2,062 research outputs found
Sketched Newton-Raphson
We propose a new globally convergent stochastic second order method. Our
starting point is the development of a new Sketched Newton-Raphson (SNR) method
for solving large scale nonlinear equations of the form with
. We then show how to design several
stochastic second order optimization methods by re-writing the optimization
problem of interest as a system of nonlinear equations and applying SNR. For
instance, by applying SNR to find a stationary point of a generalized linear
model (GLM), we derive completely new and scalable stochastic second order
methods. We show that the resulting method is very competitive as compared to
state-of-the-art variance reduced methods. Furthermore, using a variable
splitting trick, we also show that the Stochastic Newton method (SNM) is a
special case of SNR, and use this connection to establish the first global
convergence theory of SNM.
We establish the global convergence of SNR by showing that it is a variant of
the stochastic gradient descent (SGD) method, and then leveraging proof
techniques of SGD. As a special case, our theory also provides a new global
convergence theory for the original Newton-Raphson method under strictly weaker
assumptions as compared to the classic monotone convergence theory.Comment: Accepted for SIAM Journal on Optimization. 47 pages, 4 figure
Numerical methods and computers used in elastohydrodynamic lubrication
Some of the methods of obtaining approximate numerical solutions to boundary value problems that arise in elastohydrodynamic lubrication are reviewed. The highlights of four general approaches (direct, inverse, quasi-inverse, and Newton-Raphson) are sketched. Advantages and disadvantages of these approaches are presented along with a flow chart showing some of the details of each. The basic question of numerical stability of the elastohydrodynamic lubrication solutions, especially in the pressure spike region, is considered. Computers used to solve this important class of lubrication problems are briefly described, with emphasis on supercomputers
Dissipative vortex solitons in 2D-lattices
We report the existence of stable symmetric vortex-type solutions for
two-dimensional nonlinear discrete dissipative systems governed by a
cubic-quintic complex Ginzburg-Landau equation. We construct a whole family of
vortex solitons with a topological charge S = 1. Surprisingly, the dynamical
evolution of unstable solutions of this family does not alter significantly
their profile, instead their phase distribution completely changes. They
transform into two-charges swirl-vortex solitons. We dynamically excite this
novel structure showing its experimental feasibility.Comment: 4 pages, 20 figure
Investigating Multiple Solutions in the Constrained Minimal Supersymmetric Standard Model
Recent work has shown that the Constrained Minimal Supersymmetric Standard
Model (CMSSM) can possess several distinct solutions for certain values of its
parameters. The extra solutions were not previously found by public
supersymmetric spectrum generators because fixed point iteration (the algorithm
used by the generators) is unstable in the neighbourhood of these solutions.
The existence of the additional solutions calls into question the robustness of
exclusion limits derived from collider experiments and cosmological
observations upon the CMSSM, because limits were only placed on one of the
solutions. Here, we map the CMSSM by exploring its multi-dimensional parameter
space using the shooting method, which is not subject to the stability issues
which can plague fixed point iteration. We are able to find multiple solutions
where in all previous literature only one was found. The multiple solutions are
of two distinct classes. One class, close to the border of bad electroweak
symmetry breaking, is disfavoured by LEP2 searches for neutralinos and
charginos. The other class has sparticles that are heavy enough to evade the
LEP2 bounds. Chargino masses may differ by up to around 10% between the
different solutions, whereas other sparticle masses differ at the sub-percent
level. The prediction for the dark matter relic density can vary by a hundred
percent or more between the different solutions, so analyses employing the dark
matter constraint are incomplete without their inclusion.Comment: 30 pages, 12 figures, 2 tables; v2: added discussion on speed of
shooting method, fixed typos, matches published versio
Integration algorithms of elastoplasticity for ceramic powder compaction
Inelastic deformation of ceramic powders (and of a broad class of rock-like
and granular materials), can be described with the yield function proposed by
Bigoni and Piccolroaz (2004, Yield criteria for quasibrittle and frictional
materials. Int. J. Solids and Structures, 41, 2855-2878). This yield function
is not defined outside the yield locus, so that 'gradient-based' integration
algorithms of elastoplasticity cannot be directly employed. Therefore, we
propose two ad hoc algorithms: (i.) an explicit integration scheme based on a
forward Euler technique with a 'centre-of-mass' return correction and (ii.) an
implicit integration scheme based on a 'cutoff-substepping' return algorithm.
Iso-error maps and comparisons of the results provided by the two algorithms
with two exact solutions (the compaction of a ceramic powder against a rigid
spherical cup and the expansion of a thick spherical shell made up of a green
body), show that both the proposed algorithms perform correctly and accurately.Comment: 21 pages. Journal of the European Ceramic Society, 201
A theory of two-dimensional airfoils with strong inlet flow on the upper surface
The two-dimensional theory of airfoils with arbitrarily strong inlet
flow into the upper surface was examined with the aim of developing a thin-airfoil
theory which is valid for this condition. Such a theory has, in fact,
been developed and reduces uniformly to the conventional thin-wing theory
when the inlet flow vanishes. The integrals associated with the arbitrary
shape, corresponding to the familiar Munk integrals, are somewhat more
complex but not so as to make calculations difficult.
To examine the limit for very high ratios of inlet to free-stream
velocity, the theory of the Joukowski airfoil was extended to incorporate
an arbitrary inlet on the upper surface. Because this calculation is exact,
phenomena observed in the limit cannot be attributed to the linearized calculation. These results showed that airfoil theory, in the conventional sense,
breaks down at very large ratios of inlet to free-stream velocity. This
occurs where the strong induced field of the inlet dominates the free-stream
flow so overwhelmingly that the flow no longer leaves the trailing edge but
flows toward it. Then the trailing edge becomes, in fact a leading edge
and the Kutta condition is physically inapplicable. For the example in this
work, this breakdown occurred at a ratio of inlet to free-stream velocity
of about 10. This phenomena suggests that for ratios in excess of the
critical value, the flow separates from the trailing edge and the circulation
is dominated by conditions at the edges of the inlet
Noise-induced phase transitions: Effects of the noises' statistics and spectrum
The local, uncorrelated multiplicative noises driving a second-order, purely
noise-induced, ordering phase transition (NIPT) were assumed to be Gaussian and
white in the model of [Phys. Rev. Lett. \textbf{73}, 3395 (1994)]. The
potential scientific and technological interest of this phenomenon calls for a
study of the effects of the noises' statistics and spectrum. This task is
facilitated if these noises are dynamically generated by means of stochastic
differential equations (SDE) driven by white noises. One such case is that of
Ornstein--Uhlenbeck noises which are stationary, with Gaussian pdf and a
variance reduced by the self-correlation time (\tau), and whose effect on the
NIPT phase diagram has been studied some time ago. Another such case is when
the stationary pdf is a (colored) Tsallis' (q)--\emph{Gaussian} which, being a
\emph{fat-tail} distribution for (q>1) and a \emph{compact-support} one for
(q<1), allows for a controlled exploration of the effects of the departure from
Gaussian statistics. As done before with stochastic resonance and other
phenomena, we now exploit this tool to study--within a simple mean-field
approximation and with an emphasis on the \emph{order parameter} and the
``\emph{susceptibility}''--the combined effect on NIPT of the noises'
statistics and spectrum. Even for relatively small (\tau), it is shown that
whereas fat-tail noise distributions ((q>1)) counteract the effect of
self-correlation, compact-support ones ((q<1)) enhance it. Also, an interesting
effect on the susceptibility is seen in the last case.Comment: 6 pages, 10 figures, uses aipproc.cls, aip-8s.clo and aipxfm.sty. To
appear in AIP Conference Proceedings. Invited talk at MEDYFINOL'06 (XV
Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics
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