8,804 research outputs found
Efficient high order algorithms for fractional integrals and fractional differential equations
We propose an efficient algorithm for the approximation of fractional
integrals by using Runge--Kutta based convolution quadrature. The algorithm is
based on a novel integral representation of the convolution weights and a
special quadrature for it. The resulting method is easy to implement, allows
for high order, relies on rigorous error estimates and its performance in terms
of memory and computational cost is among the best to date. Several numerical
results illustrate the method and we describe how to apply the new algorithm to
solve fractional diffusion equations. For a class of fractional diffusion
equations we give the error analysis of the full space-time discretization
obtained by coupling the FEM method in space with Runge--Kutta based
convolution quadrature in time
The numerical solution of fractional differential equations: Speed versus accuracy
This paper discusses the development of efficient algorithms for a certain fractional differential equation.Manchester Centre for Computational Mathematic
Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion
We consider continuous-time diffusion models driven by fractional Brownian
motion. Observations are assumed to possess a non-trivial likelihood given the
latent path. Due to the non-Markovianity and high-dimensionality of the latent
paths, estimating posterior expectations is a computationally challenging
undertaking. We present a reparameterization framework based on the Davies and
Harte method for sampling stationary Gaussian processes and use this framework
to construct a Markov chain Monte Carlo algorithm that allows computationally
efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based
on a version of hybrid Monte Carlo that delivers increased efficiency when
applied on the high-dimensional latent variables arising in this context. We
specify the methodology on a stochastic volatility model allowing for memory in
the volatility increments through a fractional specification. The methodology
is illustrated on simulated data and on the S&P500/VIX time series and is shown
to be effective. Contrary to a long range dependence attribute of such models
often assumed in the literature, with Hurst parameter larger than 1/2, the
posterior distribution favours values smaller than 1/2, pointing towards medium
range dependence
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