1,106 research outputs found
Long term nonlinear propagation of uncertainties in perturbed geocentric dynamics using automatic domain splitting
Current approaches to uncertainty propagation in astrodynamics mainly refer tolinearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationallyintensive. Differential algebra has already proven to be an efficient compromiseby replacing thousands of pointwise integrations of Monte Carlo runs with thefast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics and long term propagations. Furthermore, valuable additional information about the dynamical system is available from the pattern in which those automatic splits occur. From this pattern it is immediately visible where the system behaves chaotically and where its evolution is smooth. Furthermore, it is possible to deduce the behavior of the system for each region, yielding further insight into the dynamics. In this work, the method is applied to the analysis of an end-of-life disposal trajectory of the INTEGRAL spacecraft
Differentiable Genetic Programming
We introduce the use of high order automatic differentiation, implemented via
the algebra of truncated Taylor polynomials, in genetic programming. Using the
Cartesian Genetic Programming encoding we obtain a high-order Taylor
representation of the program output that is then used to back-propagate errors
during learning. The resulting machine learning framework is called
differentiable Cartesian Genetic Programming (dCGP). In the context of symbolic
regression, dCGP offers a new approach to the long unsolved problem of constant
representation in GP expressions. On several problems of increasing complexity
we find that dCGP is able to find the exact form of the symbolic expression as
well as the constants values. We also demonstrate the use of dCGP to solve a
large class of differential equations and to find prime integrals of dynamical
systems, presenting, in both cases, results that confirm the efficacy of our
approach
An automatic domain splitting technique to propagate uncertainties in highly nonlinear orbital dynamics
Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics. The method is tested on the propagation of (99942) Apophis post-encounter motion
Lazy Multivariate Higher-Order Forward-Mode AD
A method is presented for computing all higher-order partial
derivatives of a multivariate function Rn → R. This method works
by evaluating the function under a nonstandard interpretation, lifting
reals to multivariate power series. Multivariate power series,
with potentially an infinite number of terms with nonzero coefficients,
are represented using a lazy data structure constructed
out of linear terms. A complete implementation of this method
in SCHEME is presented, along with a straightforward exposition,
based on Taylor expansions, of the method’s correctness
Multivariate Residues and Maximal Unitarity
We extend the maximal unitarity method to amplitude contributions whose cuts
define multidimensional algebraic varieties. The technique is valid to all
orders and is explicitly demonstrated at three loops in gauge theories with any
number of fermions and scalars in the adjoint representation. Deca-cuts
realized by replacement of real slice integration contours by
higher-dimensional tori encircling the global poles are used to factorize the
planar triple box onto a product of trees. We apply computational algebraic
geometry and multivariate complex analysis to derive unique projectors for all
master integral coefficients and obtain compact analytic formulae in terms of
tree-level data.Comment: 34 pages, 3 figure
Two-Loop Vertices in Quantum Field Theory: Infrared and Collinear Divergent Configurations
A comprehensive study is performed of two-loop Feynman diagrams with three
external legs which, due to the exchange of massless gauge-bosons, give raise
to infrared and collinear divergencies. Their relevance in assembling realistic
computations of next-to-next-to-leading corrections to physical observables is
emphasised. A classification of infrared singular configurations, based on
solutions of Landau equations, is introduced. Algorithms for the numerical
evaluation of the residues of the infrared poles and of the infrared finite
parts of diagrams are introduced and discussed within the scheme of dimensional
regularization. Integral representations of Feynman diagrams which form a
generalization of Nielsen - Goncharov polylogarithms are introduced and their
numerical evaluation discussed. Numerical results are shown for all different
families of multi-scale, two-loop, three-point infrared divergent diagrams and
successful comparisons with analytical results, whenever available, are
performed. Part of these results has already been included in a recent
evaluation of electroweak pseudo-observables at the two-loop level.Comment: 62 pages, 15 figures, 16 table
Energy flow polynomials: A complete linear basis for jet substructure
We introduce the energy flow polynomials: a complete set of jet substructure
observables which form a discrete linear basis for all infrared- and
collinear-safe observables. Energy flow polynomials are multiparticle energy
correlators with specific angular structures that are a direct consequence of
infrared and collinear safety. We establish a powerful graph-theoretic
representation of the energy flow polynomials which allows us to design
efficient algorithms for their computation. Many common jet observables are
exact linear combinations of energy flow polynomials, and we demonstrate the
linear spanning nature of the energy flow basis by performing regression for
several common jet observables. Using linear classification with energy flow
polynomials, we achieve excellent performance on three representative jet
tagging problems: quark/gluon discrimination, boosted W tagging, and boosted
top tagging. The energy flow basis provides a systematic framework for complete
investigations of jet substructure using linear methods.Comment: 41+15 pages, 13 figures, 5 tables; v2: updated to match JHEP versio
Lower order terms in the full moment conjecture for the Riemann zeta function
We describe an algorithm for obtaining explicit expressions for lower terms
for the conjectured full asymptotics of the moments of the Riemann zeta
function, and give two distinct methods for obtaining numerical values of these
coefficients. We also provide some numerical evidence in favour of the
conjecture.Comment: 37 pages, 4 figure
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