298 research outputs found
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
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
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
Stochastic Galerkin methods for non-affine coefficient representations are
known to cause major difficulties from theoretical and numerical points of
view. In this work, an adaptive Galerkin FE method for linear parametric PDEs
with lognormal coefficients discretized in Hermite chaos polynomials is
derived. It employs problem-adapted function spaces to ensure solvability of
the variational formulation. The inherently high computational complexity of
the parametric operator is made tractable by using hierarchical tensor
representations. For this, a new tensor train format of the lognormal
coefficient is derived and verified numerically. The central novelty is the
derivation of a reliable residual-based a posteriori error estimator. This can
be regarded as a unique feature of stochastic Galerkin methods. It allows for
an adaptive algorithm to steer the refinements of the physical mesh and the
anisotropic Wiener chaos polynomial degrees. For the evaluation of the error
estimator to become feasible, a numerically efficient tensor format
discretization is developed. Benchmark examples with unbounded lognormal
coefficient fields illustrate the performance of the proposed Galerkin
discretization and the fully adaptive algorithm
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
MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM
We introduce the multivariate decomposition finite element method for
elliptic PDEs with lognormal diffusion coefficient where is a
Gaussian random field defined by an infinite series expansion
with and a given sequence of functions . We
use the MDFEM to approximate the expected value of a linear functional of the
solution of the PDE which is an infinite-dimensional integral over the
parameter space. The proposed algorithm uses the multivariate decomposition
method to compute the infinite-dimensional integral by a decomposition into
finite-dimensional integrals, which we resolve using quasi-Monte Carlo methods,
and for which we use the finite element method to solve different instances of
the PDE.
We develop higher-order quasi-Monte Carlo rules for integration over the
finite-dimensional Euclidean space with respect to the Gaussian distribution by
use of a truncation strategy. By linear transformations of interlaced
polynomial lattice rules from the unit cube to a multivariate box of the
Euclidean space we achieve higher-order convergence rates for functions
belonging to a class of anchored Gaussian Sobolev spaces while taking into
account the truncation error.
Under appropriate conditions, the MDFEM achieves higher-order convergence
rates in term of error versus cost, i.e., to achieve an accuracy of
the computational cost is where and
are respectively the cost of the quasi-Monte Carlo
cubature and the finite element approximations, with
for some and the physical dimension, and is a parameter representing the sparsity of .Comment: 48 page
- …