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Peter Benner

Publications and source records attributed to Peter Benner.

At least 163 records · Page 9Linked to original sources
arXiv2019

Data-Driven Identification of Rayleigh-Damped Second-Order Systems

In this paper, we present a data-driven approach to identify second-order systems, having internal Rayleigh damping. This means that the damping matrix is given as a linear combination of the mass and stiffness matrices. These systems typically appear when performing various engineering studies, e.g., vibrational and structural analysis. In an experimental setup, the frequency response of a system can be measured via various approaches, for instance, by measuring the vibrations using an accelerometer. As a consequence, given frequency samples, the identification of the underlying system relies on rational approximation. To that aim, we propose an identification of the corresponding second-order system, extending the Loewner framework for this class of systems. The efficiency of the proposed method is demonstrated by means of various numerical benchmarks.

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arXiv2019

Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method

This paper extends the algorithm of Benner, Heinkenschloss, Saak, and Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142, doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.

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arXiv2019

Gramians, Energy Functionals and Balanced Truncation for Linear Dynamical Systems with Quadratic Outputs

Model order reduction is a technique that is used to construct low-order approximations of large-scale dynamical systems. In this paper, we investigate a balancing based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. To this aim, we propose a new algebraic observability Gramian for the system based on Hilbert space adjoint theory. We then show the proposed Gramians satisfy a particular type of generalized Lyapunov equations and we investigate their connections to energy functionals, namely, the controllability and observability. This allows us to find the states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order systems. Finally, based on $\mathcal H_2$ energy considerations, we, furthermore, derive error bounds, depending on the neglected singular values. The efficiency of the proposed method is demonstrated by means of two semi-discretized partial differential equations and is compared with the existing model reduction techniques in the literature.

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arXiv2019

Faster and more accurate computation of the $\mathcal{H}_\infty$ norm via optimization

In this paper, we propose an improved method for computing the $\mathcal{H}_\infty$ norm of linear dynamical systems that results in a code that is often several times faster than existing methods. By using standard optimization tools to rebalance the work load of the standard algorithm due to Boyd, Balakrishnan, Bruinsma, and Steinbuch, we aim to minimize the number of expensive eigenvalue computations that must be performed. Unlike the standard algorithm, our modified approach can also calculate the $\mathcal{H}_\infty$ norm to full precision with little extra work, and also offers more opportunity to further accelerate its performance via parallelization. Finally, we demonstrate that the local optimization we have employed to speed up the standard globally-convergent algorithm can also be an effective strategy on its own for approximating the $\mathcal{H}_\infty$ norm of large-scale systems.

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arXiv2019

Extended and improved criss-cross algorithms for computing the spectral value set abscissa and radius

In this paper, we extend the original criss-cross algorithms for computing the $\varepsilon$-pseudospectral abscissa and radius to general spectral value sets. By proposing new root-finding-based strategies for the horizontal/radial search subphases, we significantly reduce the number of expensive Hamiltonian eigenvalue decompositions incurred, which typically translates to meaningful speedups in overall computation times. Furthermore, and partly necessitated by our root-finding approach, we develop a new way of handling the singular pencils or problematic interior searches that can arise when computing the $\varepsilon$-spectral value set radius. Compared to would-be direct extensions of the original algorithms, that is, without our additional modifications, our improved criss-cross algorithms are not only noticeably faster but also more robust and numerically accurate, for both spectral value set and pseudospectral problems.

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arXiv2019

Analysis of switching strategies for the optimization of periodic chemical reactions with controlled flow-rate

An isoperimetric optimal control problem with non-convex cost is considered for a class of nonlinear control systems with periodic boundary conditions. This problem arises in chemical engineering as the maximization of the product of non-isothermal reactions by consuming a fixed amount of input reactants. It follows from the Pontryagin maximum principle that the optimal controls are piecewise constant in the considered case. We focus on a parametrization of optimal controls in terms of switching times in order to estimate the cost under different switching strategies. We exploit the Chen-Fliess functional expansion of solutions to the considered nonlinear system with bang-bang controls to satisfy the boundary conditions and evaluate the cost analytically for small periods. In contrast to the previous results in this area, the system under consideration is not control-affine, and the integrand of the cost depends on the state. This approach is applied to non-isothermal chemical reactions with simultaneous modulation of the input concentration and the volumetric flow-rate.

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arXiv2019

Cross-Gramian-Based Dominant Subspaces

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspace projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

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arXiv2019

An Adaptive Pole-Matching Method for Interpolating Reduced-Order Models

An adaptive parametric reduced-order modeling method based on interpolating poles of reduced-order models is proposed in this paper. To guarantee correct interpolation, a pole-matching process is conducted to determine which poles of two reduced-order models correspond to the same parametric pole. First, the pole-matching in the scenario of parameter perturbation is discussed. It is formulated as a combinatorial optimization problem and solved by a branch and bound algorithm. Then, an adaptive framework is proposed to build repository ROMs at adaptively chosen parameter values, which well represent the parameter domain of interest. To achieve this, we propose techniques including a predictor-corrector strategy and an adaptive refinement strategy, which enable us to use larger steps to explore the parameter domain of interest with good accuracy. The framework also consists of regression as an optional post-processing phase to further reduce the data storage. The advantages over other parametric reduced-order modeling approaches are, e.g., compatibility with any model order reduction method, constant size of the parametric reduced-order model with respect to the number of parameters, and capability to deal with complicated parameter dependency.

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arXiv2019

A Subspace Framework for ${\mathcal H}_\infty$-Norm Minimization

We deal with the minimization of the ${\mathcal H}_\infty$-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatory approaches inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38(4):1496--1516, 2017] for the computation of the ${\mathcal H}_\infty$-norm. In this work, we minimize the ${\mathcal H}_\infty$-norm of a reduced-order parameter-dependent system obtained by two-sided restrictions onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced order system. We formally establish the superlinear convergence of the subspace frameworks under some smoothness assumptions. The fast convergence of the proposed frameworks in practice is illustrated by several large-scale systems.

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arXiv2019

Interpolation-Based Model Order Reduction for Polynomial Parametric Systems

In this work, we investigate a model order reduction scheme for polynomial parametric systems. We begin with defining the generalized multivariate transfer functions for the system. Based on this, we aim at constructing a reduced-order system, interpolating the defined generalized transfer functions at a given set of interpolation points. Furthermore, we provide a method, inspired by the Loewner approach for linear and (quadratic-)bilinear systems, to determine a good-quality reduced-order system in an automatic way. We also discuss the computational issues related to the proposed method and a potential application of CUR matrix approximation in order to further speed-up simulations of reduced-order systems. We test the efficiency of the proposed methods via several numerical examples.

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arXiv2019

Computing electrostatic potentials using regularization based on the range-separated tensor format

In this paper, we apply the range-separated (RS) tensor format [6] for the construction of new regularization scheme for the Poisson-Boltzmann equation (PBE) describing the electrostatic potential in biomolecules. In our approach, we use the RS tensor representation to the discretized Dirac delta [21] to construct an efficient RS splitting of the PBE solution in the solute (molecular) region. The PBE then needs to be solved with a regularized source term, and thus black-box solvers can be applied. The main computational benefits are due to the localization of the modified right-hand side within the molecular region and automatic maintaining of the continuity in the Cauchy data on the interface. Moreover, this computational scheme only includes solving a single system of FDM/FEM equations for the smooth long-range (i.e., regularized) part of the collective potential represented by a low-rank RS-tensor with a controllable precision. The total potential is obtained by adding this solution to the directly precomputed rank-structured tensor representation for the short-range contribution. Enabling finer grids in PBE computations is another advantage of the proposed techniques. In the numerical experiments, we consider only the free space electrostatic potential for proof of concept. We illustrate that the classical Poisson equation (PE) model does not accurately capture the solution singularities in the numerical approximation as compared to the new approach by the RS tensor format.

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arXiv2018

Efficient Numerical Methods for Gas Network Modeling and Simulation

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). With our modeling, we reduce the number of algebraic constraints, which correspond to the $(2,2)$ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $(1, 1)$ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

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arXiv2018

Solution Formulas for Differential Sylvester and Lyapunov Equations

The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches if applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $\mathcal S(X)=AX+XB$ and derive a formula for its norm using an induced operator norm based on the spectrum of $A$ and $B$. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

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arXiv2018

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier-Stokes Equations

We consider the control problem of the stochastic Navier-Stokes equations in multidimensional domains introduced in \cite{ocpc} restricted to noise terms defined by Q-Wiener processes. Using a stochastic maximum principle, we derive a necessary optimality condition to design the optimal control based on an adjoint equation, which is given by a backward SPDE. Moreover, we show that the optimal control satisfies a sufficient optimality condition. As a consequence, we can solve uniquely control problems constrained by the stochastic Navier-Stokes equations especially for two-dimensional as well as for three-dimensional domains.

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arXiv2018

Optimal Distributed and Tangential Boundary Control for the Unsteady Stochastic Stokes Equations

We consider a control problem constrained by the unsteady stochastic Stokes equations with nonhomogeneous boundary conditions in connected and bounded domains. In this paper, controls are defined inside the domain as well as on the boundary. Using a stochastic maximum principle, we derive necessary and sufficient optimality conditions such that explicit formulas for the optimal controls are derived. As a consequence, we are able to control the stochastic Stokes equations using distributed controls as well as boundary controls in a desired way.

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arXiv2018

Hankel-Norm Approximation of Large-Scale Descriptor Systems

The Hankel-norm approximation is a model reduction method which provides the best approximation in the Hankel semi-norm. In this paper the computation of the optimal Hankel-norm approximation is generalized to the case of linear time-invariant continuous-time descriptor systems. An efficient algorithm is developed by refining the generalized balanced truncation square root method. For a wide practical usage, adaptations of the introduced algorithm towards stable computations and sparse systems are made as well as an approach for a projection-free algorithm. To show the approximation behavior of the introduced method, numerical examples are presented.

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arXiv2018

Peer Methods for the Solution of Large-Scale Differential Matrix Equations

We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The performance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential equations.

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