569 research outputs found
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example
Coarse timesteppers provide a bridge between microscopic / stochastic system
descriptions and macroscopic tasks such as coarse stability/bifurcation
computations. Exploiting this computational enabling technology, we present a
framework for designing observers and controllers based on microscopic
simulations, that can be used for their coarse control. The proposed
methodology provides a bridge between traditional numerical analysis and
control theory on the one hand and microscopic simulation on the other
Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary Value Problems
In "equation-free" multiscale computation a dynamic model is given at a fine,
microscopic level; yet we believe that its coarse-grained, macroscopic dynamics
can be described by closed equations involving only coarse variables. These
variables are typically various low-order moments of the distributions evolved
through the microscopic model. We consider the problem of integrating these
unavailable equations by acting directly on kinetic Monte Carlo microscopic
simulators, thus circumventing their derivation in closed form. In particular,
we use projective multi-step integration to solve the coarse initial value
problem forward in time as well as backward in time (under certain conditions).
Macroscopic trajectories are thus traced back to unstable, source-type, and
even sometimes saddle-like stationary points, even though the microscopic
simulator only evolves forward in time. We also demonstrate the use of such
projective integrators in a shooting boundary value problem formulation for the
computation of "coarse limit cycles" of the macroscopic behavior, and the
approximation of their stability through estimates of the leading "coarse
Floquet multipliers".Comment: Submitted to Journal of Computational Physic
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
Adaptive Detection of Instabilities: An Experimental Feasibility Study
We present an example of the practical implementation of a protocol for
experimental bifurcation detection based on on-line identification and feedback
control ideas. The idea is to couple the experiment with an on-line
computer-assisted identification/feedback protocol so that the closed-loop
system will converge to the open-loop bifurcation points. We demonstrate the
applicability of this instability detection method by real-time,
computer-assisted detection of period doubling bifurcations of an electronic
circuit; the circuit implements an analog realization of the Roessler system.
The method succeeds in locating the bifurcation points even in the presence of
modest experimental uncertainties, noise and limited resolution. The results
presented here include bifurcation detection experiments that rely on
measurements of a single state variable and delay-based phase space
reconstruction, as well as an example of tracing entire segments of a
codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format
(eps), need psfig macro to include them. Submitted to Physica
History of the department of medical biology and genetics (1931-2021)
The article discusses the history of the formation, the present and the future of the Department of Medical Biology and Genetics of the Ural State Medical UniversityВ статье рассмотрена история становления, настоящее и перспективы кафедры медицинской биологии и генетики Уральского государственного медицинского университет
Tipping Points of Evolving Epidemiological Networks: Machine Learning-Assisted, Data-Driven Effective Modeling
We study the tipping point collective dynamics of an adaptive
susceptible-infected-susceptible (SIS) epidemiological network in a
data-driven, machine learning-assisted manner. We identify a
parameter-dependent effective stochastic differential equation (eSDE) in terms
of physically meaningful coarse mean-field variables through a deep-learning
ResNet architecture inspired by numerical stochastic integrators. We construct
an approximate effective bifurcation diagram based on the identified drift term
of the eSDE and contrast it with the mean-field SIS model bifurcation diagram.
We observe a subcritical Hopf bifurcation in the evolving network's effective
SIS dynamics, that causes the tipping point behavior; this takes the form of
large amplitude collective oscillations that spontaneously -- yet rarely --
arise from the neighborhood of a (noisy) stationary state. We study the
statistics of these rare events both through repeated brute force simulations
and by using established mathematical/computational tools exploiting the
right-hand-side of the identified SDE. We demonstrate that such a collective
SDE can also be identified (and the rare events computations also performed) in
terms of data-driven coarse observables, obtained here via manifold learning
techniques, in particular Diffusion Maps. The workflow of our study is
straightforwardly applicable to other complex dynamics problems exhibiting
tipping point dynamics.Comment: 22 pages, 12 figure
Learning effective stochastic differential equations from microscopic simulations: combining stochastic numerics and deep learning
We identify effective stochastic differential equations (SDE) for coarse
observables of fine-grained particle- or agent-based simulations; these SDE
then provide coarse surrogate models of the fine scale dynamics. We approximate
the drift and diffusivity functions in these effective SDE through neural
networks, which can be thought of as effective stochastic ResNets. The loss
function is inspired by, and embodies, the structure of established stochastic
numerical integrators (here, Euler-Maruyama and Milstein); our approximations
can thus benefit from error analysis of these underlying numerical schemes.
They also lend themselves naturally to "physics-informed" gray-box
identification when approximate coarse models, such as mean field equations,
are available. Our approach does not require long trajectories, works on
scattered snapshot data, and is designed to naturally handle different time
steps per snapshot. We consider both the case where the coarse collective
observables are known in advance, as well as the case where they must be found
in a data-driven manner.Comment: 19 pages, includes supplemental materia
Dvoretzky type theorems for multivariate polynomials and sections of convex bodies
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type
theorem) for homogeneous polynomials on , and improve bounds on
the number in the analogous conjecture for odd degrees (this case
is known as the Birch theorem) and complex polynomials. We also consider a
stronger conjecture on the homogeneous polynomial fields in the canonical
bundle over real and complex Grassmannians. This conjecture is much stronger
and false in general, but it is proved in the cases of (for 's of
certain type), odd , and the complex Grassmannian (for odd and even and
any ). Corollaries for the John ellipsoid of projections or sections of a
convex body are deduced from the case of the polynomial field conjecture
Critical Behavior in Light Nuclear Systems: Experimental Aspects
An extensive experimental survey of the features of the disassembly of a
small quasi-projectile system with 36, produced in the reactions of 47
MeV/nucleon Ar + Al, Ti and Ni, has been carried
out. Nuclei in the excitation energy range of 1-9 MeV/u have been investigated
employing a new method to reconstruct the quasi-projectile source. At an
excitation energy 5.6 MeV/nucleon many observables indicate the presence
of maximal fluctuations in the de-excitation processes. The fragment
topological structure shows that the rank sorted fragments obey Zipf's law at
the point of largest fluctuations providing another indication of a liquid gas
phase transition. The caloric curve for this system shows a monotonic increase
of temperature with excitation energy and no apparent plateau. The temperature
at the point of maximal fluctuations is MeV. Taking this
temperature as the critical temperature and employing the caloric curve
information we have extracted the critical exponents , and
from the data. Their values are also consistent with the values of the
universality class of the liquid gas phase transition. Taken together, this
body of evidence strongly suggests a phase change in an equilibrated mesoscopic
system at, or extremely close to, the critical point.Comment: Physical Review C, in press; some discussions about the validity of
excitation energy in peripheral collisions have been added; 24 pages and 32
figures; longer abstract in the preprin
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