3,281 research outputs found
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
We study the approximation of by a Monte Carlo algorithm,
where is the solution of a stochastic differential equation and is a
given function. We introduce a new variance reduction method, which can be
viewed as a statistical analogue of Romberg extrapolation method. Namely, we
use two Euler schemes with steps and . This
leads to an algorithm which, for a given level of the statistical error, has a
complexity significantly lower than the complexity of the standard Monte Carlo
method. We analyze the asymptotic error of this algorithm in the context of
general (possibly degenerate) diffusions. In order to find the optimal
(which turns out to be ), we establish a central limit type theorem,
based on a result of Jacod and Protter for the asymptotic distribution of the
error in the Euler scheme. We test our method on various examples. In
particular, we adapt it to Asian options. In this setting, we have a CLT and,
as a by-product, an explicit expansion of the discretization error.Comment: Published at http://dx.doi.org/10.1214/105051605000000511 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Performance Evaluation of an Extrapolation Method for Ordinary Differential Equations with Error-free Transformation
The application of error-free transformation (EFT) is recently being
developed to solve ill-conditioned problems. It can reduce the number of
arithmetic operations required, compared with multiple precision arithmetic,
and also be applied by using functions supported by a well-tuned BLAS library.
In this paper, we propose the application of EFT to explicit extrapolation
methods to solve initial value problems of ordinary differential equations.
Consequently, our implemented routines can be effective for large-sized linear
ODE and small-sized nonlinear ODE, especially in the case when harmonic
sequence is used
Accurate and efficient spin integration for particle accelerators
Accurate spin tracking is a valuable tool for understanding spin dynamics in
particle accelerators and can help improve the performance of an accelerator.
In this paper, we present a detailed discussion of the integrators in the spin
tracking code gpuSpinTrack. We have implemented orbital integrators based on
drift-kick, bend-kick, and matrix-kick splits. On top of the orbital
integrators, we have implemented various integrators for the spin motion. These
integrators use quaternions and Romberg quadratures to accelerate both the
computation and the convergence of spin rotations. We evaluate their
performance and accuracy in quantitative detail for individual elements as well
as for the entire RHIC lattice. We exploit the inherently data-parallel nature
of spin tracking to accelerate our algorithms on graphics processing units.Comment: 43 pages, 17 figure
Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
A common problem in cosmology is to integrate the product of two or more
spherical Bessel functions (sBFs) with different configuration-space arguments
against the power spectrum or its square, weighted by powers of wavenumber.
Naively computing them scales as with the number of
configuration space arguments and the grid size, and they cannot be
done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the
sBFs as sums of products of sine and cosine and then using the product to sum
identities, these integrals can then be performed using 1-D FFTs with scaling. This "rotation" method has the potential to
accelerate significantly a number of calculations in cosmology, such as
perturbation theory predictions of loop integrals, higher order correlation
functions, and analytic templates for correlation function covariance matrices.
We implement this approach numerically both in a free-standing,
publicly-available \textsc{Python} code and within the larger,
publicly-available package \texttt{mcfit}. The rotation method evaluated with
direct integrations already offers a factor of 6-10 speed-up over the
naive approach in our test cases. Using FFTs, which the rotation method
enables, then further improves this to a speed-up of
over the naive approach. The rotation method should be useful in light of
upcoming large datasets such as DESI or LSST. In analysing these datasets
recomputation of these integrals a substantial number of times, for instance to
update perturbation theory predictions or covariance matrices as the input
linear power spectrum is changed, will be one piece in a Monte Carlo Markov
Chain cosmological parameter search: thus the overall savings from our method
should be significant
A Multi-Step Richardson-Romberg Extrapolation Method For Stochastic Approximation
We obtain an expansion of the implicit weak discretization error for the
target of stochastic approximation algorithms introduced and studied in
[Frikha2013]. This allows us to extend and develop the Richardson-Romberg
extrapolation method for Monte Carlo linear estimator (introduced in [Talay &
Tubaro 1990] and deeply studied in [Pag{\`e}s 2007]) to the framework of
stochastic optimization by means of stochastic approximation algorithm. We
notably apply the method to the estimation of the quantile of diffusion
processes. Numerical results confirm the theoretical analysis and show a
significant reduction in the initial computational cost.Comment: 31 pages, 1 figur
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