417 research outputs found
Analysis of circulant embedding methods for sampling stationary random fields
In this paper we prove, under mild conditions, that the positive definiteness
of the circulant matrix appearing in the circulant embedding method is always
guaranteed, provided the enclosing cube is sufficiently large. We examine in
detail the case of the Mat\'ern covariance, and prove (for fixed correlation
length) that, as , positive definiteness is guaranteed when
the random field is sampled on a cube of size order times larger than the size of the physical domain. (Here is
the mesh spacing of the regular grid and the Mat\'ern smoothness
parameter.) We show that the sampling cube can become smaller as the
correlation length decreases when and are fixed. Our results are
confirmed by numerical experiments. We prove several results about the decay of
the eigenvalues of the circulant matrix. These lead to the conjecture, verified
by numerical experiment, that they decay with the same rate as the
Karhunen--Lo\`{e}ve eigenvalues of the covariance operator
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Smoothed Circulant Embedding with Applications to Multilevel Monte Carlo Methods for PDEs with Random Coefficients
We consider the computational efficiency of Monte Carlo (MC) and Multilevel
Monte Carlo (MLMC) methods applied to partial differential equations with
random coefficients. These arise, for example, in groundwater flow modelling,
where a commonly used model for the unknown parameter is a random field. We
make use of the circulant embedding procedure for sampling from the
aforementioned coefficient. To improve the computational complexity of the MLMC
estimator in the case of highly oscillatory random fields, we devise and
implement a smoothing technique integrated into the circulant embedding method.
This allows to choose the coarsest mesh on the first level of MLMC
independently of the correlation length of the covariance function of the
random field, leading to considerable savings in computational cost. We
illustrate this with numerical experiments, where we see a saving of factor
5-10 in computational cost for accuracies of practical interest.Comment: 33 pages, 10 figures, submitted to IMA Journal of Numerical Analysi
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