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 (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-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 $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (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 $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ 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 $A \sim$ 36, produced in the reactions of 47 MeV/nucleon $^{40}$Ar + $^{27}$Al, $^{48}$Ti and $^{58}$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 $\sim$ 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 $8.3 \pm 0.5$ MeV. Taking this temperature as the critical temperature and employing the caloric curve information we have extracted the critical exponents $\beta$, $\gamma$ and $\sigma$ 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