4,719 research outputs found
Moments of spectral functions: Monte Carlo evaluation and verification
The subject of the present study is the Monte Carlo path-integral evaluation
of the moments of spectral functions. Such moments can be computed by formal
differentiation of certain estimating functionals that are
infinitely-differentiable against time whenever the potential function is
arbitrarily smooth. Here, I demonstrate that the numerical differentiation of
the estimating functionals can be more successfully implemented by means of
pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial
interpolant), which utilize information from the entire interval . The algorithmic detail that leads to robust numerical
approximations is the fact that the path integral action and not the actual
estimating functional are interpolated. Although the resulting approximation to
the estimating functional is non-linear, the derivatives can be computed from
it in a fast and stable way by contour integration in the complex plane, with
the help of the Cauchy integral formula (e.g., by Lyness' method). An
interesting aspect of the present development is that Hamburger's conditions
for a finite sequence of numbers to be a moment sequence provide the necessary
and sufficient criteria for the computed data to be compatible with the
existence of an inversion algorithm. Finally, the issue of appearance of the
sign problem in the computation of moments, albeit in a milder form than for
other quantities, is addressed.Comment: 13 pages, 2 figure
Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments
In this paper, we address the risk estimation problem where one aims at
estimating the probability of violation of safety constraints for a robot in
the presence of bounded uncertainties with arbitrary probability distributions.
In this problem, an unsafe set is described by level sets of polynomials that
is, in general, a non-convex set. Uncertainty arises due to the probabilistic
parameters of the unsafe set and probabilistic states of the robot. To solve
this problem, we use a moment-based representation of probability
distributions. We describe upper and lower bounds of the risk in terms of a
linear weighted sum of the moments. Weights are coefficients of a univariate
Chebyshev polynomial obtained by solving a sum-of-squares optimization problem
in the offline step. Hence, given a finite number of moments of probability
distributions, risk can be estimated in real-time. We demonstrate the
performance of the provided approach by solving probabilistic collision
checking problems where we aim to find the probability of collision of a robot
with a non-convex obstacle in the presence of probabilistic uncertainties in
the location of the robot and size, location, and geometry of the obstacle.Comment: 57th IEEE Conference on Decision and Control 201
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
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