---
_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>. 2023. 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. (2023). 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, 2023. <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>. Springer Nature, 2023.'
  ista: 'Clozeau N, Josien M, Otto F, Xu Q. 2023. Bias in the representative volume
    element method: Periodize the ensemble instead of its realizations. Foundations
    of Computational Mathematics.'
  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>, Springer Nature, 2023, 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
    (2023).
date_created: 2023-06-11T22:00:40Z
date_published: 2023-05-30T00:00:00Z
date_updated: 2023-08-02T06:12:39Z
day: '30'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s10208-023-09613-y
external_id:
  isi:
  - '000999623100001'
has_accepted_license: '1'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s10208-023-09613-y
month: '05'
oa: 1
oa_version: Published Version
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: epub_ahead
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
year: '2023'
...
---
_id: '10211'
abstract:
- lang: eng
  text: "We study the problem of recovering an unknown signal \U0001D465\U0001D465
    given measurements obtained from a generalized linear model with a Gaussian sensing
    matrix. Two popular solutions are based on a linear estimator \U0001D465\U0001D465^L
    and a spectral estimator \U0001D465\U0001D465^s. The former is a data-dependent
    linear combination of the columns of the measurement matrix, and its analysis
    is quite simple. The latter is the principal eigenvector of a data-dependent matrix,
    and a recent line of work has studied its performance. In this paper, we show
    how to optimally combine \U0001D465\U0001D465^L and \U0001D465\U0001D465^s. At
    the heart of our analysis is the exact characterization of the empirical joint
    distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s)
    in the high-dimensional limit. This allows us to compute the Bayes-optimal combination
    of \U0001D465\U0001D465^L and \U0001D465\U0001D465^s, given the limiting distribution
    of the signal \U0001D465\U0001D465. When the distribution of the signal is Gaussian,
    then the Bayes-optimal combination has the form \U0001D703\U0001D465\U0001D465^L+\U0001D465\U0001D465^s
    and we derive the optimal combination coefficient. In order to establish the limiting
    distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s),
    we design and analyze an approximate message passing algorithm whose iterates
    give \U0001D465\U0001D465^L and approach \U0001D465\U0001D465^s. Numerical simulations
    demonstrate the improvement of the proposed combination with respect to the two
    methods considered separately."
acknowledgement: M. Mondelli would like to thank Andrea Montanari for helpful discussions.
  All the authors would like to thank the anonymous reviewers for their helpful comments.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Marco
  full_name: Mondelli, Marco
  id: 27EB676C-8706-11E9-9510-7717E6697425
  last_name: Mondelli
  orcid: 0000-0002-3242-7020
- first_name: Christos
  full_name: Thrampoulidis, Christos
  last_name: Thrampoulidis
- first_name: Ramji
  full_name: Venkataramanan, Ramji
  last_name: Venkataramanan
citation:
  ama: Mondelli M, Thrampoulidis C, Venkataramanan R. Optimal combination of linear
    and spectral estimators for generalized linear models. <i>Foundations of Computational
    Mathematics</i>. 2021. doi:<a href="https://doi.org/10.1007/s10208-021-09531-x">10.1007/s10208-021-09531-x</a>
  apa: Mondelli, M., Thrampoulidis, C., &#38; Venkataramanan, R. (2021). Optimal combination
    of linear and spectral estimators for generalized linear models. <i>Foundations
    of Computational Mathematics</i>. Springer. <a href="https://doi.org/10.1007/s10208-021-09531-x">https://doi.org/10.1007/s10208-021-09531-x</a>
  chicago: Mondelli, Marco, Christos Thrampoulidis, and Ramji Venkataramanan. “Optimal
    Combination of Linear and Spectral Estimators for Generalized Linear Models.”
    <i>Foundations of Computational Mathematics</i>. Springer, 2021. <a href="https://doi.org/10.1007/s10208-021-09531-x">https://doi.org/10.1007/s10208-021-09531-x</a>.
  ieee: M. Mondelli, C. Thrampoulidis, and R. Venkataramanan, “Optimal combination
    of linear and spectral estimators for generalized linear models,” <i>Foundations
    of Computational Mathematics</i>. Springer, 2021.
  ista: Mondelli M, Thrampoulidis C, Venkataramanan R. 2021. Optimal combination of
    linear and spectral estimators for generalized linear models. Foundations of Computational
    Mathematics.
  mla: Mondelli, Marco, et al. “Optimal Combination of Linear and Spectral Estimators
    for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>,
    Springer, 2021, doi:<a href="https://doi.org/10.1007/s10208-021-09531-x">10.1007/s10208-021-09531-x</a>.
  short: M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational
    Mathematics (2021).
date_created: 2021-11-03T10:59:08Z
date_published: 2021-08-17T00:00:00Z
date_updated: 2023-09-05T14:13:57Z
day: '17'
ddc:
- '510'
department:
- _id: MaMo
doi: 10.1007/s10208-021-09531-x
external_id:
  arxiv:
  - '2008.03326'
  isi:
  - '000685721000001'
file:
- access_level: open_access
  checksum: 9ea12dd8045a0678000a3a59295221cb
  content_type: application/pdf
  creator: alisjak
  date_created: 2021-12-13T15:47:54Z
  date_updated: 2021-12-13T15:47:54Z
  file_id: '10542'
  file_name: 2021_Springer_Mondelli.pdf
  file_size: 2305731
  relation: main_file
  success: 1
file_date_updated: 2021-12-13T15:47:54Z
has_accepted_license: '1'
isi: 1
keyword:
- Applied Mathematics
- Computational Theory and Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal combination of linear and spectral estimators for generalized linear
  models
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2021'
...