7,991 research outputs found
Guidance, flight mechanics and trajectory optimization. Volume 6 - The N-body problem and special perturbation techniques
Analytical formulations and numerical integration methods for many body problem and special perturbative technique
Inference for Differential Equation Models using Relaxation via Dynamical Systems
Statistical regression models whose mean functions are represented by
ordinary differential equations (ODEs) can be used to describe phenomenons
dynamical in nature, which are abundant in areas such as biology, climatology
and genetics. The estimation of parameters of ODE based models is essential for
understanding its dynamics, but the lack of an analytical solution of the ODE
makes the parameter estimation challenging. The aim of this paper is to propose
a general and fast framework of statistical inference for ODE based models by
relaxation of the underlying ODE system. Relaxation is achieved by a properly
chosen numerical procedure, such as the Runge-Kutta, and by introducing
additive Gaussian noises with small variances. Consequently, filtering methods
can be applied to obtain the posterior distribution of the parameters in the
Bayesian framework. The main advantage of the proposed method is computation
speed. In a simulation study, the proposed method was at least 14 times faster
than the other methods. Theoretical results which guarantee the convergence of
the posterior of the approximated dynamical system to the posterior of true
model are presented. Explicit expressions are given that relate the order and
the mesh size of the Runge-Kutta procedure to the rate of convergence of the
approximated posterior as a function of sample size
Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems
In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of
arbitrarily high order of accuracy are introduced in closed form. The stability
domain of RKG polynomials extends in the the real direction with the square of
polynomial degree, and in the imaginary direction as an increasing function of
Gegenbauer parameter. Consequently, the polynomials are naturally suited to the
construction of high order stabilized Runge-Kutta (SRK) explicit methods for
systems of PDEs of mixed hyperbolic-parabolic type.
We present SRK methods composed of ordered forward Euler stages, with
complex-valued stepsizes derived from the roots of RKG stability polynomials of
degree . Internal stability is maintained at large stage number through an
ordering algorithm which limits internal amplification factors to .
Test results for mildly stiff nonlinear advection-diffusion-reaction problems
with moderate () mesh P\'eclet numbers are provided at second,
fourth, and sixth orders, with nonlinear reaction terms treated by complex
splitting techniques above second order.Comment: 20 pages, 7 figures, 3 table
Efficient implementation of Radau collocation methods
In this paper we define an efficient implementation of Runge-Kutta methods of
Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems.
The proposed implementation relies on an alternative low-rank formulation of
the methods, for which a splitting procedure is easily defined. The linear
convergence analysis of this splitting procedure exhibits excellent properties,
which are confirmed by its performance on a few numerical tests.Comment: 19 pages, 3 figures, 9 table
Probabilistic Numerics and Uncertainty in Computations
We deliver a call to arms for probabilistic numerical methods: algorithms for
numerical tasks, including linear algebra, integration, optimization and
solving differential equations, that return uncertainties in their
calculations. Such uncertainties, arising from the loss of precision induced by
numerical calculation with limited time or hardware, are important for much
contemporary science and industry. Within applications such as climate science
and astrophysics, the need to make decisions on the basis of computations with
large and complex data has led to a renewed focus on the management of
numerical uncertainty. We describe how several seminal classic numerical
methods can be interpreted naturally as probabilistic inference. We then show
that the probabilistic view suggests new algorithms that can flexibly be
adapted to suit application specifics, while delivering improved empirical
performance. We provide concrete illustrations of the benefits of probabilistic
numeric algorithms on real scientific problems from astrometry and astronomical
imaging, while highlighting open problems with these new algorithms. Finally,
we describe how probabilistic numerical methods provide a coherent framework
for identifying the uncertainty in calculations performed with a combination of
numerical algorithms (e.g. both numerical optimisers and differential equation
solvers), potentially allowing the diagnosis (and control) of error sources in
computations.Comment: Author Generated Postprint. 17 pages, 4 Figures, 1 Tabl
BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams
We present the program BOKASUN for fast and precise evaluation of the Master
Integrals of the two-loop self-mass sunrise diagram for arbitrary values of the
internal masses and the external four-momentum. We use a combination of two
methods: a Bernoulli accelerated series expansion and a Runge-Kutta numerical
solution of a system of linear differential equations
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