15,694 research outputs found
Stochastic partial differential equation based modelling of large space-time data sets
Increasingly larger data sets of processes in space and time ask for
statistical models and methods that can cope with such data. We show that the
solution of a stochastic advection-diffusion partial differential equation
provides a flexible model class for spatio-temporal processes which is
computationally feasible also for large data sets. The Gaussian process defined
through the stochastic partial differential equation has in general a
nonseparable covariance structure. Furthermore, its parameters can be
physically interpreted as explicitly modeling phenomena such as transport and
diffusion that occur in many natural processes in diverse fields ranging from
environmental sciences to ecology. In order to obtain computationally efficient
statistical algorithms we use spectral methods to solve the stochastic partial
differential equation. This has the advantage that approximation errors do not
accumulate over time, and that in the spectral space the computational cost
grows linearly with the dimension, the total computational costs of Bayesian or
frequentist inference being dominated by the fast Fourier transform. The
proposed model is applied to postprocessing of precipitation forecasts from a
numerical weather prediction model for northern Switzerland. In contrast to the
raw forecasts from the numerical model, the postprocessed forecasts are
calibrated and quantify prediction uncertainty. Moreover, they outperform the
raw forecasts, in the sense that they have a lower mean absolute error
PoisFFT - A Free Parallel Fast Poisson Solver
A fast Poisson solver software package PoisFFT is presented. It is available
as a free software licensed under the GNU GPL license version 3. The package
uses the fast Fourier transform to directly solve the Poisson equation on a
uniform orthogonal grid. It can solve the pseudo-spectral approximation and the
second order finite difference approximation of the continuous solution. The
paper reviews the mathematical methods for the fast Poisson solver and
discusses the software implementation and parallelization. The use of PoisFFT
in an incompressible flow solver is also demonstrated
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