14,417 research outputs found
Another approach to Runge-Kutta methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct optimization, avoiding the necessity to introduce simplifying assumptions upon the Runge-Kutta coefficients. More favourable coefficients, in view of rounding errors, are found
A Variational Assimilation Method for Satellite and Conventional Data: a Revised Basic Model 2B
A variational objective analysis technique that modifies observations of temperature, height, and wind on the cyclone scale to satisfy the five 'primitive' model forecast equations is presented. This analysis method overcomes all of the problems that hindered previous versions, such as over-determination, time consistency, solution method, and constraint decoupling. A preliminary evaluation of the method shows that it converges rapidly, the divergent part of the wind is strongly coupled in the solution, fields of height and temperature are well-preserved, and derivative quantities such as vorticity and divergence are improved. Problem areas are systematic increases in the horizontal velocity components, and large magnitudes of the local tendencies of the horizontal velocity components. The preliminary evaluation makes note of these problems but detailed evaluations required to determine the origin of these problems await future research
Uncertainty Principles and Vector Quantization
Given a frame in C^n which satisfies a form of the uncertainty principle (as
introduced by Candes and Tao), it is shown how to quickly convert the frame
representation of every vector into a more robust Kashin's representation whose
coefficients all have the smallest possible dynamic range O(1/\sqrt{n}). The
information tends to spread evenly among these coefficients. As a consequence,
Kashin's representations have a great power for reduction of errors in their
coefficients, including coefficient losses and distortions.Comment: Final version, to appear in IEEE Trans. Information Theory.
Introduction updated, minor inaccuracies corrected
A Short Tale of Long Tail Integration
Integration of the form , where is either
or , is widely
encountered in many engineering and scientific applications, such as those
involving Fourier or Laplace transforms. Often such integrals are approximated
by a numerical integration over a finite domain , leaving a truncation
error equal to the tail integration in addition
to the discretization error. This paper describes a very simple, perhaps the
simplest, end-point correction to approximate the tail integration, which
significantly reduces the truncation error and thus increases the overall
accuracy of the numerical integration, with virtually no extra computational
effort. Higher order correction terms and error estimates for the end-point
correction formula are also derived. The effectiveness of this one-point
correction formula is demonstrated through several examples
Spectral Ewald Acceleration of Stokesian Dynamics for polydisperse suspensions
In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics
(SEASD), a novel computational method for dynamic simulations of polydisperse
colloidal suspensions with full hydrodynamic interactions. SEASD is based on
the framework of Stokesian Dynamics (SD) with extension to compressible
solvents, and uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J.
Comput. Phys. 229 (2010) 8994] for the wave-space mobility computation. To meet
the performance requirement of dynamic simulations, we use Graphic Processing
Units (GPU) to evaluate the suspension mobility, and achieve an order of
magnitude speedup compared to a CPU implementation. For further speedup, we
develop a novel far-field block-diagonal preconditioner to reduce the far-field
evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of
the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118
(2003) 10323]. We extensively discuss implementation and parameter selection
strategies in SEASD, and demonstrate the spectral accuracy in the mobility
evaluation and the overall computation scaling. We
present three computational examples to further validate SEASD and SEASD-nf in
monodisperse and bidisperse suspensions: the short-time transport properties,
the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear
Brownian rheology. Our validation results show that the agreement between SEASD
and SEASD-nf is satisfactory over a wide range of parameters, and also provide
significant insight into the dynamics of polydisperse colloidal suspensions.Comment: 39 pages, 21 figure
Computing Tails of Compound Distributions Using Direct Numerical Integration
An efficient adaptive direct numerical integration (DNI) algorithm is
developed for computing high quantiles and conditional Value at Risk (CVaR) of
compound distributions using characteristic functions. A key innovation of the
numerical scheme is an effective tail integration approximation that reduces
the truncation errors significantly with little extra effort. High precision
results of the 0.999 quantile and CVaR were obtained for compound losses with
heavy tails and a very wide range of loss frequencies using the DNI, Fast
Fourier Transform (FFT) and Monte Carlo (MC) methods. These results,
particularly relevant to operational risk modelling, can serve as benchmarks
for comparing different numerical methods. We found that the adaptive DNI can
achieve high accuracy with relatively coarse grids. It is much faster than MC
and competitive with FFT in computing high quantiles and CVaR of compound
distributions in the case of moderate to high frequencies and heavy tails
- ā¦