224 research outputs found
Model Reduction of Descriptor Systems by Interpolatory Projection Methods
In this paper, we investigate interpolatory projection framework for model
reduction of descriptor systems. With a simple numerical example, we first
illustrate that employing subspace conditions from the standard state space
settings to descriptor systems generically leads to unbounded H2 or H-infinity
errors due to the mismatch of the polynomial parts of the full and
reduced-order transfer functions. We then develop modified interpolatory
subspace conditions based on the deflating subspaces that guarantee a bounded
error. For the special cases of index-1 and index-2 descriptor systems, we also
show how to avoid computing these deflating subspaces explicitly while still
enforcing interpolation. The question of how to choose interpolation points
optimally naturally arises as in the standard state space setting. We answer
this question in the framework of the H2-norm by extending the Iterative
Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical
examples are used to illustrate the theoretical discussion.Comment: 22 page
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energyadaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes
Optimal control of robot-guided laser material treatment
In this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel
Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem
Symplectic eigenvalues are conventionally defined for symmetric
positive-definite matrices via Williamson's diagonal form. Many properties of
standard eigenvalues, including the trace minimization theorem, are extended to
the case of symplectic eigenvalues. In this note, we will generalize
Williamson's diagonal form for symmetric positive-definite matrices to the case
of symmetric positive-semidefinite matrices, which allows us to define
symplectic eigenvalues, and prove the trace minimization theorem in the new
setting.Comment: 9 page
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Optimal control of robot guided laser material treatment
In this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems
such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational
physics and chemistry. These problems characterize critical points of energy
minimization problems on the infinite-dimensional Stiefel manifold. To
efficiently compute minimizers, we propose a novel Riemannian gradient descent
method induced by an energy-adaptive metric. Quantified convergence of the
methods is established under suitable assumptions on the underlying problem. A
non-monotone line search and the inexact evaluation of Riemannian gradients
substantially improve the overall efficiency of the method. Numerical
experiments illustrate the performance of the method and demonstrates its
competitiveness with well-established schemes.Comment: accepted for publication in M2A
Model Reduction of Descriptor Systems
Model reduction is of fundamental importance in many control applications. We consider model reduction methods for linear continuous-time descriptor systems. The methods are based on balanced truncation techniques and closely related to the controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving the generalized Lyapunov equations with special right-hand sides. The numerical solution of generalized Lyapunov equations is also discussed. A numerical example is given
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