42,649 research outputs found
Calculation of aggregate loss distributions
Estimation of the operational risk capital under the Loss Distribution
Approach requires evaluation of aggregate (compound) loss distributions which
is one of the classic problems in risk theory. Closed-form solutions are not
available for the distributions typically used in operational risk. However
with modern computer processing power, these distributions can be calculated
virtually exactly using numerical methods. This paper reviews numerical
algorithms that can be successfully used to calculate the aggregate loss
distributions. In particular Monte Carlo, Panjer recursion and Fourier
transformation methods are presented and compared. Also, several closed-form
approximations based on moment matching and asymptotic result for heavy-tailed
distributions are reviewed
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
Quantum probabilistic sampling of multipartite 60-qubit Bell inequality violations
We show that violation of genuine multipartite Bell inequalities can be
obtained with sampled, probabilistic phase space methods. These genuine Bell
violations cannot be replicated if any part of the system is described by a
local hidden variable theory. The Bell violations are simulated
probabilistically using quantum phase-space representations. We treat
mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60
qubits, using both a multipartite SU(2) Q-representation and the positive
P-representation. Surprisingly, we find that sampling with phase-space
distributions can be exponentially faster than experiment. This is due to the
classical parallelism inherent in the simulation of quantum measurements using
phase-space methods. Our probabilistic sampling method predicts a contradiction
with local realism of "Schr\"odinger-cat" states that can be realized as a GHZ
spin state, either in ion traps or with photonic qubits. We also present a
quantum simulation of the observed super-decoherence of the ion-trap "cat"
state, using a phenomenological noise model
Simplified Onsager theory for isotropic-nematic phase equilibria of length polydisperse hard rods
Polydispersity is believed to have important effects on the formation of
liquid crystal phases in suspensions of rod-like particles. To understand such
effects, we analyse the phase behaviour of thin hard rods with length
polydispersity. Our treatment is based on a simplified Onsager theory, obtained
by truncating the series expansion of the angular dependence of the excluded
volume. We describe the model and give the full phase equilibrium equations;
these are then solved numerically using the moment free energy method which
reduces the problem from one with an infinite number of conserved densities to
one with a finite number of effective densities that are moments of the full
density distribution. The method yields exactly the onset of nematic ordering.
Beyond this, results are approximate but we show that they can be made
essentially arbitrarily precise by adding adaptively chosen extra moments,
while still avoiding the numerical complications of a direct solution of the
full phase equilibrium conditions.
We investigate in detail the phase behaviour of systems with three different
length distributions: a (unimodal) Schulz distribution, a bidisperse
distribution and a bimodal mixture of two Schulz distributions which
interpolates between these two cases. A three-phase isotropic-nematic-nematic
coexistence region is shown to exist for the bimodal and bidisperse length
distributions if the ratio of long and short rod lengths is sufficiently large,
but not for the unimodal one. We systematically explore the topology of the
phase diagram as a function of the width of the length distribution and of the
rod length ratio in the bidisperse and bimodal cases.Comment: 18 pages, 16 figure
Ensemble Transport Adaptive Importance Sampling
Markov chain Monte Carlo methods are a powerful and commonly used family of
numerical methods for sampling from complex probability distributions. As
applications of these methods increase in size and complexity, the need for
efficient methods increases. In this paper, we present a particle ensemble
algorithm. At each iteration, an importance sampling proposal distribution is
formed using an ensemble of particles. A stratified sample is taken from this
distribution and weighted under the posterior, a state-of-the-art ensemble
transport resampling method is then used to create an evenly weighted sample
ready for the next iteration. We demonstrate that this ensemble transport
adaptive importance sampling (ETAIS) method outperforms MCMC methods with
equivalent proposal distributions for low dimensional problems, and in fact
shows better than linear improvements in convergence rates with respect to the
number of ensemble members. We also introduce a new resampling strategy,
multinomial transformation (MT), which while not as accurate as the ensemble
transport resampler, is substantially less costly for large ensemble sizes, and
can then be used in conjunction with ETAIS for complex problems. We also focus
on how algorithmic parameters regarding the mixture proposal can be quickly
tuned to optimise performance. In particular, we demonstrate this methodology's
superior sampling for multimodal problems, such as those arising from inference
for mixture models, and for problems with expensive likelihoods requiring the
solution of a differential equation, for which speed-ups of orders of magnitude
are demonstrated. Likelihood evaluations of the ensemble could be computed in a
distributed manner, suggesting that this methodology is a good candidate for
parallel Bayesian computations
Monte Carlo techniques for real-time quantum dynamics
The stochastic-gauge representation is a method of mapping the equation of
motion for the quantum mechanical density operator onto a set of equivalent
stochastic differential equations. One of the stochastic variables is termed
the "weight", and its magnitude is related to the importance of the stochastic
trajectory. We investigate the use of Monte Carlo algorithms to improve the
sampling of the weighted trajectories and thus reduce sampling error in a
simulation of quantum dynamics. The method can be applied to calculations in
real time, as well as imaginary time for which Monte Carlo algorithms are
more-commonly used. The method is applicable when the weight is guaranteed to
be real, and we demonstrate how to ensure this is the case. Examples are given
for the anharmonic oscillator, where large improvements over stochastic
sampling are observed.Comment: 28 pages, submitted to J. Comp. Phy
A new, efficient algorithm for the Forest Fire Model
The Drossel-Schwabl Forest Fire Model is one of the best studied models of
non-conservative self-organised criticality. However, using a new algorithm,
which allows us to study the model on large statistical and spatial scales, it
has been shown to lack simple scaling. We thereby show that the considered
model is not critical. This paper presents the algorithm and its parallel
implementation in detail, together with large scale numerical results for
several observables. The algorithm can easily be adapted to related problems
such as percolation.Comment: 38 pages, 28 figures, REVTeX 4, RMP style; V2 is for clarifications
as well as corrections and update of reference
Analytical Solutions to the Mass-Anisotropy Degeneracy with Higher Order Jeans Analysis: A General Method
The Jeans analysis is often used to infer the total density of a system by
relating the velocity moments of an observable tracer population to the
underlying gravitational potential. This technique has recently been applied in
the search for Dark Matter in objects such as dwarf spheroidal galaxies where
the presence of Dark Matter is inferred via stellar velocities. A precise
account of the density is needed to constrain the expected gamma ray flux from
DM self-annihilation and to distinguish between cold and warm dark matter
models. Unfortunately the traditional method of fitting the second order Jeans
equation to the tracer dispersion suffers from an unbreakable degeneracy of
solutions due to the unknown velocity anisotropy of the projected system. To
tackle this degeneracy one can appeal to higher moments of the Jeans equation.
By introducing an analog to the Binney anisotropy parameter at fourth order,
beta' we create a framework that encompasses all solutions to the fourth order
Jeans equations rather than those in the literature that impose unnecessary
correlations between anisotropy of second and fourth order moments. The
condition beta' = f(beta) ensures that the degeneracy is lifted and we
interpret the separable augmented density system as the order-independent case
beta'= beta. For a generic choice of beta' we present the line of sight
projection of the fourth moment and how it could be incorporated into a joint
likelihood analysis of the dispersion and kurtosis. Having presented the
mathematical framework, we then use it to develop a statistical method for the
purpose of placing constraints on dark matter density parameters from discrete
velocity data. The method is tested on simulated dwarf spheroidal data sets
leading to results which motivate study of real dwarf spheroidal data sets.Comment: 21 pages, 15 figures. Accepted by MNRAS. Typo corrected in eq. 3
Benchmarking of Gaussian boson sampling using two-point correlators
Gaussian boson sampling is a promising scheme for demonstrating a quantum
computational advantage using photonic states that are accessible in a
laboratory and, thus, offer scalable sources of quantum light. In this
contribution, we study two-point photon-number correlation functions to gain
insight into the interference of Gaussian states in optical networks. We
investigate the characteristic features of statistical signatures which enable
us to distinguish classical from quantum interference. In contrast to the
typical implementation of boson sampling, we find additional contributions to
the correlators under study which stem from the phase dependence of Gaussian
states and which are not observable when Fock states interfere. Using the first
three moments, we formulate the tools required to experimentally observe
signatures of quantum interference of Gaussian states using two outputs only.
By considering the current architectural limitations in realistic experiments,
we further show that a statistically significant discrimination between quantum
and classical interference is possible even in the presence of loss, noise, and
a finite photon-number resolution. Therefore, we formulate and apply a
theoretical framework to benchmark the quantum features of Gaussian boson
sampling under realistic conditions
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