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Fast Low-Rank Empirical Cross Gramians

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PAMM 2018年第1期17卷

作者： Christian Himpe Tobias Leibner Stephan Rave Jens Saak Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg Germany Applied Mathematics University of Münster Einsteinstr. 62 48149 Münster Germany

The cross Gramian matrix encodes the input-output coherence of linear control systems and is used in projection-based model reduction. The empirical cross Gramian is a data-driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large-scale systems is its full order and dense structure; yet, it may be computed column-wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Numerical solution to low rank perturbed lyapunov equations by the sign function method

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Numerical solution to low rank perturbed lyapunov equations ...

Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Benner, Peter Denißen, Jonas Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 Magdeburg39106 Germany

We investigate the numerical solution to a low rank perturbed Lyapunov equation AT X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i.e. A = B + UCVT, where B is a blockdiagonal matrix and UCVT is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Matrix algebra

Adjoint-based optimal boundary control of a stefan problem system fully coupled with navier-stokes equations

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Adjoint-based optimal boundary control of a stefan problem s...

Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Baran, Björn Heiland, Jan Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the "optimize-then-discretize" paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional. By means of an example setup of a container with an in-and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Crystal growth

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

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arXiv 2017年

作者： Benner, Peter Qiu, Yue Stoll, Martin Computational Methods in Systems and Control Theory Group Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Numerical Linear Algebra for Dynamical Systems Group Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

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 O(nxnt) complexity for both, computation and storage, where nx is the dimension of the spatial domain and nt 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 O(nxnt) to O(nx + nt). We use numerical experiments to illustrate the advantages of our approach. Copyright © 2017, The Authors. All rights reserved.

关键词： Constrained optimization

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

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arXiv 2017年

作者： Benner, Peter Dolgov, Sergey Onwunta, Akwum Stoll, Martin University of Bath Department of Mathematical Sciences Claverton Down BathBA2 7AY United Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

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. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

Solving differential matrix equations using parareal

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Solving differential matrix equations using parareal

Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Köhler, Martin Lang, Norman Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Computational Methods in Systems and Control Theory Germany Technische Universität Chemnitz Faculty of Mathematics Germany

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time-varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Ordinary differential equations

An inexact Newton-Krylov method for stochastic eigenvalue problems

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arXiv 2017年

作者： Benner, Peter Onwunta, Akwum Stoll, Martin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstrasse 1 Magdeburg39106 Germany Numerical Linear Algebra for Dynamical Systems Group Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstrasse 1 Magdeburg39106 Germany Technische Universität Chemnitz Faculty of Mathematics Professorship Scientific Computing Chemnitz09107 Germany

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. Copyright © 2017, The Authors. All rights reserved.

关键词： Galerkin methods

AFast Solver for an H_(1) Regularized PDE-Constrained Optimization Problem

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Communications in computational Physics 2016年第1期19卷 143-167页

作者： Andrew T.Barker Tyrone Rees Martin Stoll Center for Applied Scientific Computing Lawrence Livermore National LaboratoryMail Stop L-561LivermoreCA 94551USA Numerical Analysis Group Scientific Computing DepartmentRutherford Appleton LaboratoryChiltonDidcotOxfordshireOX110QXUnited Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr.139106MagdeburgGermany

In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control *** focus on a time-dependent PDE,and consider both distributed and boundary *** problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle *** propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

关键词： Preconditioning Krylov methods PDE-constrained optimization optimal control of PDEs

Model reduction of controlled fokker-planck and liouville-von neumann equations

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arXiv 2017年

作者： BENNER, PETER BREITEN, TOBIAS HARTMANN, CARSTEN SCHMIDT, BURKHARD Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 MagdeburgD-39106 Germany Institut fur Mathematik Karl-Franzens-Universitat Heinrichstr. 36/III GrazA-8010 Austria Institut fur Mathematik Brandenburgische Technische Universitat Konrad-Wachsmann-Allee 1 CottbusD-03046 Germany Institut fur Mathematik Freie Universitat Berlin Arnimallee 6 BerlinD-14195 Germany

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. Copyright © 2017, The Authors. All rights reserved.

关键词： Lyapunov functions