92 research outputs found
Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods
This paper is concerned with the comparison of semi-analytical and
non-averaged propagation methods for Earth satellite orbits. We analyse the
total integration error for semi-analytical methods and propose a novel
decomposition into dynamical, model truncation, short-periodic, and numerical
error components. The first three are attributable to distinct approximations
required by the method of averaging, which fundamentally limit the attainable
accuracy. In contrast, numerical error, the only component present in
non-averaged methods, can be significantly mitigated by employing adaptive
numerical algorithms and regularized formulations of the equations of motion.
We present a collection of non-averaged methods based on the integration of
existing regularized formulations of the equations of motion through an
adaptive solver. We implemented the collection in the orbit propagation code
THALASSA, which we make publicly available, and we compared the non-averaged
methods to the semi-analytical method implemented in the orbit propagation tool
STELA through numerical tests involving long-term propagations (on the order of
decades) of LEO, GTO, and high-altitude HEO orbits. For the test cases
considered, regularized non-averaged methods were found to be up to two times
slower than semi-analytical for the LEO orbit, to have comparable speed for the
GTO, and to be ten times as fast for the HEO (for the same accuracy). We show
for the first time that efficient implementations of non-averaged regularized
formulations of the equations of motion, and especially of non-singular element
methods, are attractive candidates for the long-term study of high-altitude and
highly elliptical Earth satellite orbits.Comment: 33 pages, 10 figures, 7 tables. Part of the CMDA Topical Collection
on "50 years of Celestial Mechanics and Dynamical Astronomy". Comments and
feedback are encourage
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor
Coding on Flag Manifolds for Limited Feedback MIMO Systems
The efficiency of the physical layer in modern communication systems using multi-input multi-output (MIMO) techniques is largely based on the availability of channel state information (CSI) at the transmitter. In many practical systems, CSI needs to be quantized at the receiver side before transmission through a limited rate feedback channel. This is typically done using a codebook-based precoding transmission, where the receiver transmits the index of a codeword from a pre-designed codebook shared with the transmitter. To construct such codes one has to discretize complex flag manifolds. For single-user MIMO with a maximum likelihood receiver, the spaces of interest are Grassmann manifolds. With a linear receiver and network MIMO, the codebook design is related to discretization of Stiefel manifolds and more general flag manifolds.
In this thesis, coding in flag manifolds is studied. In a first part, flag manifolds are defined as metric spaces corresponding to subsurfaces of hyperspheres. The choice of distance defines the geometry of the space and impacts clustering and averaging (centroid computation) in vector quantization, as well as coding theoretical packing bounds and optimum constructions.
For two transmitter antenna systems, the problem reduces to designing spherical codes. A simple isomorphism enables to analytically derive closed-form codebooks with inherent low-implementation complexity. For more antennas, the concept of orbits of symmetry groups is investigated. Optimum codebooks, having desirable implementation properties as described in industry standardization, can be obtained using orbits of specific groups.
For large antenna systems and base station cooperation, a product codebook strategy is also considered. Such a design requires to jointly discretize the Grassmann and Stiefel manifolds. A vector quantization algorithm for joint Grassmann-Stiefel quantization is proposed. Finally, the pertinence of flag codebook design is illustrated for a MIMO system with linear receiver
Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics
We provide an explicit formula for the Levi-Civita connection and Riemannian
Hessian for a Riemannian manifold that is a quotient of a manifold embedded in
an inner product space with a non-constant metric function. Together with a
classical formula for projection, this allows us to evaluate Riemannian
gradient and Hessian for several families of metrics on classical manifolds,
including a family of metrics on Stiefel manifolds connecting both the constant
and canonical ambient metrics with closed-form geodesics. Using these formulas,
we derive Riemannian optimization frameworks on quotients of Stiefel manifolds,
including flag manifolds, and a new family of complete quotient metrics on the
manifold of positive-semidefinite matrices of fixed rank, considered as a
quotient of a product of Stiefel and positive-definite matrix manifold with
affine-invariant metrics. The method is procedural, and in many instances, the
Riemannian gradient and Hessian formulas could be derived by symbolic calculus.
The method extends the list of potential metrics that could be used in manifold
optimization and machine learning
Nonlinear eigenvalue problems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 211-217).by Ross Adams Lippert.Ph.D
The Lyapunov Characteristic Exponents and their computation
We present a survey of the theory of the Lyapunov Characteristic Exponents
(LCEs) for dynamical systems, as well as of the numerical techniques developed
for the computation of the maximal, of few and of all of them. After some
historical notes on the first attempts for the numerical evaluation of LCEs, we
discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68},
which provides the theoretical basis for the computation of the LCEs. Then, we
analyze the algorithm for the computation of the maximal LCE, whose value has
been extensively used as an indicator of chaos, and the algorithm of the
so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for
the computation of many LCEs. We also consider different discrete and
continuous methods for computing the LCEs based on the QR or the singular value
decomposition techniques. Although, we are mainly interested in
finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems
and symplectic maps, we also briefly refer to the evaluation of LCEs of
dissipative systems and time series. The relation of two chaos detection
techniques, namely the fast Lyapunov indicator (FLI) and the generalized
alignment index (GALI), to the computation of the LCEs is also discussed.Comment: 74 pages, 8 figures, accepted for publication in Lecture Notes in
Physic
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Tools for discovering and characterizing extrasolar planets
Among the group of extrasolar planets, transiting planets provide a great
opportunity to obtain direct measurements for the basic physical properties,
such as mass and radius of these objects. These planets are therefore highly
important in the understanding of the evolution and formation of planetary
systems: from the observations of photometric transits, the interior structure
of the planet and atmospheric properties can also be constrained. The most
efficient way to search for transiting extrasolar planets is based on
wide-field surveys by hunting for short and shallow periodic dips in light
curves covering quite long time intervals. These surveys monitor fields with
several degrees in diameter and tens or hundreds of thousands of objects
simultaneously. In the practice of astronomical observations, surveys of large
field-of-view are rather new and therefore require special methods for
photometric data reduction that have not been used before. In this PhD thesis,
I summarize my efforts related to the development of a complete software
solution for high precision photometric reduction of astronomical images. I
also demonstrate the role of this newly developed package and the related
algorithms in the case of particular discoveries of the HATNet project.
[abridged]Comment: PhD thesis, Eotvos Lorand University (June 18, 2009), 68 pages in
journal style, 41 figures, 18 table
- …