731 research outputs found
Fifty Years of Stiffness
The notion of stiffness, which originated in several applications of a
different nature, has dominated the activities related to the numerical
treatment of differential problems for the last fifty years. Contrary to what
usually happens in Mathematics, its definition has been, for a long time, not
formally precise (actually, there are too many of them). Again, the needs of
applications, especially those arising in the construction of robust and
general purpose codes, require nowadays a formally precise definition. In this
paper, we review the evolution of such a notion and we also provide a precise
definition which encompasses all the previous ones.Comment: 24 pages, 11 figure
Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes.
The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory.
At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms
Black resonators and geons in AdS
We construct dynamical black hole solutions with a helical symmetry in
AdS, called black resonators, as well as their horizonless limits, called
geons. We introduce a cohomogeneity-1 metric describing a class of black
resonators and geons whose isometry group is . This allows us to
study them in a wide range of parameters. We obtain the phase diagram for the
black resonators, geons, and Myers-Perry-AdS, where the black resonators
emerge from the onset of a superradiant instability of the Myers-Perry-AdS
with equal angular momenta and are connected to the geons in the small horizon
limit. The angular velocities of the black resonators always satisfy
in units of the AdS radius. A black resonator is shown to have higher entropy
than a Myers-Perry-AdS black hole with the same asymptotic charges. This
implies that the Myers-Perry-AdS can dynamically evolve into the black
resonator under the exact -symmetry although its endpoint will be
further unstable to -violating perturbations.Comment: 27 pages, 9 figure
Frequency-domain algorithm for the Lorenz-gauge gravitational self-force
State-of-the-art computations of the gravitational self-force (GSF) on
massive particles in black hole spacetimes involve numerical evolution of the
metric perturbation equations in the time-domain, which is computationally very
costly. We present here a new strategy, based on a frequency-domain treatment
of the perturbation equations, which offers considerable computational saving.
The essential ingredients of our method are (i) a Fourier-harmonic
decomposition of the Lorenz-gauge metric perturbation equations and a numerical
solution of the resulting coupled set of ordinary equations with suitable
boundary conditions; (ii) a generalized version of the method of extended
homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)] used to circumvent
the Gibbs phenomenon that would otherwise hamper the convergence of the Fourier
mode-sum at the particle's location; and (iii) standard mode-sum
regularization, which finally yields the physical GSF as a sum over regularized
modal contributions. We present a working code that implements this strategy to
calculate the Lorenz-gauge GSF along eccentric geodesic orbits around a
Schwarzschild black hole. The code is far more efficient than existing
time-domain methods; the gain in computation speed (at a given precision) is
about an order of magnitude at an eccentricity of 0.2, and up to three orders
of magnitude for circular or nearly circular orbits. This increased efficiency
was crucial in enabling the recently reported calculation of the long-term
orbital evolution of an extreme mass ratio inspiral [Phys. Rev. D {\bf 85},
061501(R) (2012)]. Here we provide full technical details of our method to
complement the above report.Comment: 27 pages, 4 figure
Fast derivatives of likelihood functionals for ODE based models using adjoint-state method
We consider time series data modeled by ordinary differential equations
(ODEs), widespread models in physics, chemistry, biology and science in
general. The sensitivity analysis of such dynamical systems usually requires
calculation of various derivatives with respect to the model parameters.
We employ the adjoint state method (ASM) for efficient computation of the
first and the second derivatives of likelihood functionals constrained by ODEs
with respect to the parameters of the underlying ODE model. Essentially, the
gradient can be computed with a cost (measured by model evaluations) that is
independent of the number of the ODE model parameters and the Hessian with a
linear cost in the number of the parameters instead of the quadratic one. The
sensitivity analysis becomes feasible even if the parametric space is
high-dimensional.
The main contributions are derivation and rigorous analysis of the ASM in the
statistical context, when the discrete data are coupled with the continuous ODE
model. Further, we present a highly optimized implementation of the results and
its benchmarks on a number of problems.
The results are directly applicable in (e.g.) maximum-likelihood estimation
or Bayesian sampling of ODE based statistical models, allowing for faster, more
stable estimation of parameters of the underlying ODE model.Comment: 5 figure
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