Percolation-Based Approaches For Ray-Optical Propagation in Inhomogeneous Random Distribution of Discrete Scatterers

We address the problem of optical ray propagation in an inhomogeneous half�]plane lattice, where each cell can be occupied according to a known one�]dimensional obstacles density distribution. A monochromatic plane wave impinges on the random grid with a known angle and undergoes specular reflections on the occupied cells. We present two different approaches for evaluating the propagation depth inside the lattice. The former is based on the theory of the Martingale random processes, while in the latter ray propagation is modelled in terms of a Markov chain. A numerical validation assesses the proposed solutions, while validation through experimental data shows that the percolation model, in spite of its simplicity, can be applied to model real propagation problems

A Hybrid Approach for Modeling Stochastic Ray Propagation in Stratified Random Lattices

The present contribution deals with ray propagation in semi-innite percolation lattices consisting of a succession of uniform density layers. The problem of analytically evaluating the probability that a single ray penetrates up to a prescribed level before being reected back into the above empty half-plane is addressed. A hybrid approach, exploiting the complementarity of two mathematical models in dealing with uniform congurations, is presented and assessed through numerical ray-tracing-based experiments in order to show improvements upon previous predictions techniques. "The definitive version is available at www3.interscience.wiley.com

A Hamiltonian Monte Carlo method for Bayesian Inference of Supermassive Black Hole Binaries

We investigate the use of a Hamiltonian Monte Carlo to map out the posterior density function for supermassive black hole binaries. While previous Markov Chain Monte Carlo (MCMC) methods, such as Metropolis-Hastings MCMC, have been successfully employed for a number of different gravitational wave sources, these methods are essentially random walk algorithms. The Hamiltonian Monte Carlo treats the inverse likelihood surface as a "gravitational potential" and by introducing canonical positions and momenta, dynamically evolves the Markov chain by solving Hamilton's equations of motion. We present an implementation of the Hamiltonian Markov Chain that is faster, and more efficient by a factor of approximately the dimension of the parameter space, than the standard MCMC.Comment: 16 pages, 8 figure

Fastest mixing Markov chain on graphs with symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.Comment: 39 pages, 15 figure

Characterising poroelastic materials in the ultrasonic range - A Bayesian approach

Acoustic fields scattered by poroelastic materials contain key information about the materials' pore structure and elastic properties. Therefore, such materials are often characterised with inverse methods that use acoustic measurements. However, it has been shown that results from many existing inverse characterisation methods agree poorly. One reason is that inverse methods are typically sensitive to even small uncertainties in a measurement setup, but these uncertainties are difficult to model and hence often neglected. In this paper, we study characterising poroelastic materials in the Bayesian framework, where measurement uncertainties can be taken into account, and which allows us to quantify uncertainty in the results. Using the finite element method, we simulate measurements where ultrasonic waves are incident on a water-saturated poroelastic material in normal and oblique angles. We consider uncertainties in the incidence angle and level of measurement noise, and then explore the solution of the Bayesian inverse problem, the posterior density, with an adaptive parallel tempering Markov chain Monte Carlo algorithm. Results show that both the elastic and pore structure parameters can be feasibly estimated from ultrasonic measurements.Comment: Published in JSV. https://doi.org/10.1016/j.jsv.2019.05.02

Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard specular reflection rule is replaced with a Markov transition probabilities operator P that, at each collision of the billiard particle with the boundary of the billiard domain, gives the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure (RBM) is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain. RBMs provide simple and explicit mechanical models of particle-surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. We focus on the operator P itself and how it relates to the mechanical/geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we characterize the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine law, arise naturally as generalized invariant billiard measures; (2) we obtain some basic functional theoretic properties of P. Under very general conditions, we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert space. In a simple but illustrative example, we show that P is a compact (Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the features of the microstructure;(3) we explore the latter issue both analytically and numerically in a few representative examples;(4) we present a general algorithm for simulating these Markov chains based on a geometric description of the invariant volumes of classical statistical mechanics

The partition function zeroes of quantum critical points

The Lee–Yang theorem for the zeroes of the partition function is not strictly applicable to quantum systems because the zeroes are defined in units of the fugacity ehΔτ, and the Euclidean-time lattice spacing Δτ can be divergent in the infrared (IR). We recently presented analytic arguments describing how a new space-Euclidean time zeroes expansion can be defined, which reproduces Lee and Yang's scaling but avoids the unresolved branch points associated with the breaking of nonlocal symmetries such as Parity. We now present a first numerical analysis for this new zeroes approach for a quantum spin chain system. We use our scheme to quantify the renormalization group flow of the physical lattice couplings to the IR fixed point of this system. We argue that the generic Finite-Size Scaling (FSS) function of our scheme is identically the entanglement entropy of the lattice partition function and, therefore, that we are able to directly extract the central charge, c, of the quantum spin chain system using conformal predictions for the scaling of the entanglement entropy

Some simple but challenging Markov processes

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems