3,658 research outputs found
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
Fast convex optimization via inertial dynamics with Hessian driven damping
We first study the fast minimization properties of the trajectories of the
second-order evolution equation where
is a smooth convex function acting on a real
Hilbert space , and , are positive parameters. This
inertial system combines an isotropic viscous damping which vanishes
asymptotically, and a geometrical Hessian driven damping, which makes it
naturally related to Newton's and Levenberg-Marquardt methods. For , , along any trajectory, fast convergence of the values
is
obtained, together with rapid convergence of the gradients
to zero. For , just assuming that has minimizers, we show that
any trajectory converges weakly to a minimizer of , and . Strong convergence is
established in various practical situations. For the strongly convex case,
convergence can be arbitrarily fast depending on the choice of . More
precisely, we have . We extend the results to the case of a general
proper lower-semicontinuous convex function . This is based on the fact that the inertial
dynamic with Hessian driven damping can be written as a first-order system in
time and space. By explicit-implicit time discretization, this opens a gate to
new possibly more rapid inertial algorithms, expanding the field of
FISTA methods for convex structured optimization problems
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
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