426 research outputs found
Implementation of two-step Runge-Kutta methods for ordinary differential equations
AbstractWe investigate the potential for efficient implementation of two-step Runge-Kutta methods (TSRK), a new class of methods introduced recently by Jackiewicz and Tracogna for numerical integration of ordinary differential equations. The implementation issues addressed are the local error estimation, changing stepsize using Nordsieck technique and construction of interpolants. The numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment
Performance Improvements for Nuclear Reaction Network Integration
Aims: The aim of this work is to compare the performance of three reaction
network integration methods used in stellar nucleosynthesis calculations. These
are the Gear's backward differentiation method, Wagoner's method (a 2nd-order
Runge-Kutta method), and the Bader-Deuflehard semi-implicit multi-step method.
Methods: To investigate the efficiency of each of the integration methods
considered here, a test suite of temperature and density versus time profiles
is used. This suite provides a range of situations ranging from constant
temperature and density to the dramatically varying conditions present in white
dwarf mergers, novae, and x-ray bursts. Some of these profiles are obtained
separately from full hydrodynamic calculations. The integration efficiencies
are investigated with respect to input parameters that constrain the desired
accuracy and precision.
Results: Gear's backward differentiation method is found to improve accuracy,
performance, and stability in integrating nuclear reaction networks. For
temperature-density profiles that vary strongly with time, it is found to
outperform the Bader-Deuflehard method (although that method is very powerful
for more smoothly varying profiles). Wagoner's method, while relatively fast
for many scenarios, exhibits hard-to-predict inaccuracies for some choices of
integration parameters owing to its lack of error estimations.Comment: 13 pages, 12 figures, accepted to Astronomy and Astrophysics (section
15) - corrected units in Figs. 6-1
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
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
Calibrated Adaptive Probabilistic ODE Solvers
Probabilistic solvers for ordinary differential equations assign a posterior
measure to the solution of an initial value problem. The joint covariance of
this distribution provides an estimate of the (global) approximation error. The
contraction rate of this error estimate as a function of the solver's step size
identifies it as a well-calibrated worst-case error, but its explicit numerical
value for a certain step size is not automatically a good estimate of the
explicit error. Addressing this issue, we introduce, discuss, and assess
several probabilistically motivated ways to calibrate the uncertainty estimate.
Numerical experiments demonstrate that these calibration methods interact
efficiently with adaptive step-size selection, resulting in descriptive, and
efficiently computable posteriors. We demonstrate the efficiency of the
methodology by benchmarking against the classic, widely used Dormand-Prince 4/5
Runge-Kutta method.Comment: 17 pages, 10 figures
Spitzer 24 micron Survey of Debris Disks in the Pleiades
We performed a 24 micron 2 Deg X 1 Deg survey of the Pleiades cluster, using
the MIPS instrument on Spitzer. Fifty four members ranging in spectral type
from B8 to K6 show 24 micron fluxes consistent with bare photospheres. All Be
stars show excesses attributed to free-free emission in their gaseous
envelopes. Five early-type stars and four solar-type stars show excesses
indicative of debris disks. We find a debris disk fraction of 25 % for B-A
members and 10 % for F-K3 ones. These fractions appear intermediate between
those for younger clusters and for the older field stars. They indicate a decay
with age of the frequency of the dust-production events inside the planetary
zone, with similar time scales for solar-mass stars as have been found
previously for A-stars.Comment: accepted to Ap
Economic MPC with Modifier Adaptation using Transient Measurements
Producción CientíficaThis paper presents a method to estimate process dynamic gradients along the transient that combined with the idea of Modifier Adaptation (MA) improves the economic cost fuction of the examples presented. The gradient estimation method, called TMA, aims to reduce the large convergence time required to traditional MA in processes of slow dynamics. TMA is used with an economic predictive control with MA (eMPC+TMA) and was applied in two case studies: a simulation of the Williams-Otto reactor and a hybrid laboratory plant based on the Van de Vusse reactor. The results show that eMPC+TMA could reach the plant real steady-state optimum despite process-model mismatch, due to the inclusion of the effect of process dynamics in the TMA algorithm. Despite the estimation errors, the proposed methodology improved the profit of the experimental case study, with respect to the use of an eMPC with no modifiers, by about 20% for the unconstrained case, and by 130% in the constrained case.Junta de Castilla y León (CLU 2017-09 and UIC 233)FEDER - AEI (PGC2018-099312-B-C31
- …