7,264 research outputs found
On the determination of near body orbits using mass concentration models
Mathematical model for near-body orbit calculation using mass concentration, perturbation theory, nonlinear equations, geopotentials, and least squares metho
Numerical solution of perturbed Kepler problem using a split operator technique
An efficient geometric integrator is proposed for solving the perturbed
Kepler motion. This method is stable and accurate over long integration time,
which makes it appropriate for treating problems in astrophysics, like solar
system simulations, and atomic and molecular physics, like classical
simulations of highly excited atoms in external fields. The key idea is to
decompose the hamiltonian in solvable parts and propagate the system according
to each term. Two case studies, the Kepler atom in an uniform field and in a
monochromatic field, are presented and the errors are analyzed.Comment: 17 pages, 5 figures, submitted to the Journal of Computational
Physic
Newton-Raphson Consensus for Distributed Convex Optimization
We address the problem of distributed uncon- strained convex optimization
under separability assumptions, i.e., the framework where each agent of a
network is endowed with a local private multidimensional convex cost, is
subject to communication constraints, and wants to collaborate to compute the
minimizer of the sum of the local costs. We propose a design methodology that
combines average consensus algorithms and separation of time-scales ideas. This
strategy is proved, under suitable hypotheses, to be globally convergent to the
true minimizer. Intuitively, the procedure lets the agents distributedly
compute and sequentially update an approximated Newton- Raphson direction by
means of suitable average consensus ratios. We show with numerical simulations
that the speed of convergence of this strategy is comparable with alternative
optimization strategies such as the Alternating Direction Method of
Multipliers. Finally, we propose some alternative strategies which trade-off
communication and computational requirements with convergence speed.Comment: 18 pages, preprint with proof
Convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces
The computation of robust regression estimates often relies on minimization of a convex functional on a convex set. In this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization-minimization algorithms. Our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case. We prove convergence on convex function spaces for general coercive and convex functionals F and derive geometric convergence in certain unconstrained settings. The algorithm is applied to TV penalized quantile regression and is compared with a step size corrected Newton-Raphson algorithm. It is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better, whereas the Newton type method outpaces the former only after many iterations. Finally, in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows to utilize highly efficient algorithms for special quadratic programs in more complex settings. --regression analysis,monotone regression,quantile regression,shape constraints,L1 regression,nonparametric regression,total variation semi-norm,reweighted least squares,Fermat's problem,convex approximation,quadratic approximation,pool adjacent violators algorithm
Towards a new generation of multi-dimensional stellar evolution models: development of an implicit hydrodynamic code
This paper describes the first steps of development of a new multidimensional
time implicit code devoted to the study of hydrodynamical processes in stellar
interiors. The code solves the hydrodynamical equations in spherical geometry
and is based on the finite volume method. Radiation transport is taken into
account within the diffusion approximation. Realistic equation of state and
opacities are implemented, allowing the study of a wide range of problems
characteristic of stellar interiors. We describe in details the numerical
method and various standard tests performed to validate the method. We present
preliminary results devoted to the description of stellar convection. We first
perform a local simulation of convection in the surface layers of a A-type star
model. This simulation is used to test the ability of the code to address
stellar conditions and to validate our results, since they can be compared to
similar previous simulations based on explicit codes. We then present a global
simulation of turbulent convective motions in a cold giant envelope, covering
80% in radius of the stellar structure. Although our implicit scheme is
unconditionally stable, we show that in practice there is a limitation on the
time step which prevent the flow to move over several cells during a time step.
Nevertheless, in the cold giant model we reach a hydro CFL number of 100. We
also show that we are able to address flows with a wide range of Mach numbers
(10^-3 < Ms< 0.5), which is impossible with an anelastic approach. Our first
developments are meant to demonstrate that the use of an implicit scheme
applied to a stellar evolution context is perfectly thinkable and to provide
useful guidelines to optimise the development of an implicit multi-D
hydrodynamical code.Comment: 21 pages, 18 figures, accepted for publication in A&
Fast and accurate clothoid fitting
An effective solution to the problem of Hermite interpolation with a
clothoid curve is provided. At the beginning the problem is naturally
formulated as a system of nonlinear equations with multiple solutions that is
generally difficult to solve numerically. All the solutions of this nonlinear
system are reduced to the computation of the zeros of a single nonlinear
equation. A simple strategy, together with the use of a good and simple guess
function, permits to solve the single nonlinear equation with a few iterations
of the Newton--Raphson method.
The computation of the clothoid curve requires the computation of Fresnel and
Fresnel related integrals. Such integrals need asymptotic expansions near
critical values to avoid loss of precision. This is necessary when, for
example, the solution of interpolation problem is close to a straight line or
an arc of circle. Moreover, some special recurrences are deduced for the
efficient computation of asymptotic expansion.
The reduction of the problem to a single nonlinear function in one variable
and the use of asymptotic expansions make the solution algorithm fast and
robust.Comment: 14 pages, 3 figures, 9 Algorithm Table
Implementing Quantum Gates by Optimal Control with Doubly Exponential Convergence
We introduce a novel algorithm for the task of coherently controlling a
quantum mechanical system to implement any chosen unitary dynamics. It performs
faster than existing state of the art methods by one to three orders of
magnitude (depending on which one we compare to), particularly for quantum
information processing purposes. This substantially enhances the ability to
both study the control capabilities of physical systems within their coherence
times, and constrain solutions for control tasks to lie within experimentally
feasible regions. Natural extensions of the algorithm are also discussed.Comment: 4+2 figures; to appear in PR
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