158 research outputs found
Some variance reduction methods for numerical stochastic homogenization
We overview a series of recent works devoted to variance reduction techniques
for numerical stochastic homogenization. Numerical homogenization requires
solving a set of problems at the micro scale, the so-called corrector problems.
In a random environment, these problems are stochastic and therefore need to be
repeatedly solved, for several configurations of the medium considered. An
empirical average over all configurations is then performed using the
Monte-Carlo approach, so as to approximate the effective coefficients necessary
to determine the macroscopic behavior. Variance severely affects the accuracy
and the cost of such computations. Variance reduction approaches, borrowed from
other contexts of the engineering sciences, can be useful. Some of these
variance reduction techniques are presented, studied and tested here
Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media
This paper proposes to address the issue of complexity reduction for the
numerical simulation of multiscale media in a quasi-periodic setting. We
consider a stationary elliptic diffusion equation defined on a domain such
that is the union of cells and we
introduce a two-scale representation by identifying any function defined
on with a bi-variate function , where relates to the
index of the cell containing the point and relates to a local
coordinate in a reference cell . We introduce a weak formulation of the
problem in a broken Sobolev space using a discontinuous Galerkin
framework. The problem is then interpreted as a tensor-structured equation by
identifying with a tensor product space of
functions defined over the product set . Tensor numerical methods
are then used in order to exploit approximability properties of quasi-periodic
solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to
regularity) and the argumentation thereof. Corrected proof of proposition 3.
Corrected wrong multiplicative factor in proposition 4 and its proof (was 2
instead of 1). Added remark 6 at the end of section 2. Extended remark 7.
Added references. Some minor improvements (typos, typesetting
Special Quasirandom Structures: a selection approach for stochastic homogenization
We adapt and study a variance reduction approach for the homogenization of
elliptic equations in divergence form. The approach, borrowed from atomistic
simulations and solid-state science [von Pezold et al, Physical Review B 2010;
Wei et al, Physical Review B 1990; Zunger et al, Physical Review Letters 1990],
consists in selecting random realizations that best satisfy some statistical
properties (such as the volume fraction of each phase in a composite material)
usually only obtained asymptotically.
We study the approach theoretically in some simplified settings
(one-dimensional setting, perturbative setting in higher dimensions), and
numerically demonstrate its efficiency in more general cases
Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization
The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities
The choice of representative volumes in the approximation of effective properties of random materials
The effective large-scale properties of materials with random heterogeneities
on a small scale are typically determined by the method of representative
volumes: A sample of the random material is chosen - the representative volume
- and its effective properties are computed by the cell formula. Intuitively,
for a fixed sample size it should be possible to increase the accuracy of the
method by choosing a material sample which captures the statistical properties
of the material particularly well: For example, for a composite material
consisting of two constituents, one would select a representative volume in
which the volume fraction of the constituents matches closely with their volume
fraction in the overall material. Inspired by similar attempts in material
science, Le Bris, Legoll, and Minvielle have designed a selection approach for
representative volumes which performs remarkably well in numerical examples of
linear materials with moderate contrast. In the present work, we provide a
rigorous analysis of this selection approach for representative volumes in the
context of stochastic homogenization of linear elliptic equations. In
particular, we prove that the method essentially never performs worse than a
random selection of the material sample and may perform much better if the
selection criterion for the material samples is chosen suitably.Comment: 84 page
- …