arXiv subjects

Peter Benner

Publications and source records attributed to Peter Benner.

At least 145 records · Page 8Linked to original sources

arXiv2020

Structure-Preserving Interpolation of Bilinear Control Systems

In this paper, we extend the structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples.

math.NA↗

arXiv2020

On Error Estimation for Reduced-order Modeling of Linear Non-parametric and Parametric Systems

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Furthermore, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It has been shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate output error of reduced-order modeling for steady linear parametric systems.

math.NA↗

arXiv2020

Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues

This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence.

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arXiv2020

Low-rank Linear Fluid-structure Interaction Discretizations

Fluid-structure interaction models involve parameters that describe the solid and the fluid behavior. In simulations, there often is a need to vary these parameters to examine the behavior of a fluid-structure interaction model for different solids and different fluids. For instance, a shipping company wants to know how the material, a ship's hull is made of, interacts with fluids at different Reynolds and Strouhal numbers before the building process takes place. Also, the behavior of such models for solids with different properties is considered before the prototype phase. A parameter-dependent linear fluid-structure interaction discretization provides approximations for a bundle of different parameters at one step. Such a discretization with respect to different material parameters leads to a big block-diagonal system matrix that is equivalent to a matrix equation as discussed in [KressnerTobler 2011]. The unknown is then a matrix which can be approximated using a low-rank approach that represents the iterate by a tensor. This paper discusses a low-rank GMRES variant and a truncated variant of the Chebyshev iteration. Bounds for the error resulting from the truncation operations are derived. Numerical experiments show that such truncated methods applied to parameter-dependent discretizations provide approximations with relative residual norms smaller than $10^{-8}$ within a twentieth of the time used by individual standard approaches.

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arXiv2020

A Low-rank Approach for Nonlinear Parameter-dependent Fluid-structure Interaction Problems

Parameter-dependent discretizations of linear fluid-structure interaction problems can be approached with low-rank methods. When discretizing with respect to a set of parameters, the resulting equations can be translated to a matrix equation since all operators involved are linear. If nonlinear FSI problems are considered, a direct translation to a matrix equation is not possible. We present a method that splits the parameter set into disjoint subsets and, on each subset, computes an approximation of the problem related to the upper median parameter by means of the Newton iteration. This approximation is then used as initial guess for one Newton step on a subset of problems.

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arXiv2020

High Performance Solution of Skew-symmetric Eigenvalue Problems with Applications in Solving the Bethe-Salpeter Eigenvalue Problem

We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the so-called Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library, such as GPU support. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 higher than the reference method using Intel's ScaLAPACK implementation. The runtime to solve the Bethe-Salpeter-Eigenvalue problem can be improved by a factor of 10. Our method is freely available in the current release of the ELPA library.

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arXiv2020

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.

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arXiv2020

Low-Rank and Total Variation Regularization and Its Application to Image Recovery

In this paper, we study the problem of image recovery from given partial (corrupted) observations. Recovering an image using a low-rank model has been an active research area in data analysis and machine learning. But often, images are not only of low-rank but they also exhibit sparsity in a transformed space. In this work, we propose a new problem formulation in such a way that we seek to recover an image that is of low-rank and has sparsity in a transformed domain. We further discuss various non-convex non-smooth surrogates of the rank function, leading to a relaxed problem. Then, we present an efficient iterative scheme to solve the relaxed problem that essentially employs the (weighted) singular value thresholding at each iteration. Furthermore, we discuss the convergence properties of the proposed iterative method. We perform extensive experiments, showing that the proposed algorithm outperforms state-of-the-art methodologies in recovering images.

cs.CV↗

arXiv2020

On the Solution of the Nonsymmetric T-Riccati Equation

The nonsymmetric T-Riccati equation is a quadratic matrix equation where the linear part corresponds to the so-called T-Sylvester or T-Lyapunov operator that has previously been studied in the literature. It has applications in macroeconomics and policy dynamics. So far, it presents an unexplored problem in numerical analysis, and both, theoretical results and computational methods, are lacking in the literature. In this paper we provide some sufficient conditions for the existence and uniqueness of a nonnegative minimal solution and its efficient computation is deeply analyzed. Both the small-scale and the large-scale setting are addressed and Newton-Kleinman-like methods are derived. The convergence of these procedures to the minimal solution is proved and several numerical results illustrate the computational efficiency of the proposed methods.

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arXiv2020

Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain

Interpolatory methods offer a powerful framework for generating reduced-order models (ROMs) for non-parametric or parametric systems with time-varying inputs. Choosing the interpolation points adaptively remains an area of active interest. A greedy framework has been introduced in Feng et al. [ESAIM: Math. Model. Numer. Anal. 51(6), 2017] and in Feng and Benner [IEEE Trans. Microw. Theory Techn. 67(12), 2019] to choose interpolation points automatically using a posteriori error estimators. Nevertheless, when the parameter range is large or if the parameter space dimension is larger than two, the greedy algorithm may take considerable time, since the training set needs to include a considerable number of parameters. As a remedy, we introduce an adaptive training technique by learning an efficient a posteriori error estimator over the parameter domain. A fast learning process is created by interpolating the error estimator using radial basis functions (RBF) over a fine parameter training set, representing the whole parameter domain. The error estimator is evaluated only on a coarse training set including a few parameter samples. The algorithm is an extension of the work in Chellappa et al. [arXiv e-prints 1910.00298] to interpolatory model order reduction (MOR) in frequency domain. Beyond this work, we use a newly proposed inf-sup-constant-free error estimator in the frequency domain, which is often much tighter than the error estimator using the inf-sup constant.

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arXiv2020

MORLAB -- The Model Order Reduction LABoratory

For an easy use of model order reduction techniques in applications, software solutions are needed. In this paper, we describe the MORLAB, Model Order Reduction LABoratory, toolbox as an efficient implementation of model reduction techniques for dense, medium-scale linear time-invariant systems. Giving an introduction to the underlying programming principles of the toolbox, we show the basic idea of spectral splitting and present an overview about implemented model reduction techniques. Two numerical examples are used to illustrate different use cases of the MORLAB toolbox.

cs.MS↗

arXiv2020

Automatic Decoupling and Index-aware Model-Order Reduction for Nonlinear Differential-Algebraic Equations

We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term f(Ex), where E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet accurately represent the original large-scale system behavior. However, direct reduction and simulation of nonlinear differential-algebraic equations is difficult due to hidden constraints which affect the choice of numerical integration methods and model-order reduction techniques. We propose an extension of index-aware model-order reduction methods to nonlinear differential-algebraic equations without any kind of linearization. The proposed model-order reduction approach involves automatic decoupling of nonlinear differential-algebraic equations into nonlinear ordinary differential equations and algebraic equations. This allows applying standard model-order reduction techniques to both parts without worrying about the index. The same procedure can also be used to simulate nonlinear differential-algebraic equations using standard integration schemes. We illustrate the performance of our proposed method for nonlinear differential-algebraic equations arising from gas flow models in pipeline networks.

math.NA↗

arXiv2020

Frequency- and Time-Limited Balanced Truncation for Large-Scale Second-Order Systems

Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefor, it usually pays off to compute local approximations of the used dynamical systems in the frequency and time domain. In this paper, we consider a structure-preserving extension of the frequency- and time-limited balanced truncation methods to second-order dynamical systems. We give a full overview about the first-order limited balanced truncation methods and extend those methods to second-order systems by using the different second-order balanced truncation formulas from the literature. Also, we present numerical methods for solving the arising large-scale sparse matrix equations and give numerical modifications to deal with the problematic case of second-order systems. The results are then illustrated on three numerical examples.

math.OC↗

arXiv2019

Implicit Higher-Order Moment Matching Technique for Model Reduction of Quadratic-bilinear Systems

We propose a projection based multi-moment matching method for model order reduction of quadratic-bilinear systems. The goal is to construct a reduced system that ensures higher-order moment matching for the multivariate transfer functions appearing in the input-output representation of the nonlinear system. An existing technique achieves this for the first two multivariate transfer functions, in what is called the symmetric form of the multivariate transfer functions. We extend this framework to an equivalent and simplified form, the regular form, which allows us to show moment matching for the first three multivariate transfer functions. Numerical results for three benchmark examples of quadratic-bilinear systems show that the proposed framework exhibits better performance with reduced computational cost in comparison to existing techniques.

eess.SY↗

arXiv2019

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al., 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for connectome reconstruction problem in very high dimensions. The algorithm approximates the solution by a sequence of rank-one updates which exploit the sparse and positive definite problem structure. This algorithm was described previously (Kressner and Sirković, 2015) but never implemented for this connectome problem, leading to a number of challenges. We have had to design judicious stopping criteria and employ efficient solvers for the three main sub-problems of the algorithm, including an efficient GPU implementation that alleviates the main bottleneck for large datasets. The performance of the method is evaluated on three examples: an artificial "toy" dataset and two whole-cortex instances using data from the Allen Mouse Brain Connectivity Atlas. We find that the method is significantly faster than previous methods and that moderate ranks offer good approximation. This speedup allows for the estimation of increasingly large-scale connectomes across taxa as these data become available from tracing experiments. The data and code are available online.

math.NA↗

arXiv2019

Identification of Port-Hamiltonian Systems from Frequency Response Data

In this paper, we study the identification problem of a passive system from tangential interpolation data. We present a simple construction approach based on the Mayo-Antoulas generalized realization theory that automatically yields a port-Hamiltonian realization for every strictly passive system with simple spectral zeros. Furthermore, we discuss the construction of a frequency-limited port-Hamiltonian realization. We illustrate the proposed method by means of several examples.

eess.SY↗

arXiv2019

Identification of Dominant Subspaces for Linear Structured Parametric Systems and Model Reduction

In this paper, we discuss a novel model reduction framework for generalized linear systems. The transfer functions of these systems are assumed to have a special structure, e.g., coming from second-order linear systems and time-delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation-based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation-based model reduction encode these subspaces. As a result, we are able to identify the dominant reachable and observable subspaces of the underlying system. Based on this, we propose a new model reduction algorithm combining these features leading to reduced-order systems. Furthermore, we pay special attention to computational aspects of the approach and discuss its applicability to a large-scale setting. We illustrate the efficiency of the proposed approach with several numerical large-scale benchmark examples.

math.NA↗

arXiv2019

Invariant Galerkin Ansatz Spaces and Davison-Maki Methods for the Numerical Solution of Differential Riccati Equations

The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. In view of memory efficient approximation, we review and extend known solution formulas and identify invariant subspaces for a possibly low-dimensional solution representation. Based on these theoretical findings, we propose a Galerkin projection onto a space related to a low-rank approximation of the algebraic Riccati equation. For the numerical implementation, we provide an alternative interpretation of the modified \emph{Davison-Maki method} via the transformed flow of the differential Riccati equation, which enables us to rule out known stability issues of the method in combination with the proposed projection scheme. We present numerical experiments for large-scale autonomous differential Riccati equations and compare our approach with high-order splitting schemes.

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