An Exactly Solvable Model for the Integrability-Chaos Transition in Rough Quantum Billiards

A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability-chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization-delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.Comment: 29 pages, 5 figure

Multicanonical Parallel Tempering

We present a novel implementation of the parallel tempering Monte Carlo method in a multicanonical ensemble. Multicanonical weights are derived by a self-consistent iterative process using a Boltzmann inversion of global energy histograms. This procedure gives rise to a much broader overlap of thermodynamic-property histograms; fewer replicas are necessary in parallel tempering simulations, and the acceptance of trial swap moves can be made arbitrarily high. We demonstrate the usefulness of the method in the context of a grand-multicanonical ensemble, where we use multicanonical simulations in energy space with the addition of an unmodified chemical potential term in particle-number space. Several possible implementations are discussed, and the best choice is presented in the context of the liquid-gas phase transition of the Lennard-Jones fluid. A substantial decrease in the necessary number of replicas can be achieved through the proposed method, thereby providing a higher efficiency and the possibility of parallelization.Comment: 8 pages, 3 figure, accepted by J Chem Phy

Reduced-Order Monte Carlo Modeling of Radiation Transport in Random Media

The ability to perform radiation transport computations in stochastic media is essential for predictive capabilities in applications such as weather modeling, radiation shielding involving non-homogeneous materials, atmospheric radiation transport computations, and transport in plasma-air structures. Due to the random nature of such media, it is often not clear how to model or otherwise compute on many forms of stochastic media. Several approaches to evaluation of transport quantities for some stochastic media exist, though such approaches often either yield considerable error or are quite computationally expensive. We model stochastic media using the Karhunen-Loève (KL) expansion, seek to improve efficiency through use of stochastic collocation (SC), and provide higher-order information of output values using the polynomial chaos expansion (PCE). We study and demonstrate method convergence and apply the new methods to both spatially continuous and spatially discontinuous stochastic media. New methods are shown to produce accurate solutions for reasonable computational cost for several problem when compared with existing solution methods. Spatially random media are modeled using transformations of the Gaussian-distributed KL expansion--continuous random media with a lognormal transformation and discontinuous random media with a Nataf transformation. Each transformation preserves second-order statistics for the quantity--atom density or material index, respectively--being modeled. The Nyström method facilitates numerical solution of the KL eigenvalues and eigenvectors, and a variety of methods are investigated for sampling KL eigenfunctions as a function of solved eigenvectors. The infinite KL expansion is truncated to a finite number of terms each containing a random variable, and material realizations are created by either randomly or deterministically sampling from the random variables. Deterministic sampling is performed with either isotropic or anisotropic stochastic collocation (SC), the latter of which takes advantage of the monotonic decay of KL terms. Transport quantities are solved on realizations using Monte Carlo particle transport with Woodcock sampling (WMC). Surrogate models of system responses are constructed from SC solutions using the polynomial chaos expansion (PCE) from which probability density functions (PDFs) of response quantities are constructed. The error convergence of solution methods is examined as a validation of the choice of methods, a verification of method implementation, and to give insight towards parameter selection for efficient computation. Solutions are compared against benchmark values generated in a variety of ways including analytic solutions, computational solutions of simpler models, expensive benchmark computations, and published benchmark values

Confidence regions for images observed under the Radon transform

Recovering a function f from its integrals over hyperplanes (or line integrals in the two-dimensional case), that is, recovering f from the Radon transform Rf of f, is a basic problem with important applications in medical imaging such as computerized tomography (CT). In the presence of stochastic noise in the observed function Rf, we shall construct asymptotic uniform confidence regions for the function f of interest, which allows to draw conclusions regarding global features of f. Speci cally, in a white noise model as well as a fixed-design regression model, we prove a Bickel-Rosenblatt-type theorem for the maximal deviation of a kernel-type estimator from its mean, and give uniform estimates for the bias for f in a Sobolev smoothness class. The finite sample properties of the proposed methods are investigated in a simulation study