---
OA_place: publisher
OA_type: hybrid
_id: '13129'
abstract:
- lang: eng
  text: "We study the representative volume element (RVE) method, which is a method
    to approximately infer the effective behavior ahom of a stationary random medium.
    The latter is described by a coefficient field a(x) generated from a given ensemble
    ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory
    of homogenization, the method proceeds by computing d=3 correctors (d denoting
    the space dimension). To be numerically tractable, this computation has to be
    done on a finite domain: the so-called representative volume element, i.e., a
    large box with, say, periodic boundary conditions. The main message of this article
    is: Periodize the ensemble instead of its realizations. By this, we mean that
    it is better to sample from a suitably periodized ensemble than to periodically
    extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩.
    We make this point by investigating the bias (or systematic error), i.e., the
    difference between ahom and the expected value of the RVE method, in terms of
    its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x),
    we heuristically argue that this error is generically O(L−1). In case of a suitable
    periodization of ⟨⋅⟩\r\n, we rigorously show that it is O(L−d). In fact, we give
    a characterization of the leading-order error term for both strategies and argue
    that even in the isotropic case it is generically non-degenerate. We carry out
    the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩\r\n of Gaussian
    type, which allow for a straightforward periodization, passing via the (integrable)
    covariance function. This setting has also the advantage of making the Price theorem
    and the Malliavin calculus available for optimal stochastic estimates of correctors.
    We actually need control of second-order correctors to capture the leading-order
    error term. This is due to inversion symmetry when applying the two-scale expansion
    to the Green function. As a bonus, we present a stream-lined strategy to estimate
    the error in a higher-order two-scale expansion of the Green function."
acknowledgement: Open access funding provided by Institute of Science and Technology
  (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Nicolas
  full_name: Clozeau, Nicolas
  id: fea1b376-906f-11eb-847d-b2c0cf46455b
  last_name: Clozeau
- first_name: Marc
  full_name: Josien, Marc
  last_name: Josien
- first_name: Felix
  full_name: Otto, Felix
  last_name: Otto
- first_name: Qiang
  full_name: Xu, Qiang
  last_name: Xu
citation:
  ama: 'Clozeau N, Josien M, Otto F, Xu Q. Bias in the representative volume element
    method: Periodize the ensemble instead of its realizations. <i>Foundations of
    Computational Mathematics</i>. 2024;24:1305-1387. doi:<a href="https://doi.org/10.1007/s10208-023-09613-y">10.1007/s10208-023-09613-y</a>'
  apa: 'Clozeau, N., Josien, M., Otto, F., &#38; Xu, Q. (2024). Bias in the representative
    volume element method: Periodize the ensemble instead of its realizations. <i>Foundations
    of Computational Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s10208-023-09613-y">https://doi.org/10.1007/s10208-023-09613-y</a>'
  chicago: 'Clozeau, Nicolas, Marc Josien, Felix Otto, and Qiang Xu. “Bias in the
    Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.”
    <i>Foundations of Computational Mathematics</i>. Springer Nature, 2024. <a href="https://doi.org/10.1007/s10208-023-09613-y">https://doi.org/10.1007/s10208-023-09613-y</a>.'
  ieee: 'N. Clozeau, M. Josien, F. Otto, and Q. Xu, “Bias in the representative volume
    element method: Periodize the ensemble instead of its realizations,” <i>Foundations
    of Computational Mathematics</i>, vol. 24. Springer Nature, pp. 1305–1387, 2024.'
  ista: 'Clozeau N, Josien M, Otto F, Xu Q. 2024. Bias in the representative volume
    element method: Periodize the ensemble instead of its realizations. Foundations
    of Computational Mathematics. 24, 1305–1387.'
  mla: 'Clozeau, Nicolas, et al. “Bias in the Representative Volume Element Method:
    Periodize the Ensemble Instead of Its Realizations.” <i>Foundations of Computational
    Mathematics</i>, vol. 24, Springer Nature, 2024, pp. 1305–87, doi:<a href="https://doi.org/10.1007/s10208-023-09613-y">10.1007/s10208-023-09613-y</a>.'
  short: N. Clozeau, M. Josien, F. Otto, Q. Xu, Foundations of Computational Mathematics
    24 (2024) 1305–1387.
corr_author: '1'
date_created: 2023-06-11T22:00:40Z
date_published: 2024-08-01T00:00:00Z
date_updated: 2025-01-09T07:37:50Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s10208-023-09613-y
external_id:
  isi:
  - '000999623100001'
file:
- access_level: open_access
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  creator: dernst
  date_created: 2025-01-09T07:36:57Z
  date_updated: 2025-01-09T07:36:57Z
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has_accepted_license: '1'
intvolume: '        24'
isi: 1
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: 1305-1387
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Bias in the representative volume element method: Periodize the ensemble instead
  of its realizations'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2024'
...