6,026 research outputs found
Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM
Elliptic boundary value problems which are posed on a random domain can be
mapped to a fixed, nominal domain. The randomness is thus transferred to the
diffusion matrix and the loading. While this domain mapping method is quite
efficient for theory and practice, since only a single domain discretisation is
needed, it also requires the knowledge of the domain mapping.
However, in certain applications, the random domain is only described by its
random boundary, while the quantity of interest is defined on a fixed,
deterministic subdomain. In this setting, it thus becomes necessary to compute
a random domain mapping on the whole domain, such that the domain mapping is
the identity on the fixed subdomain and maps the boundary of the chosen fixed,
nominal domain on to the random boundary.
To overcome the necessity of computing such a mapping, we therefore couple
the finite element method on the fixed subdomain with the boundary element
method on the random boundary. We verify the required regularity of the
solution with respect to the random domain mapping for the use of multilevel
quadrature, derive the coupling formulation, and show by numerical results that
the approach is feasible
Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial
This article provides a high-level overview of some recent works on the
application of quasi-Monte Carlo (QMC) methods to PDEs with random
coefficients. It is based on an in-depth survey of a similar title by the same
authors, with an accompanying software package which is also briefly discussed
here. Embedded in this article is a step-by-step tutorial of the required
analysis for the setting known as the uniform case with first order QMC rules.
The aim of this article is to provide an easy entry point for QMC experts
wanting to start research in this direction and for PDE analysts and
practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661
Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems
In this paper we present a rigorous cost and error analysis of a multilevel
estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for
lognormal diffusion problems. These problems are motivated by uncertainty
quantification problems in subsurface flow. We extend the convergence analysis
in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite
element discretizations and give a constructive proof of the
dimension-independent convergence of the QMC rules. More precisely, we provide
suitable parameters for the construction of such rules that yield the required
variance reduction for the multilevel scheme to achieve an -error
with a cost of with , and in
practice even , for sufficiently fast decaying covariance
kernels of the underlying Gaussian random field inputs. This confirms that the
computational gains due to the application of multilevel sampling methods and
the gains due to the application of QMC methods, both demonstrated in earlier
works for the same model problem, are complementary. A series of numerical
experiments confirms these gains. The results show that in practice the
multilevel QMC method consistently outperforms both the multilevel MC method
and the single-level variants even for non-smooth problems.Comment: 32 page
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