arXiv subjects

Georgi Medvedev

Publications and source records attributed to Georgi Medvedev.

4 recordsLinked to original sources

arXiv2026

From PDEs on standard domains to self-similar particle systems on fractals

We construct transported PDEs on self-similar fractal domains from reference equations posed on the unit interval, and derive explicit self-similar interacting particle systems that approximate the resulting dynamics. The construction combines a measure-preserving isometry between $L^2$-spaces on $[0,1]$ and on the fractal \cite{Med2026}, a nonlocal-to-local approximation of differential operators \cite{PauTre2025}, and a Galerkin discretization on the canonical self-similar partitions. This yields a two-parameter approximation scheme whose error separates a nonlocal consistency term from a Galerkin network term. We work out the transport, Burgers, and heat equations, discuss the relation with intrinsic operators on fractals, and outline extensions to local charts and to pullbacks of nonlocal equations on fractal domains. Moreover, the reverse mapping transforms a nonlocal evolution equation on a fractal domains into the evolution equation on the unit interval, where the methods of classical numerical analysis can be applied. This suggests a promising direction for the development of numerical methods for nonlocal models on fractals, including fractional heat equation, fractal scattering, and related models.

math.AP↗

arXiv2022

A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and Q-Wiener process describing the interactions among the particles in the network and stochastic forcing respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct a semidiscrete (discrete in space and continuous in time) and a fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler-Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence estimate is expressed in terms of the regularity of the graphon, Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity including small-world and power-law are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler-Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

math.NA↗

arXiv2021

The large deviation principle for interacting dynamical systems on random graphs

Using the weak convergence approach to large deviations, we formulate and prove the large deviation principle (LDP) for W-random graphs in the cut-norm topology. This generalizes the LDP for Erdős-R{\' e}nyi random graphs by Chatterjee and Varadhan. Furthermore, we translate the LDP for random graphs to a class of interacting dynamical systems on such graphs. To this end, we demonstrate that the solutions of the dynamical models depend continuously on the underlying graphs with respect to the cut-norm and apply the contraction principle.

math.PR↗

arXiv2007

Multimodal oscillations in systems with strong contraction

One- and two-parameter families of flows in $R^3$ near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a weakly unstable saddle-focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle-focus. Our analysis covers both cases of sub- and supercritical AHB. For the supercritical case, we find that the periodic orbits born from the AHB are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium. If the AHB is subcritical, it is accompanied by the appearance of multimodal orbits, which consist of long series of nearly harmonic oscillations separated by large amplitude spikes. We analyze the dependence of the interspike intervals (which can be extremely long) on the control parameters. In particular, we show that the interspike intervals grow logarithmically as the boundary between regions of sub- and supercritical AHB is approached in the parameter space. We also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub- and supercritical AHB and explain the mechanism generating these oscillations. This work is motivated by the numerical results for a finite-dimensional approximation of a free boundary problem modeling solid fuel combustion.

math.CA↗