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Parametric model reduction via rational interpolation along parameters

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arXiv 2021年

作者： Gosea, Ion Victor Gugercin, Serkan Unger, Benjamin Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science VirginiaTech BlacksburgVA24061 United States SC Simulation Technology University of Stuttgart Germany

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 *** Codes 93C05, 93C35, 32E30 Copyright © 2021, The Authors. All rights reserved.

关键词： Transfer functions

RANDOMIZATION for the EFFICIENT COMPUTATION of PARAMETRIC REDUCED ORDER MODELS for INVERSION

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arXiv 2020年

作者： Aslan, Selin de Sturler, Eric Gugercin, Serkan Advanced Photon Source Argonne National Laboratory LemontIL60439 United States Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA2406 United States

Nonlinear parametric inverse problems appear in many applications. In this work, we focus on diffuse optical tomography (DOT) in medical image reconstruction, in which we aim to recover an unknown image of interest (defined by a set of parameters), 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 computational bottleneck in such inverse 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. In addition, as Newton-type methods are the method of choice for these problems, one needs to solve additional linear systems with the adjoint to efficiently compute derivative information. As rapid advances in technology allow for large numbers of sources and detectors, these problems become computationally prohibitively expensive. In the past, the use of reduced order models (ROM) has been proposed to drastically reduce the size of the linear systems solved in each optimization step, while still solving the inverse problem accurately. However, as the number of sources and detectors increases, even the construction of the ROM bases incurs a substantial cost. Interpolatory model reduction, when performed in the setting of full matrix-interpolation to match the full parameter gradient matrix, requires the solution of large linear systems for all sources and frequencies as well as for all detectors and frequencies for each interpolation point in parameter space to build a candidate basis for the ROM projection space. 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. Since only the ROM projection space matters, not its basis, we propose to use randomization to approximate this basis efficientl

关键词： Inverse problems

Structured vector fitting framework for mechanical systems

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arXiv 2021年

作者： Werner, Steffen W.R. Gosea, Ion Victor Gugercin, Serkan Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics and Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

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. Copyright © 2021, The Authors. All rights reserved.

关键词： Vectors

Contour integral methods for nonlinear eigenvalue problems: A systems theoretic approach

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arXiv 2020年

作者： Brennan, Michael C. Embree, Mark Gugercin, Serkan Center for Computational Engineering Massachusetts Institute of Technology CambridgeMA02139-4307 United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States

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 *** Codes 15A18, 65F15, 93B20, 93B30 15A18, 65F15, 93B20, 93B30 15A18, 65F15, 93B20, 93B30 Copyright © 2020, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

The AAA framework for modeling linear dynamical systems with quadratic output

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arXiv 2020年

作者： Gosea, Ion Victor Gugercin, Serkan Data-Driven System Reduction and Identification Group Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Department of Mathematics and Division of Computational Modeling and Data Analytics Virginia Tech BlacksburgVA24061 United States

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. Copyright © 2020, The Authors. All rights reserved.

关键词： Invariance

On the balanced truncation error bound and sign parameters from arrowhead realizations

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arXiv 2020年

作者： Reiter, Sean Damm, Tobias Embree, Mark Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Kaiserslautern67653 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced-order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling, and illustrate these results with a power system model that motivated this *** Codes 93B10, 93B11, 93C05 Copyright © 2020, The Authors. All rights reserved.

关键词： Dynamical systems

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

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arXiv 2021年

作者： Krishnan, Manu Gugercin, Serkan Tarazaga, Pablo A. Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University BlacksburgVA24060 United States Department of Mathematics Division of Computational Modeling & Data Analytics Virginia Polytechnic Institute and State University BlacksburgVA24060 United States Department of Mechanical Engineering Texas A & M College StationTX77843 United States Department of Mechanical Engineering Virginia Polytechnic Institute and State University

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). The wavelet-based dynamic mode decomposition (WDMD) builds on the ioDMD framework without the restrictive assumption of full state measurements. 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 an Euler-Bernoulli 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 can then be used to simulate the beam’s response towards different types of forcing functions. The method is further validated on a real (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 havi

关键词： Dynamical systems

Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors

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arXiv 2020年

作者： Cho, Taewon Chung, Julianne Jiang, Jiahua Department of Mathematics Virginia Tech BlacksburgVA United States Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics Virginia Tech BlacksburgVA United States

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Differential equations

Pseudospectra of Loewner Matrix Pencils

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arXiv 2019年

作者： Embree, Mark Cosmin Ioniţă, A. Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States MathWorks Inc. NatickMA01760 United States

Loewner matrix pencils play a central role in the system realization theory of Mayo and Antoulas, an important development in data-driven modeling. The eigenvalues of these pencils reveal system poles. How robust are the poles recovered via Loewner realization? With several simple examples, we show how pseudospectra of Loewner pencils can be used to investigate the influence of interpolation point location and partitioning on pole stability, the transient behavior of the realized system, and the effect of noisy measurement data. We include an algorithm to efficiently compute such pseudospectra by exploiting Loewner structure. Copyright © 2019, The Authors. All rights reserved.

关键词： Poles