On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_\sigma$, for $\mathcal{N}_\sigma\triangleq\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$, by $\hat{P}_n\ast\mathcal{N}_\sigma$, where $\hat{P}_n$ is the empirical measure, under different statistical distances. The convergence is examined in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $\chi^2$-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf{W}_1$) converges at rate $e^{O(d)}n^{-\frac{1}{2}}$ in remarkable contrast to a typical $n^{-\frac{1}{d}}$ rate for unsmoothed $\mathsf{W}_1$ (and $d\ge 3$). For the KL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and $\chi^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$ achieves finite input-output $\chi^2$ mutual information across the additive white Gaussian noise channel. If the latter condition is not met, the rate changes to $\omega(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, while the $\chi^2$-divergence becomes infinite - a curious dichotomy. As a main application we consider estimating the differential entropy $h(P\ast\mathcal{N}_\sigma)$ in the high-dimensional regime. The distribution $P$ is unknown but $n$ i.i.d samples from it are available. We first show that any good estimator of $h(P\ast\mathcal{N}_\sigma)$ must have sample complexity that is exponential in $d$. Using the empirical approximation results we then show that the absolute-error risk of the plug-in estimator converges at the parametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimax rate-optimality of the plug-in. Numerical results that demonstrate a significant empirical superiority of the plug-in approach to general-purpose differential entropy estimators are provided.Comment: arXiv admin note: substantial text overlap with arXiv:1810.1158

Stochastic turbulence modeling in RANS simulations via Multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al.(2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.Comment: 40 page

On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method