arXiv subjects

Peter Benner

Publications and source records attributed to Peter Benner.

At least 181 records · Page 10Linked to original sources

arXiv2018

New Gramians for Linear Switched Systems: Reachability, Observability, and Model Reduction

In this paper, we propose new algebraic Gramians for continuous-time linear switched systems, which satisfy generalized Lyapunov equations. The main contribution of this work is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a linear switched system. As a consequence, a simple Gramian-based criterion for reachability and observability is established. Second, a balancing-based model order reduction technique is proposed and, under some sufficient conditions, stability preservation and an error bound are shown. Finally, the efficiency of the proposed method is illustrated by means of numerical examples.

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arXiv2018

Low-Order Control Design using a Reduced-Order Model with a Stability Constraint on the Full-Order Model

We consider low-order controller design for large-scale linear time-invariant dynamical systems with inputs and outputs. Model order reduction is a popular technique, but controllers designed for reduced-order models may result in unstable closed-loop plants when applied to the full-order system. We introduce a new method to design a fixed-order controller by minimizing the $L_\infty$ norm of a reduced-order closed-loop transfer matrix function subject to stability constraints on the closed-loop systems for both the reduced-order and the full-order models. Since the optimization objective and the constraints are all nonsmooth and nonconvex we use a sequential quadratic programming method based on quasi-Newton updating that is intended for this problem class, available in the open-source software package GRANSO. Using a publicly available test set, the controllers obtained by the new method are compared with those computed by the HIFOO (H-Infinity Fixed-Order Optimization) toolbox applied to a reduced-order model alone, which frequently fail to stabilize the closed-loop system for the associated full-order model.

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arXiv2018

Synchronization and Aggregation of Nonlinear Power Systems with Consideration of Bus Network Structures

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power system.

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arXiv2018

Computing the density of states for optical spectra by low-rank and QTT tensor approximation

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter equation (BSE). The presented approach for approximating the DOS is based on two main techniques. First, we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block-diagonal plus low-rank matrix structure described in [6, 3] for the BSE/TDA problem. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size $N$ can be accurately represented by a low rank quantized tensor train (QTT) tensor that can be determined through a least squares fitting procedure. The latter provides good approximation properties for strictly oscillating DOS functions with multiple gaps, and requires asymptotically much fewer ($O(\log N)$) functional calls compared with the full grid size $N$. This approach allows us to overcome the computational difficulties of the traditional schemes by avoiding both the need of stochastic sampling and interpolation by problem independent functions like polynomials etc. Numerical tests indicate that the QTT approach yields accurate recovery of DOS associated with problems that contain relatively large spectral gaps. The QTT tensor rank only weakly depends on the size of a molecular system which paves the way for treating large-scale spectral problems.

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arXiv2017

On Reduced Input-Output Dynamic Mode Decomposition

The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems.

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arXiv2017

An inexact Newton-Krylov method for stochastic eigenvalue problems

This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments.

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arXiv2017

Fast Solution of the Linearized Poisson-Boltzmann Equation with nonaffine Parametrized Boundary Conditions Using the Reduced Basis Method

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to FOM. The discrete empirical interpolation method (DEIM) is applied to the Dirichlet boundary conditions which are nonaffine with the parameter, to reduce the complexity of the reduced order model (ROM). From the numerical results, we notice that the RBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ at an accuracy of $10^{-9}$ and reduces computational time by a factor of approximately $7,600$. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of $20$ for a single iteration of the greedy algorithm.

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arXiv2017

Example Setups of Navier-Stokes Equations with Control and Observation: Spatial Discretization and Representation via Linear-quadratic Matrix Coefficients

We provide spatial discretizations of nonlinear incompressible Navier-Stokes equations with inputs and outputs in the form of matrices ready to use in any numerical linear algebra package. We discuss the assembling of the system operators and the realization of boundary conditions and inputs and outputs. We describe the two benchmark problems - the driven cavity and the cylinder wake - and provide the corresponding data. The use of the data is illustrated by numerous example setups. The test cases are provided as plain PYTHON or OCTAVE/MATLAB script files for immediate replication.

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arXiv2017

Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.

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arXiv2017

Large-Scale Computation of ${\mathcal L}_\infty$-Norms by a Greedy Subspace Method

We are concerned with the computation of the ${\mathcal L}_\infty$-norm for an ${\mathcal L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$, where the middle factor is the inverse of a meromorphic matrix-valued function, and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function $H$ where the middle factor is of much smaller dimension. The ${\mathcal L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the ${\mathcal L}_\infty$-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

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arXiv2017

$\mathcal H_2$-Quasi-Optimal Model Order Reduction for Quadratic-Bilinear Control Systems

We investigate the optimal model reduction problem for large-scale quadratic-bilinear (QB) control systems. Our contributions are threefold. First, we discuss the variational analysis and the Volterra series formulation for QB systems. We then define the $\mathcal H_2$-norm for a QB system based on the kernels of the underlying Volterra series and also propose a truncated $\mathcal H_2$-norm. Next, we derive first-order necessary conditions for an optimal approximation, where optimality is measured in term of the truncated $\mathcal H_2$-norm of the error system. We then propose an iterative model reduction algorithm, which upon convergence yields a reduced-order system that approximately satisfies the newly derived optimality conditions. We also discuss an efficient computation of the reduced Hessian, using the special Kronecker structure of the Hessian of the system. We illustrate the efficiency of the proposed method by means of several numerical examples resulting from semi-discretized nonlinear partial differential equations and show its competitiveness with the existing model reduction schemes for QB systems such as moment-matching methods and balanced truncation.

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arXiv2017

An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations

We discuss model reduction for a particular class of quadratic-bilinear (QB) descriptor systems. The main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QBODEs [Benner et al. '16] to a class of descriptor systems in an efficient and reliable way. Recently, it has been shown in the case of linear or bilinear systems that a direct extension of interpolation-based model reduction techniques to descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we aim at transforming the considered QB descriptor system into an equivalent QBODE system by means of projectors for which standard model reduction techniques for QBODEs can be employed, including aforementioned interpolation scheme. Subsequently, we discuss related computational issues, thus resulting in a modified algorithm that allows us to construct \emph{near}--optimal reduced-order systems without explicitly computing the projectors used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier-Stokes equations.

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arXiv2017

Balanced Truncation Model Order Reduction For Quadratic-Bilinear Control Systems

We discuss balanced truncation model order reduction for large-scale quadratic-bilinear (QB) systems. Balanced truncation for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (1993), where it is shown that Gramians for nonlinear systems are the solutions of state-dependent nonlinear Hamilton-Jacobi equations. Therefore, they are not only difficult to compute for large-scale systems but also hard to utilize in the model reduction framework. In this paper, we propose algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relations with a certain type of generalized quadratic Lyapunov equation. Furthermore, we present how these algebraic Gramians and energy functionals relate to each other. Moreover, we characterize the reachability and observability of QB systems based on the proposed algebraic Gramians. This allows us to find those states that are hard to control and hard to observe via an appropriate transformation based on the Gramians. Truncating such states yields reduced-order systems. Additionally, we present a truncated version of the Gramians for QB systems and discuss their advantages in the model reduction framework. We also investigate the Lyapunov stability of the reduced-order systems. We finally illustrate the efficiency of the proposed balancing-type model reduction for QB systems by means of various semi-discretized nonlinear partial differential equations and show its competitiveness with the existing moment-matching methods for QB systems.

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arXiv2017

Finite-Frequency Model Order Reduction of Linear Systems via Parameterized Frequency-dependent Balanced Truncation

Balanced truncation is one of the most common model order reduction schemes. In this paper, we study finite-frequency model order reduction (FF-MOR) problems of linear continuous-time systems within the framework of balanced truncation method. Firstly, we construct a family of parameterized frequency-dependent (PFD) mappings which generate discrete-time PFD mapped systems and continuous-time PFD mapped systems of the given continuous-time system. The relationships between the maximum singular value of the given system over pre-specified frequency ranges and the maximum singular value of the PFD mapped systems over entire frequency range are established. By exploiting the properties of the discrete-time PFD mapped systems, a new parameterized frequency-dependent balanced truncation (PFDBT) method providing finite-frequency type error bound with respect to the maximum singular value of the approximation error systems are developed. Examples are included for illustration.

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arXiv2017

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

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arXiv2017

Low-rank computation of posterior covariance matrices in Bayesian inverse problems

We consider the problem of estimating the uncertainty in statistical inverse problems using Bayesian inference. When the probability density of the noise and the prior are Gaussian, the solution of such a statistical inverse problem is also Gaussian. Therefore, the underlying solution is characterized by the mean and covariance matrix of the posterior probability density. However, the covariance matrix of the posterior probability density is full and large. Hence, the computation of such a matrix is impossible for large dimensional parameter spaces. It is shown that for many ill-posed problems, the Hessian matrix of the data misfit part has low numerical rank and it is therefore possible to perform a low-rank approach to approximate the posterior covariance matrix. For such a low-rank approximation, one needs to solve a forward partial differential equation (PDE) and the adjoint PDE in both space and time. This in turn gives $\mathcal{O}(n_x n_t)$ complexity for both, computation and storage, where $n_x$ is the dimension of the spatial domain and $n_t$ is the dimension of the time domain. Such computations and storage demand are infeasible for large problems. To overcome this obstacle, we develop a new approach that utilizes a recently developed low-rank in time algorithm together with the low-rank Hessian method. We reduce both the computational complexity and storage requirement from $\mathcal{O}(n_x n_t)$ to $\mathcal{O}(n_x + n_t)$. We use numerical experiments to illustrate the advantages of our approach.

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arXiv2017

Computing the stochastic $H^\infty$-norm

The stochastic $H^\infty$-norm is defined as the $L^2$-induced norm of the input-output operator of a stochastic linear system. Like the deterministic $H^\infty$-norm it is characterised by a version of the bounded real lemma, but without a frequency domain description or a Hamiltonian condition. Therefore, we base its computation on a parametrised algebraic Riccati-type matrix equation.

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arXiv2016

Space-time Galerkin POD with application in optimal control of semi-linear parabolic partial differential equations

In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -- probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time \textit{is} a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.

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