493,151 research outputs found
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems
We consider projection algorithms for solving (nonconvex) feasibility
problems in Euclidean spaces. Of special interest are the Method of Alternating
Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection
Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of
projection mappings is a global property that yields global convergence of MAP
and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets
developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of
firm nonexpansiveness with respect to the intersection is introduced for
consistent feasibility problems. Together with a coercivity condition that
relates to the regularity of the intersection, this yields local linear
convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference
Determination of chaotic behaviour in time series generated by charged particle motion around magnetized Schwarzschild black holes
We study behaviour of ionized region of a Keplerian disk orbiting a
Schwarzschild black hole immersed in an asymptotically uniform magnetic field.
In dependence on the magnetic parameter , and inclination angle
of the disk plane with respect to the magnetic field direction, the
charged particles of the ionized disk can enter three regimes: a) regular
oscillatory motion, b) destruction due to capture by the magnetized black hole,
c) chaotic regime of the motion. In order to study transition between the
regular and chaotic type of the charged particle motion, we generate time
series of the solution of equations of motion under various conditions, and
study them by non-linear (box counting, correlation dimension, Lyapunov
exponent, recurrence analysis, machine learning) methods of chaos
determination. We demonstrate that the machine learning method appears to be
the most efficient in determining the chaotic region of the space.
We show that the chaotic character of the ionized particle motion increases
with the inclination angle. For the inclination angles whole
the ionized internal part of the Keplerian disk is captured by the black hole.Comment: 21 pages, 9 figure
Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility
The problem of finding a vector with the fewest nonzero elements that
satisfies an underdetermined system of linear equations is an NP-complete
problem that is typically solved numerically via convex heuristics or
nicely-behaved nonconvex relaxations. In this work we consider elementary
methods based on projections for solving a sparse feasibility problem without
employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang
(2014) showed that, locally, the fundamental method of alternating projections
must converge linearly to a solution to the sparse feasibility problem with an
affine constraint. In this paper we apply different analytical tools that allow
us to show global linear convergence of alternating projections under familiar
constraint qualifications. These analytical tools can also be applied to other
algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm
where we establish local linear convergence of this method applied to the
sparse affine feasibility problem.Comment: 29 pages, 2 figures, 37 references. Much expanded version from last
submission. Title changed to reflect new development
- …