3 research outputs found
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
Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs
By combining a certain approximation property in the spatial domain, and
weighted -summability of the Hermite polynomial expansion coefficients
in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G.
Migliorati, ESAIM Math. Model. Numer. Anal. (2017), 341-363] and [M.
Bachmayr, A. Cohen, D. D\~ung and C. Schwab, SIAM J. Numer. Anal. (2017), 2151-2186], we investigate linear non-adaptive methods of fully
discrete polynomial interpolation approximation as well as fully discrete
weighted quadrature methods of integration for parametric and stochastic
elliptic PDEs with lognormal inputs. We explicitly construct such methods and
prove corresponding convergence rates in of the approximations by them,
where is a number characterizing computation complexity. The linear
non-adaptive methods of fully discrete polynomial interpolation approximation
are sparse-grid collocation methods. Moreover, they generate in a natural way
discrete weighted quadrature formulas for integration of the solution to
parametric and stochastic elliptic PDEs and its linear functionals, and the
error of the corresponding integration can be estimated via the error in the
Bochner space norm of the generating methods
where is the Gaussian probability measure on and
is the energy space. We also briefly consider similar problems for
parametric and stochastic elliptic PDEs with affine inputs, and by-product
problems of non-fully discrete polynomial interpolation approximation and
integration. In particular, the convergence rate of non-fully discrete obtained
in this paper improves the known one
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