N. Chernov

Least Squares Fitting of Circles

Fitting standard shapes or curves to incomplete data (which represent only a small part of the curve) is a notoriously difficult problem. Even if the curve is quite simple, such as an ellipse or a circle, it is hard to reconstruct it from noisy data sampled along a short arc. Here we study the least squares fit (LSF) of circular arcs to incomplete scattered data. We analyze theoretical aspects of the problem and reveal the cause of unstable behavior of conventional algorithms. We also find a remedy that allows us to build another algorithm that accurately fits circles to data sampled along arbitrarily short arcs.

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a “thermostat” constructed according to Gauss’ principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Youngs expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohms transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.

Decay of Correlations and Dispersing Billiards

We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points

Error analysis for circle fitting algorithms

We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods – geometric fit, Kasa fit, Pratt fit, and Taubin fit – is presented. Our error analysis goes deeper than the traditional expansion to the leading order. We obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (non-iterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unbeatable) geometric fit.

Statistical efficiency of curve fitting algorithms

We study the problem of fitting parameterized curves to noisy data. Under certain assumptions (known as Cartesian and radial functional models), we derive asymptotic expressions for the bias and the covariance matrix of the parameter estimates. We also extend Kanatanis version of the Cramer-Rao lower bound, which he proved for unbiased estimates only, to more general estimates that include many popular algorithms (most notably, the orthogonal least squares and algebraic fits). We then show that the gradient-weighted algebraic fit is statistically efficient and describe all other statistically efficient algebraic fits.

Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic—enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here, we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.

Limit theorems and Markov approximations for chaotic dynamical systems

SummaryWe prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsAn⊃ X and μ-almost every pointx∈X the inclusionTnx∈An holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

Markov approximations and decay of correlations for Anosov flows

We develop Markov approximations for very general suspension flows. Based on this, we obtain a stretched exponential bound on time correlation functions for 3-D Anosov flows that verify ‘uniform nonintegrability of foliations’. These include contact Anosov flows and geodesic flows on compact surfaces of variable negative curvature. Our bound on correlations is stable under small smooth perturbations.

Entropy, Lyapunov exponents, and mean free path for billiards

We review known results and derive some new ones about the mean free path, Kolmogorov-Sinai entropy, and Lyapunov exponents for billiard-type dynamical systems. We focus on exact and asymptotic formulas for these quantities. The dynamical systems covered in this paper include the priodic Lorentz gas, the stadium and its modifications, and the gas of hard balls. Some open questions and numerical observations are discussed.