844 research outputs found
On the local stability of semidefinite relaxations
We consider a parametric family of quadratically constrained quadratic
programs (QCQP) and their associated semidefinite programming (SDP)
relaxations. Given a nominal value of the parameter at which the SDP relaxation
is exact, we study conditions (and quantitative bounds) under which the
relaxation will continue to be exact as the parameter moves in a neighborhood
around the nominal value. Our framework captures a wide array of statistical
estimation problems including tensor principal component analysis, rotation
synchronization, orthogonal Procrustes, camera triangulation and resectioning,
essential matrix estimation, system identification, and approximate GCD. Our
results can also be used to analyze the stability of SOS relaxations of general
polynomial optimization problems.Comment: 23 pages, 3 figure
Exploring strong-field deviations from general relativity via gravitational waves
Two new observational windows have been opened to strong gravitational
physics: gravitational waves, and very long baseline interferometry. This
suggests observational searches for new phenomena in this regime, and in
particular for those necessary to make black hole evolution consistent with
quantum mechanics. We describe possible features of "compact quantum objects"
that replace classical black holes in a consistent quantum theory, and
approaches to observational tests for these using gravitational waves. This is
an example of a more general problem of finding consistent descriptions of
deviations from general relativity, which can be tested via gravitational wave
detection. Simple models for compact modifications to classical black holes are
described via an effective stress tensor, possibly with an effective equation
of state. A general discussion is given of possible observational signatures,
and of their dependence on properties of the colliding objects. The possibility
that departures from classical behavior are restricted to the near-horizon
regime raises the question of whether these will be obscured in gravitational
wave signals, due to their mutual interaction in a binary coalescence being
deep in the mutual gravitational well. Numerical simulation with such simple
models will be useful to clarify the sensitivity of gravitational wave
observation to such highly compact departures from classical black holes.Comment: 20 pages, 9 figures. v2: references and CERN preprint number adde
A Hybrid Systems Model for Simple Manipulation and Self-Manipulation Systems
Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and even paradoxical results. As robotics modeling moves from the consideration of isolated behaviors to the analysis of tasks requiring their composition, a mathematically tractable framework for building models that combine these simple approximations yet achieve reliable results is overdue. In this paper we present a formal hybrid dynamical system model that introduces suitably restricted compositions of these familiar abstractions with the guarantee of consistency analogous to global existence and uniqueness in classical dynamical systems. The hybrid system developed here provides a discontinuous but self-consistent approximation to the continuous (though possibly very stiff and fast) dynamics of a physical robot undergoing intermittent impacts. The modeling choices sacrifice some quantitative numerical efficiencies while maintaining qualitatively correct and analytically tractable results with consistency guarantees promoting their use in formal reasoning about mechanism, feedback control, and behavior design in robots that make and break contact with their environment.
For more information: Kod*La
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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