications and are more susceptible to noise. The vector-fitting (VF) algorithm is adopted to develop data-driven models from experimental in-plane and out-of-plane FRFs of a one-dimensional structure. The quality of the corresponding data-driven estimates is evaluated using an analytical Timoshenko beam as a baseline. The data-driven model (using the out-of-plane FRFs) estimates the anti-symmetric ($A_0$) group velocity with a maximum error of $4\%$ over a 40~kHz frequency band. In contrast, group velocities estimated from transient experiments resulted in a maximum error of $6\%$ over the same frequency band.

physics.data-an↗

arXiv2020

Contour Integral Methods for Nonlinear Eigenvalue Problems: A Systems Theoretic Approach

Contour integral methods for nonlinear eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization techniques in control theory. This connection motivates a new general framework for contour integral methods (for linear and nonlinear eigenvalue problems), building on recent developments in multi-point rational interpolation of dynamical systems. These new techniques, which replace the usual Hankel matrices with Loewner matrix pencils, incorporate general interpolation schemes and permit ready recovery of eigenvectors. Because the main computations (the solution of linear systems associated with contour integration) are identical for these Loewner methods and the traditional Hankel approach, a variety of new eigenvalue approximations can be explored with modest additional work. Numerical examples illustrate the potential of this approach. We also discuss how the concept of filter functions can be employed in this new framework, and show how contour methods enable a data-driven modal truncation method for model reduction.

math.NA↗

arXiv2020

Preconditioning Parametrized Linear Systems

Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems, it is advantageous to recycle preconditioners, that is, update a previous preconditioner and reuse the updated version. In this paper, we introduce a simple and effective method for doing this. Although our approach can be used for matrices changing slowly in any way, we focus on the important case of sequences of the type $(s_k\textbf{E}(\textbf{p}) + \textbf{A}(\textbf{p}))\textbf{x}_k = \textbf{b}_k$, where the right hand side may or may not change. More general changes in matrices will be discussed in a future paper. We update preconditioners by defining a map from a new matrix to a previous matrix, for example the first matrix in the sequence, and combine the preconditioner for this previous matrix with the map to define the new preconditioner. This approach has several advantages. The update is entirely independent from the original preconditioner, so it can be applied to any preconditioner. The possibly high cost of an initial preconditioner can be amortized over many linear solves. The cost of updating the preconditioner is more or less constant and independent of the original preconditioner. There is flexibility in balancing the quality of the map with the computational cost. In the numerical experiments section we demonstrate good results for several applications, in particular when using an algebraic multigrid preconditioner.

math.NA↗

arXiv2020

Structure-Preserving Interpolation for Model Reduction of Parametric Bilinear Systems

In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.

math.NA↗

arXiv2020

Randomization for the Efficient Computation of Parametric Reduced Order Models for Inversion

Nonlinear parametric inverse problems appear in many applications. Here, we focus on diffuse optical tomography (DOT) in medical imaging to recover unknown images of interest, such as cancerous tissue in a given medium, using a mathematical (forward) model. The forward model in DOT is a diffusion-absorption model for the photon flux. The main bottleneck in these problems is the repeated evaluation of the large-scale forward model. For DOT, this corresponds to solving large linear systems for each source and frequency at each optimization step. Moreover, Newton-type methods, often the method of choice, require additional linear solves with the adjoint to compute derivative information. Emerging technology allows for large numbers of sources and detectors, making these problems prohibitively expensive. Reduced order models (ROM) have been used to drastically reduce the system size in each optimization step, while solving the inverse problem accurately. However, for large numbers of sources and detectors, just the construction of the candidate basis for the ROM projection space incurs a substantial cost, as matching the full parameter gradient matrix in interpolatory model reduction requires large linear solves for all sources and frequencies and all detectors and frequencies for each parameter interpolation point. As this candidate basis numerically has low rank, this construction is followed by a rank-revealing factorization that typically reduces the number of vectors in the candidate basis substantially. We propose to use randomization to approximate this basis with a drastically reduced number of large linear solves. We also provide a detailed analysis for the low-rank structure of the candidate basis for our problem of interest. Even though we focus on the DOT problem, the ideas presented are relevant to many other large scale inverse problems and optimization problems.

math.NA↗

arXiv2020

The AAA framework for modeling linear dynamical systems with quadratic output

We consider linear dynamical systems with quadratic output. We first define the two transfer functions, a single-variable and a multivariate one, that fully describe the dynamics of these special nonlinear systems. Then, using the samples of these two transfer functions, we extend the AAA algorithm to model linear systems with quadratic output from data.

math.NA↗

arXiv2020

Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points

This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state field approximation from measurements. Empirical interpolation derives approximations from a few samples (measurements) via interpolation in low-dimensional spaces. It has been observed that empirical interpolation can become unstable if the samples are perturbed due to, e.g., noise, turbulence, and numerical inaccuracies. The main contribution of this work is a probabilistic analysis that shows that stable approximations are obtained if samples are randomized and if more samples than dimensions of the low-dimensional spaces are used. Oversampling, i.e., taking more sampling points than dimensions of the low-dimensional spaces, leads to approximations via regression and is known under the name of gappy proper orthogonal decomposition. Building on the insights of the probabilistic analysis, a deterministic sampling strategy is presented that aims to achieve lower approximation errors with fewer points than randomized sampling by taking information about the low-dimensional spaces into account. Numerical results of reconstructing velocity fields from noisy measurements of combustion processes and model reduction in the presence of noise demonstrate the instability of empirical interpolation and the stability of gappy proper orthogonal decomposition with oversampling.

math.NA↗