42,522 research outputs found
On convergence of the maximum block improvement method
Abstract. The MBI (maximum block improvement) method is a greedy approach to solving optimization problems where the decision variables can be grouped into a finite number of blocks. Assuming that optimizing over one block of variables while fixing all others is relatively easy, the MBI method updates the block of variables corresponding to the maximally improving block at each iteration, which is arguably a most natural and simple process to tackle block-structured problems with great potentials for engineering applications. In this paper we establish global and local linear convergence results for this method. The global convergence is established under the Lojasiewicz inequality assumption, while the local analysis invokes second-order assumptions. We study in particular the tensor optimization model with spherical constraints. Conditions for linear convergence of the famous power method for computing the maximum eigenvalue of a matrix follow in this framework as a special case. The condition is interpreted in various other forms for the rank-one tensor optimization model under spherical constraints. Numerical experiments are shown to support the convergence property of the MBI method
Saffman-Taylor fingers with kinetic undercooling
The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Analytic solution for nonlinear shock acceleration in the Bohm limit
The selfconsistent steady state solution for a strong shock, significantly
modified by accelerated particles is obtained on the level of a kinetic
description, assuming Bohm-type diffusion. The original problem that is
commonly formulated in terms of the diffusion-convection equation for the
distribution function of energetic particles, coupled with the thermal plasma
through the momentum flux continuity equation, is reduced to a nonlinear
integral equation in one variable. Its solution provides selfconsistently both
the particle spectrum and the structure of the hydrodynamic flow. A critical
system parameter governing the acceleration process is found to be , where , with a suitably
normalized injection rate , the Mach number M >> 1, and the cut-off
momentum . We particularly focus on an efficient solution, in which
almost all the energy of the flow is converted into a few energetic particles.
It was found that (i) for this efficient solution (or, equivalently, for
multiple solutions) to exist, the parameter
must exceed a critical value ( is the injection
momentum), (ii) the total shock compression ratio r increases with M and
saturates at a level that scales as $ r \propto \Lambda_1 (iii) the downstream
power-law spectrum has the universal index q=3.5 over a broad momentum range.
(iv) completely smooth shock transitions do not appear in the steady state
kinetic description.Comment: 39 pages, 3 PostScript figures, uses aasms4.sty, to appear in Aug.
20, 1997 issue ApJ, vol. 48
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