6 research outputs found
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis
Random eigenvalue problems are useful models for quantifying the uncertainty
in several applications from the physical sciences and engineering, e.g.,
structural vibration analysis, the criticality of a nuclear reactor or photonic
crystal structures. In this paper we present a simple multilevel quasi-Monte
Carlo (MLQMC) method for approximating the expectation of the minimal
eigenvalue of an elliptic eigenvalue problem with coefficients that are given
as a series expansion of countably-many stochastic parameters. The MLQMC
algorithm is based on a hierarchy of discretisations of the spatial domain and
truncations of the dimension of the stochastic parameter domain. To approximate
the expectations, randomly shifted lattice rules are employed. This paper is
primarily dedicated to giving a rigorous analysis of the error of this
algorithm. A key step in the error analysis requires bounds on the mixed
derivatives of the eigenfunction with respect to both the stochastic and
spatial variables simultaneously. An accompanying paper [Gilbert and Scheichl,
2021], focusses on practical extensions of the MLQMC algorithm to improve
efficiency, and presents numerical results
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results
Stochastic PDE eigenvalue problems often arise in the field of uncertainty
quantification, whereby one seeks to quantify the uncertainty in an eigenvalue,
or its eigenfunction. In this paper we present an efficient multilevel
quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the
smallest eigenvalue of an elliptic eigenvalue problem with stochastic
coefficients. Each sample evaluation requires the solution of a PDE eigenvalue
problem, and so tackling this problem in practice is notoriously
computationally difficult. We speed up the approximation of this expectation in
four ways: 1) we use a multilevel variance reduction scheme to spread the work
over a hierarchy of FE meshes and truncation dimensions; 2) we use QMC methods
to efficiently compute the expectations on each level; 3) we exploit the
smoothness in parameter space and reuse the eigenvector from a nearby QMC point
to reduce the number of iterations of the eigensolver; and 4) we utilise a
two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a
single linear solve. The full error analysis of a basic MLQMC algorithm is
given in the companion paper [Gilbert and Scheichl, 2021], and so in this paper
we focus on how to further improve the efficiency and provide theoretical
justification of the enhancement strategies 3) and 4). Numerical results are
presented that show the efficiency of our algorithm, and also show that the
four strategies we employ are complementary
Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
International audienceno abstrac
Computational multiscale solvers for continuum approaches
Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper