1,810 research outputs found
Modified quaternion Newton methods
We revisit the quaternion Newton method for computing roots of a class of quaternion valued functions and propose modified algorithms for finding multiple roots of simple polynomials. We illustrate the performance of these new methods by presenting several numerical experiments.The research was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the " Fundcao para a Ciencia e a Tecnologia", through the Project PEstOE/ MAT/ UI0013/ 2014
On the numerical integration of motion for rigid polyatomics: The modified quaternion approach
A revised version of the quaternion approach for numerical integration of the
equations of motion for rigid polyatomic molecules is proposed. The modified
approach is based on a formulation of the quaternion dynamics with constraints.
This allows to resolve the rigidity problem rigorously using constraint forces.
It is shown that the procedure for preservation of molecular rigidity can be
realized particularly simply within the Verlet algorithm in velocity form. We
demonstrate that the presented method leads to an improved numerical stability
with respect to the usual quaternion rescaling scheme and it is roughly as good
as the cumbersome atomic-constraint technique.Comment: 14 pages, 2 figure
Methods for suspensions of passive and active filaments
Flexible filaments and fibres are essential components of important complex
fluids that appear in many biological and industrial settings. Direct
simulations of these systems that capture the motion and deformation of many
immersed filaments in suspension remain a formidable computational challenge
due to the complex, coupled fluid--structure interactions of all filaments, the
numerical stiffness associated with filament bending, and the various
constraints that must be maintained as the filaments deform. In this paper, we
address these challenges by describing filament kinematics using quaternions to
resolve both bending and twisting, applying implicit time-integration to
alleviate numerical stiffness, and using quasi-Newton methods to obtain
solutions to the resulting system of nonlinear equations. In particular, we
employ geometric time integration to ensure that the quaternions remain unit as
the filaments move. We also show that our framework can be used with a variety
of models and methods, including matrix-free fast methods, that resolve low
Reynolds number hydrodynamic interactions. We provide a series of tests and
example simulations to demonstrate the performance and possible applications of
our method. Finally, we provide a link to a MATLAB/Octave implementation of our
framework that can be used to learn more about our approach and as a tool for
filament simulation
Towards local-global compatibility for Hilbert modular forms of low weight
We prove some new cases of local--global compatibility for the Galois
representations associated to Hilbert modular forms of low weight (that is,
partial weight one).Comment: 14 page
Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form
of Thurston's Virtual Fibration Conjecture. In particular, this manifold has
finite covers which fiber over the circle in arbitrarily many ways. More
precisely, it has a tower of finite covers where the number of fibered faces of
the Thurston norm ball goes to infinity, in fact faster than any power of the
logarithm of the degree of the cover, and we give a more precise quantitative
lower bound. The example manifold M is arithmetic, and the proof uses detailed
number-theoretic information, at the level of the Hecke eigenvalues, to drive a
geometric argument based on Fried's dynamical characterization of the fibered
faces. The origin of the basic fibration of M over the circle is the modular
elliptic curve E=X_0(49), which admits multiplication by the ring of integers
of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a
cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion
algebra D/K ramified only at the primes above 7; the fundamental group of M is
a quotient of the principal congruence subgroup of level 7 of the
multiplicative group of a maximal order of D. To analyze the topological
properties of M, we use a new practical method for computing the Thurston norm,
which is of independent interest. We also give a non-compact finite-volume
hyperbolic 3-manifold with the same properties by using a direct topological
argument.Comment: 42 pages, 7 figures; V2: minor improvements, to appear in Amer. J.
Mat
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
A symplectic method for rigid-body molecular simulation
This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/107/7/10.1063/1.474596.Rigid-body molecular dynamics simulations typically are performed in a quaternion representation. The nonseparable form of the Hamiltonian in quaternions prevents the use of a standard leapfrog (Verlet) integrator, so nonsymplectic Runge–Kutta, multistep, or extrapolation methods are generally used. This is unfortunate since symplectic methods like Verlet exhibit superior energy conservation in long-time integrations. In this article, we describe an alternative method, which we call RSHAKE (for rotation-SHAKE), in which the entire rotation matrix is evolved (using the scheme of McLachlan and Scovel [J. Nonlin. Sci. 16 233 (1995)]) in tandem with the particle positions. We employ a fast approximate Newton solver to preserve the orthogonality of the rotation matrix. We test our method on a system of soft-sphere dipoles and compare with quaternion evolution using a 4th-order predictor–corrector integrator. Although the short-time error of the quaternion algorithm is smaller for fixed time step than that for RSHAKE, the quaternion scheme exhibits an energy drift which is not observed in simulations with RSHAKE, hence a fixed energy tolerance can be achieved by using a larger time step. The superiority of RSHAKE increases with system size
Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC
We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm
for initializing pose graph optimization problems, arising in various scenarios
such as SFM (structure from motion) or SLAM (simultaneous localization and
mapping). TG-MCMC is first of its kind as it unites asymptotically global
non-convex optimization on the spherical manifold of quaternions with posterior
sampling, in order to provide both reliable initial poses and uncertainty
estimates that are informative about the quality of individual solutions. We
devise rigorous theoretical convergence guarantees for our method and
extensively evaluate it on synthetic and real benchmark datasets. Besides its
elegance in formulation and theory, we show that our method is robust to
missing data, noise and the estimated uncertainties capture intuitive
properties of the data.Comment: Published at NeurIPS 2018, 25 pages with supplement
- …