---
res:
  bibo_abstract:
  - "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.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Nicolas
      foaf_name: Clozeau, Nicolas
      foaf_surname: Clozeau
      foaf_workInfoHomepage: http://www.librecat.org/personId=fea1b376-906f-11eb-847d-b2c0cf46455b
  - foaf_Person:
      foaf_givenName: Marc
      foaf_name: Josien, Marc
      foaf_surname: Josien
  - foaf_Person:
      foaf_givenName: Felix
      foaf_name: Otto, Felix
      foaf_surname: Otto
  - foaf_Person:
      foaf_givenName: Qiang
      foaf_name: Xu, Qiang
      foaf_surname: Xu
  bibo_doi: 10.1007/s10208-023-09613-y
  bibo_volume: 24
  dct_date: 2024^xs_gYear
  dct_identifier:
  - UT:000999623100001
  dct_isPartOf:
  - http://id.crossref.org/issn/1615-3375
  - http://id.crossref.org/issn/1615-3383
  dct_language: eng
  dct_publisher: Springer Nature@
  dct_title: 'Bias in the representative volume element method: Periodize the ensemble
    instead of its realizations@'
...