arXiv subjects

M. Chertkov

Publications and source records attributed to M. Chertkov.

At least 19 recordsLinked to original sources

arXiv2013

Approximating the Permanent with Fractional Belief Propagation

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter $γ\in[-1;1]$, where $γ=-1$ corresponds to the BP limit and $γ=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to $γ$. For every non-negative matrix, we define its special value $γ_*\in[-1;0]$ to be the $γ$ for which the minimum of the $γ$-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the $γ$-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of $γ_*$ varies for different ensembles but $γ_*$ always lies within the $[-1;-1/2]$ interval. Moreover, for all ensembles considered the behavior of $γ_*$ is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.

cs.DM↗

arXiv2011

Voltage Collapse and ODE Approach to Power Flows: Analysis of a Feeder Line with Static Disorder in Consumption/Production

We consider a model of a distribution feeder connecting multiple loads to the sub-station. Voltage is controlled directly at the head of the line (sub-station), however, voltage anywhere further down the line is subject to fluctuations, caused by irregularities of real and reactive distributed power consumption/generation. The lack of a direct control of voltage along the line may result in the voltage instability, also called voltage collapse - phenomenon well known and documented in the power engineering literature. Motivated by emerging photo-voltaic technology, which brings a new source of renewable generation but also contributes significant increase in power flow fluctuations, we reexamine the phenomenon of voltage stability and collapse. In the limit where the number of consumers is large and spatial variations in power flows are smooth functions of position along the feeder, we derive a set of the power flow Ordinary Differential Equations (ODE), verify phenomenon of voltage collapse, and study the effect of disorder and irregularity in injection and consumption on the voltage profile by simulating the stochastic ODE. We observe that disorder leads to nonlinear amplification of the voltage variations at the end of the line as the point of voltage collapse is approached. We also find that the disorder, when correlated on a scale sufficiently small compared to the length of the line, self-averages, i.e. the voltage profile remains spatially smooth for any individual realization of the disorder and is correlated only at scales comparable to the length of the line. Finally, we explain why the integrated effect of disorder on the voltage at the end of the line cannot be described within a naive one-generator-one-load model.

nlin.AO↗

arXiv2010

Geometric Universality of Currents

We discuss a non-equilibrium statistical system on a graph or network. Identical particles are injected, interact with each other, traverse, and leave the graph in a stochastic manner described in terms of Poisson rates, possibly dependent on time and instantaneous occupation numbers at the nodes of the graph. We show that under the assumption of constancy of the relative rates, the system demonstrates a profound statistical symmetry, resulting in geometric universality of the statistics of the particle currents. This phenomenon applies broadly to many man-made and natural open stochastic systems, such as queuing of packages over the internet, transport of electrons and quasi-particles in mesoscopic systems, and chains of reactions in bio-chemical networks. We illustrate the utility of our general approach using two enabling examples from the two latter disciplines.

cond-mat.stat-mech↗

arXiv2010

Inference in particle tracking experiments by passing messages between images

Methods to extract information from the tracking of mobile objects/particles have broad interest in biological and physical sciences. Techniques based on simple criteria of proximity in time-consecutive snapshots are useful to identify the trajectories of the particles. However, they become problematic as the motility and/or the density of the particles increases due to uncertainties on the trajectories that particles followed during the images' acquisition time. Here, we report an efficient method for learning parameters of the dynamics of the particles from their positions in time-consecutive images. Our algorithm belongs to the class of message-passing algorithms, known in computer science, information theory and statistical physics as Belief Propagation (BP). The algorithm is distributed, thus allowing parallel implementation suitable for computations on multiple machines without significant inter-machine overhead. We test our method on the model example of particle tracking in turbulent flows, which is particularly challenging due to the strong transport that those flows produce. Our numerical experiments show that the BP algorithm compares in quality with exact Markov Chain Monte-Carlo algorithms, yet BP is far superior in speed. We also suggest and analyze a random-distance model that provides theoretical justification for BP accuracy. Methods developed here systematically formulate the problem of particle tracking and provide fast and reliable tools for its extensive range of applications.

physics.data-an↗

arXiv2009

Universal Velocity Profile for Coherent Vortices in Two-Dimensional Turbulence

Two-dimensional turbulence generated in a finite box produces large-scale coherent vortices coexisting with small-scale fluctuations. We present a rigorous theory explaining the $η=1/4$ scaling in the $V\propto r^{-η}$ law of the velocity spatial profile within a vortex, where $r$ is the distance from the vortex center. This scaling, consistent with earlier numerical and laboratory measurements, is universal in its independence of details of the small-scale injection of turbulent fluctuations and details of the shape of the box.

nlin.CD↗

arXiv2009

Approximate inference on planar graphs using Loop Calculus and Belief Propagation

We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006) allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in (Certkov et al., 2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze the performance of the algorithm for the partition function approximation for models with binary variables and pairwise interactions on grids and other planar graphs. We study in detail both the loop series and the equivalent Pfaffian series and show that the first term of the Pfaffian series for the general, intractable planar model, can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state-of-the-art methods for approximate inference.

cs.AI↗

arXiv2008

Reactive Rayleigh-Taylor Turbulence

The Rayleigh-Taylor (RT) instability develops and leads to turbulence when a heavy fluid falls under the action of gravity through a light one. We consider this phenomenon accompanied by a reactive transformation between the fluids, and study with Direct Numerical Simulations (DNS) how the reaction (flame) affects the turbulent mixing in the Boussinesq approximation. We discuss "slow" reactions where the characteristic reaction time exceeds the temporal scale of the RT instability. In the early turbulent stage, effects of the flame are distributed over a maturing mixing zone, whose development is weakly influenced by the reaction. At later times, the fully mixed zone transforms into a conglomerate of pure-fluid patches of sizes proportional to the mixing zone width. In this "stirred flame'' regime, temperature fluctuations are consumed by reactions in the regions separating the pure-fluid patches. This DNS-based qualitative description is followed by a phenomenology suggesting that thin turbulent flame is of a single-fractal character, and thus distribution of the temperature field is strongly intermittent.

physics.flu-dyn↗

arXiv2007

Searching for low weight pseudo-codewords

Belief Propagation (BP) and Linear Programming (LP) decodings of Low Density Parity Check (LDPC) codes are discussed. We summarize results of instanton/pseudo-codeword approach developed for analysis of the error-floor domain of the codes. Instantons are special, code and decoding specific, configurations of the channel noise contributing most to the Frame-Error-Rate (FER). Instantons are decoded into pseudo-codewords. Instanton/pseudo-codeword with the lowest weight describes the largest Signal-to-Noise-Ratio (SNR) asymptotic of FER, while the whole spectra of the low weight instantons is descriptive of the FER vs SNR profile in the extended error-floor domain. First, we describe a general optimization method that allows to find the instantons for any coding/decoding. Second, we introduce LP-specific pseudo-codeword search algorithm that allows efficient calculations of the pseudo-codeword spectra. Finally, we discuss results of combined BP/LP error-floor exploration experiments for two model codes.

cs.IT↗

arXiv2007

Strong effect of weak diffusion on scalar turbulence at large scales

Passive scalar turbulence forced steadily is characterized by the velocity correlation scale, $L$, injection scale, $l$, and diffusive scale, $r_d$. The scales are well separated if the diffusivity is small, $r_d\ll l,L$, and one normally says that effects of diffusion are confined to smaller scales, $r\ll r_d$. However, if the velocity is single scale one finds that a weak dependence of the scalar correlations on the molecular diffusivity persists to even larger scales, e.g. $l\gg r\gg r_d$ \cite{95BCKL}. We consider the case of $L\gg l$ and report a counter-intuitive result -- the emergence of a new range of large scales, $L\gg r\gg l^2/r_d$, where the diffusivity shows a strong effect on scalar correlations.

nlin.CD↗

arXiv2007

Growing condensate in two-dimensional turbulence

We report a numerical study, supplemented by phenomenological explanations, of ``energy condensation'' in forced 2D turbulence in a biperiodic box. Condensation is a finite size effect which occurs after the standard inverse cascade reaches the size of the system. It leads to emergence of a coherent vortex dipole. We show that the time growth of the dipole is self-similar, and it contains most of the injected energy, thus resulting in an energy spectrum which is markedly steeper than the standard $k^{-5/3}$ one. Once the coherent component is subtracted, however, the remaining fluctuations have a spectrum close to $k^{-1}$. The fluctuations decay slowly as the coherent part grows.

nlin.CD↗

arXiv2007

Statistics of Entropy Production in Linearized Stochastic System

We consider a wide class of linear stochastic problems driven off the equilibrium by a multiplicative asymmetric force. The force brakes detailed balance, maintained otherwise, thus producing entropy. The large deviation function of the entropy production in the system is calculated explicitly. The general result is illustrated using an example of a polymer immersed in a gradient flow and subject to thermal fluctuations.

nlin.CD↗

arXiv2006

Improving convergence of Belief Propagation decoding

The decoding of Low-Density Parity-Check codes by the Belief Propagation (BP) algorithm is revisited. We check the iterative algorithm for its convergence to a codeword (termination), we run Monte Carlo simulations to find the probability distribution function of the termination time, n_it. Tested on an example [155, 64, 20] code, this termination curve shows a maximum and an extended algebraic tail at the highest values of n_it. Aiming to reduce the tail of the termination curve we consider a family of iterative algorithms modifying the standard BP by means of a simple relaxation. The relaxation parameter controls the convergence of the modified BP algorithm to a minimum of the Bethe free energy. The improvement is experimentally demonstrated for Additive-White-Gaussian-Noise channel in some range of the signal-to-noise ratios. We also discuss the trade-off between the relaxation parameter of the improved iterative scheme and the number of iterations.

cs.IT↗

arXiv2006

Instanton analysis of Low-Density-Parity-Check codes in the error-floor regime

In this paper we develop instanton method introduced in [1], [2], [3] to analyze quantitatively performance of Low-Density-Parity-Check (LDPC) codes decoded iteratively in the so-called error-floor regime. We discuss statistical properties of the numerical instanton-amoeba scheme focusing on detailed analysis and comparison of two regular LDPC codes: Tanner's (155, 64, 20) and Margulis' (672, 336, 16) codes. In the regime of moderate values of the signal-to-noise ratio we critically compare results of the instanton-amoeba evaluations against the standard Monte-Carlo calculations of the Frame-Error-Rate.

cs.IT↗

arXiv2005

The error-floor of LDPC codes in the Laplacian channel

We analyze the performance of Low-Density-Parity-Check codes in the error-floor domain where the Signal-to-Noise-Ratio, s, is large, s >> 1. We describe how the instanton method of theoretical physics, recently adapted to coding theory, solves the problem of characterizing the error-floor domain in the Laplacian channel. An example of the (155,64,20) LDPC code with four iterations (each iteration consisting of two semi-steps: from bits-to-checks and from checks-to-bits) of the min-sum decoding is discussed. A generalized computational tree analysis is devised to explain the rational structure of the leading instantons. The asymptotic for the symbol Bit-Error-Rate in the error-floor domain is comprised of individual instanton contributions, each estimated as ~ \exp(-l_{inst;L} s), where the effective distances, l_{inst;L}, of the the leading instantons are 7.6, 8.0 and 8.0 respectively. (The Hamming distance of the code is 20.) The analysis shows that the instantons are distinctly different from the ones found for the same coding/decoding scheme performing over the Gaussian channel. We validate instanton results against direct simulations and offer an explanation for remarkable performance of the instanton approximation not only in the extremal, s -> \infty, limit but also at the moderate s values of practical interest.

cs.IT↗

arXiv2005

Diagnosis of weaknesses in modern error correction codes: a physics approach

One of the main obstacles to the wider use of the modern error-correction codes is that, due to the complex behavior of their decoding algorithms, no systematic method which would allow characterization of the Bit-Error-Rate (BER) is known. This is especially true at the weak noise where many systems operate and where coding performance is difficult to estimate because of the diminishingly small number of errors. We show how the instanton method of physics allows one to solve the problem of BER analysis in the weak noise range by recasting it as a computationally tractable minimization problem.

cond-mat.stat-mech↗

arXiv2004

Tumbling of Polymers in a Random Flow with Mean Shear

A polymer placed in chaotic flow with large mean shear tumbles, making a-periodic flips. We describe the statistics of angular orientation, as well as of tumbling time (separating two subsequent flips) of polymers in this flow. The probability distribution function (PDF) of the polymer orientation is peaked around a shear-preferred direction. The tails of this angular PDF are algebraic. The PDF of the tumbling time, $τ$, has a maximum at the value estimated as inverse Lyapunov exponent of the flow. This PDF shows an exponential tail for large $τ$ and a small-$τ$ tail determined by the simultaneous statistics of velocity PDF.

cond-mat.stat-mech↗

arXiv2004

Statistics of Polymer Extension in a Random Flow with Mean Shear

Considering the dynamics of a polymer with finite extensibility placed in a chaotic flow with large mean shear, we explain how the statistics of polymer extension changes with Weissenberg number, ${\it Wi}$, defined as the product of the polymer relaxation time and the Lyapunov exponent of the flow. Four regimes, of the ${\it Wi}$ number, are identified. One below the coil-stretched transition and three above the coil-stretched transition. Specific emphasis is given to explaining these regimes in terms of the polymer dynamics.

cond-mat.stat-mech↗

arXiv2003

Acceleration of chemical reaction by chaotic mixing

Theory of fast binary chemical reaction, ${\cal A}+{\cal B}\to{\cal C}$, in a statistically stationary chaotic flow at large Schmidt number ${Sc}$ and large Damköhler number ${Da}$ is developed. For stoichiometric condition we identify subsequent stages of the chemical reaction. The first stage corresponds to the exponential decay, $\propto\exp(-λt)$ (where $λ$ is the Lyapunov exponent of the flow), of the chemicals in the bulk part of the flow. The second and the third stages are related to the chemicals remaining in the boundary region. During the second stage the amounts of ${\cal A}$ and ${\cal B}$ decay $\propto 1/\sqrt{t}$, whereas the decay law during the third stage is exponential, $\propto\exp(-γt)$, where $γ\simλ/\sqrt{Sc}$.

nlin.CD↗