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Structure-preserving interpolation for model reduction of parametric bilinear systems

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arXiv 2020年

作者： Benner, Peter Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Otto von Guericke University Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

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

关键词： Interpolation

Revisiting IRKA: Connections with pole placement and backward stability

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arXiv 2019年

作者： Beattie, C. Drmač, Z. Gugercin, S. Department of Mathematics Virginia Tech BlacksburgVA24061 United States Faculty of Science Department of Mathematics University of Zagreb Zagreb10000 Croatia Department of Mathematics Division of Computational Modeling and Data Analytics Virginia Tech BlacksburgVA24061 United States

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order H2-approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid | but not always;and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive *** find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability. Copyright © 2019, The Authors. All rights reserved.

关键词： Iterative methods

Sampled limited memory methods for massive linear inverse problems

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arXiv 2019年

作者： Chung, Julianne Chung, Matthias Tanner Slagel, J. Tenorio, Luis 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 Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

In many modern imaging applications the desire to reconstruct high resolution images, coupled with the abundance of data from acquisition using ultra-fast detectors, have led to new challenges in image reconstruction. A main challenge is that the resulting linear inverse problems are massive. The size of the forward model matrix exceeds the storage capabilities of computer memory, or the observational dataset is enormous and not available all at once. Row-action methods that iterate over samples of rows can be used to approximate the solution while avoiding memory and data availability constraints. However, their overall convergence can be slow. In this paper, we introduce a sampled limited memory row-action method for linear least squares problems, where an approximation of the global curvature of the underlying least squares problem is used to speed up the initial convergence and to improve the accuracy of iterates. We show that this limited memory method is a generalization of the damped block Kaczmarz method, and we prove linear convergence of the expectation of the iterates and of the error norm up to a convergence horizon. Numerical experiments demonstrate the benefits of these sampled limited memory row-action methods for massive 2D and 3D inverse problems in tomography applications. Copyright © 2019, The Authors. All rights reserved.

关键词： Tomography

Flexible krylov methods for pregularization

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arXiv 2018年

作者： Chung, Julianne Gazzola, Silvia Department of Mathematics Computational Modeling Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States

In this paper we develop flexible Krylov methods for efficiently computing regular- ized solutions to large-scale linear inverse problems with an 2fit-to-data term and an .p penalization term, for p ≥ 1. First we approximate the p-norm penalization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion, and then we exploit flexible preconditioning tech- niques to efficiently incorporate the weight updates. To handle general (non-square) .p-regularized least-squares problems, we introduce a flexible Golub-Kahan approach and exploit it within a Krylov-Tikhonov hybrid framework. The key benefits of our approach compared to existing optimization methods for .p regularization are that effi- cient projection methods replace inner-outer schemes and that expensive regularization parameter selection techniques can be avoided. Theoretical insights are provided, and numerical results from image deblurring and tomographic reconstruction illustrate the benefits of this approach, compared to well-established methods. Furthermore, we show that our approach for p = 1 can be used to efficiently compute solutions that are sparse with respect to some transformations. Copyright © 2018, The Authors. All rights reserved.

关键词： Image enhancement

Polynomial preconditioned arnoldi

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arXiv 2018年

作者： EMBREE, MARK LOE, JENNIFER A. MORGAN, RONALD B. Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Department of Mathematics Baylor University WacoTX76798-7328 United States

Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization;for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction of communication-intensive operations. The GMRES algorithm provides a simple and effective way of generating the preconditioning polynomial. For some problems high degree polynomials are especially effective, but they can lead to stability problems that must be mitigated. A two-level "double polynomial preconditioning" strategy provides an effective way to generate high-degree preconditioners.65F15, 15A18 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Sampled Tikhonov regularization for large linear inverse problems

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arXiv 2018年

作者： Tanner Slagel, J. Chung, Julianne Chung, Matthias Kozak, David Tenorio, Luis 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 Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

In this paper, we investigate iterative methods that are based on sampling of the data for computing Tikhonov-regularized solutions. We focus on very large inverse problems where access to the entire data set is not possible all at once (e.g., for problems with streaming or massive datasets). Row-access methods provide an ideal framework for solving such problems, since they only require access to "blocks" of the data at any given time. However, when using these iterative sampling methods to solve inverse problems, the main challenges include a proper choice of the regularization parameter, appropriate sampling strategies, and a convergence analysis. To address these challenges, we first describe a family of sampled iterative methods that can incorporate data as they become available (e.g., randomly sampled). We consider two sampled iterative methods, where the iterates can be characterized as solutions to a sequence of approximate Tikhonov problems. The first method requires the regularization parameter to be fixed a priori and converges asymptotically to an unregularized solution for randomly sampled data. This is undesirable for inverse problems. Thus, we focus on the second method where the main benefits are that the regularization parameter can be updated during the iterative process and the iterates converge asymptotically to a Tikhonov-regularized solution. We describe adaptive approaches to update the regularization parameter that are based on sampled residuals, and we describe a limited-memory variant for larger problems. Numerical examples, including a large-scale super-resolution imaging example, demonstrate the potential for these *** Codes 65F22, 65F10, 15A29 Copyright © 2018, The Authors. All rights reserved.

关键词： Iterative methods

Uncertainty quantification in large Bayesian linear inverse problems using Krylov subspace methods

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arXiv 2018年

作者： Saibaba, Arvind K. Chung, Julianne Petroske, Katrina Department of Mathematics North Carolina State University RaleighNC United States Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics North Carolina State University RaleighNC United States

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not possible (e.g., those from the Matérn class). In this work, we assume that generalized Golub-Kahan based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. By exploiting the generalized Golub-Kahan bidiagonalization, we get an approximation of the posterior covariance matrix for "free." We provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback-Liebler divergence. We present two methods that use preconditioned Lanczos methods to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Compartmental process-based model for estimating ammonia emission from liquid dairy manure storage tank

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Compartmental process-based model for estimating ammonia emi...

2017 ASABE Annual International Meeting

作者： Karunarathne, Sampath A. Chung, Matthias Ogejo, Jactone A. Biological Systems Engineering Virginia Tech BlacksburgVA24061 United States Department of Mathematics Virginia Tech BlacksburgVA24061 United States Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States

The biogeochemical processes responsible for production and emission of ammonia from stored liquid dairy manure are governed by environmental factors (e.g. manure temperature, moisture) and manure characteristics (e.g. total ammoniacal nitrogen concentration, pH). These environmental factors and manure characteristics vary spatially as a result of spatially heterogeneous physical, chemical, and biological properties of manure. Existing process-based models used for estimating ammonia emission do not consider these spatial variations leading to inaccurate estimations. In this study, a one-dimensional compartmental process-based model was developed to (i) estimate spatial variation of temperature and substrate concentration (ii) estimate spatial variations and rates of biogeochemical processes, and (iii) estimate production and emission of ammonia from a liquid manure storage tank. Model performance was evaluated using experimental data obtained from National Air Emissions Monitoring Study. A sensitivity analysis was performed and air temperature, manure pH, wind speed, and manure total ammoniacal nitrogen concentration were identified as the most sensitive model inputs. The model was used to estimate ammonia emission from a liquid dairy manure storage of a dairy farm located in Rockingham County in Virginia. Ammonia emission was estimated under different management and weather scenarios: two different manure storage periods in cold season from November to April and in warm season from May to October using historical weather data of the Rockingham County. Results suggest greater ammonia emission and manure nitrogen loss for the manure storage period in warm season compared to the storage period in cold season.

关键词： Liquids

Unstable modes in projection-based reduced-order models: How many can there be, and what do they tell you?

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arXiv 2017年

作者： Embree, Mark Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States

Projection methods provide an appealing way to construct reduced-order models of large-scale linear dynamical systems: they are intuitively motivated and fairly easy to compute. Unfortunately, the resulting reduced models need not inherit the stability of the original system. How many unstable modes can these reduced models have? This note investigates this question, using theory originally motivated by iterative methods for linear algebraic systems and eigenvalue problems, and illustrating the theory with a number of small examples. From these results follow rigorous upper bounds on the number of unstable modes in reduced models generated via orthogonal projection, for both continuous- and discrete-time systems. Can anything be learned from the unstable modes in reduced-order models? Several examples illustrate how such instability can helpfully signal transient growth in the original system. Copyright © 2017, The Authors. All rights reserved.

关键词： Digital control systems