---
_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. Foundations of
Computational Mathematics. 2023. doi:10.1007/s10208-023-09613-y'
apa: '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. Springer Nature. https://doi.org/10.1007/s10208-023-09613-y'
chicago: 'Clozeau, Nicolas, Marc Josien, Felix Otto, and Qiang Xu. “Bias in the
Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.”
Foundations of Computational Mathematics. Springer Nature, 2023. https://doi.org/10.1007/s10208-023-09613-y.'
ieee: 'N. Clozeau, M. Josien, F. Otto, and Q. Xu, “Bias in the representative volume
element method: Periodize the ensemble instead of its realizations,” Foundations
of Computational Mathematics. 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.” Foundations of Computational
Mathematics, Springer Nature, 2023, doi:10.1007/s10208-023-09613-y.'
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: '9649'
abstract:
- lang: eng
text: "Isomanifolds are the generalization of isosurfaces to arbitrary dimension
and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued
smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an
isomanifold is to consider its Piecewise-Linear (PL) approximation based on a
triangulation T of the ambient space Rd. In this paper, we give conditions under
which the PL-approximation of an isomanifold is topologically equivalent to the
isomanifold. The conditions are easy to satisfy in the sense that they can always
be met by taking a sufficiently\r\nfine triangulation T . This contrasts with
previous results on the triangulation of manifolds where, in arbitrary dimensions,
delicate perturbations are needed to guarantee topological correctness, which
leads to strong limitations in practice. We further give a bound on the Fréchet
distance between the original isomanifold and its PL-approximation. Finally we
show analogous results for the PL-approximation of an isomanifold with boundary."
acknowledgement: "First and foremost, we acknowledge Siargey Kachanovich for discussions.
We thank Herbert Edelsbrunner and all members of his group, all former and current
members of the Datashape team (formerly known as Geometrica), and André Lieutier
for encouragement. We further thank the reviewers of Foundations of Computational
Mathematics and the reviewers and program committee of the Symposium on Computational
Geometry for their feedback, which improved the exposition.\r\nThis work was funded
by the European Research Council under the European Union’s ERC Grant Agreement
number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher
Dimensions). This work was also supported by the French government, through the
3IA Côte d’Azur Investments in the Future project managed by the National Research
Agency (ANR) with the reference number ANR-19-P3IA-0002. Mathijs Wintraecken also
received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement no. 754411."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat J-D, Wintraecken M. The topological correctness of PL approximations
of isomanifolds. Foundations of Computational Mathematics . 2022;22:967-1012.
doi:10.1007/s10208-021-09520-0
apa: Boissonnat, J.-D., & Wintraecken, M. (2022). The topological correctness
of PL approximations of isomanifolds. Foundations of Computational Mathematics
. Springer Nature. https://doi.org/10.1007/s10208-021-09520-0
chicago: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness
of PL Approximations of Isomanifolds.” Foundations of Computational Mathematics
. Springer Nature, 2022. https://doi.org/10.1007/s10208-021-09520-0.
ieee: J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL approximations
of isomanifolds,” Foundations of Computational Mathematics , vol. 22. Springer
Nature, pp. 967–1012, 2022.
ista: Boissonnat J-D, Wintraecken M. 2022. The topological correctness of PL approximations
of isomanifolds. Foundations of Computational Mathematics . 22, 967–1012.
mla: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness
of PL Approximations of Isomanifolds.” Foundations of Computational Mathematics
, vol. 22, Springer Nature, 2022, pp. 967–1012, doi:10.1007/s10208-021-09520-0.
short: J.-D. Boissonnat, M. Wintraecken, Foundations of Computational Mathematics 22
(2022) 967–1012.
date_created: 2021-07-14T06:44:53Z
date_published: 2022-01-01T00:00:00Z
date_updated: 2023-08-02T06:49:17Z
day: '01'
ddc:
- '516'
department:
- _id: HeEd
doi: 10.1007/s10208-021-09520-0
ec_funded: 1
external_id:
isi:
- '000673039600001'
file:
- access_level: open_access
checksum: f1d372ec3c08ec22e84f8e93e1126b8c
content_type: application/pdf
creator: mwintrae
date_created: 2021-07-14T06:44:36Z
date_updated: 2021-07-14T06:44:36Z
file_id: '9650'
file_name: Boissonnat-Wintraecken2021_Article_TheTopologicalCorrectnessOfPLA.pdf
file_size: 1455699
relation: main_file
file_date_updated: 2021-07-14T06:44:36Z
has_accepted_license: '1'
intvolume: ' 22'
isi: 1
language:
- iso: eng
month: '0'
oa: 1
oa_version: Published Version
page: 967-1012
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: 'Foundations of Computational Mathematics '
publication_identifier:
eissn:
- 1615-3383
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '7952'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: The topological correctness of PL approximations of isomanifolds
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
volume: 22
year: '2022'
...
---
_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. Foundations of Computational
Mathematics. 2021. doi:10.1007/s10208-021-09531-x
apa: Mondelli, M., Thrampoulidis, C., & Venkataramanan, R. (2021). Optimal combination
of linear and spectral estimators for generalized linear models. Foundations
of Computational Mathematics. Springer. https://doi.org/10.1007/s10208-021-09531-x
chicago: Mondelli, Marco, Christos Thrampoulidis, and Ramji Venkataramanan. “Optimal
Combination of Linear and Spectral Estimators for Generalized Linear Models.”
Foundations of Computational Mathematics. Springer, 2021. https://doi.org/10.1007/s10208-021-09531-x.
ieee: M. Mondelli, C. Thrampoulidis, and R. Venkataramanan, “Optimal combination
of linear and spectral estimators for generalized linear models,” Foundations
of Computational Mathematics. 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.” Foundations of Computational Mathematics,
Springer, 2021, doi:10.1007/s10208-021-09531-x.
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'
...
---
_id: '6662'
abstract:
- lang: eng
text: "In phase retrieval, we want to recover an unknown signal \U0001D465∈ℂ\U0001D451
from n quadratic measurements of the form \U0001D466\U0001D456=|⟨\U0001D44E\U0001D456,\U0001D465⟩|2+\U0001D464\U0001D456,
where \U0001D44E\U0001D456∈ℂ\U0001D451 are known sensing vectors and \U0001D464\U0001D456
is measurement noise. We ask the following weak recovery question: What is the
minimum number of measurements n needed to produce an estimator \U0001D465^(\U0001D466)
that is positively correlated with the signal \U0001D465? We consider the case
of Gaussian vectors \U0001D44E\U0001D44E\U0001D456. We prove that—in the high-dimensional
limit—a sharp phase transition takes place, and we locate the threshold in the
regime of vanishingly small noise. For \U0001D45B≤\U0001D451−\U0001D45C(\U0001D451),
no estimator can do significantly better than random and achieve a strictly positive
correlation. For \U0001D45B≥\U0001D451+\U0001D45C(\U0001D451), a simple spectral
estimator achieves a positive correlation. Surprisingly, numerical simulations
with the same spectral estimator demonstrate promising performance with realistic
sensing matrices. Spectral methods are used to initialize non-convex optimization
algorithms in phase retrieval, and our approach can boost the performance in this
setting as well. Our impossibility result is based on classical information-theoretic
arguments. The spectral algorithm computes the leading eigenvector of a weighted
empirical covariance matrix. We obtain a sharp characterization of the spectral
properties of this random matrix using tools from free probability and generalizing
a recent result by Lu and Li. Both the upper bound and lower bound generalize
beyond phase retrieval to measurements \U0001D466\U0001D456 produced according
to a generalized linear model. As a by-product of our analysis, we compare the
threshold of the proposed spectral method with that of a message passing algorithm."
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: Andrea
full_name: Montanari, Andrea
last_name: Montanari
citation:
ama: Mondelli M, Montanari A. Fundamental limits of weak recovery with applications
to phase retrieval. Foundations of Computational Mathematics. 2019;19(3):703-773.
doi:10.1007/s10208-018-9395-y
apa: Mondelli, M., & Montanari, A. (2019). Fundamental limits of weak recovery
with applications to phase retrieval. Foundations of Computational Mathematics.
Springer. https://doi.org/10.1007/s10208-018-9395-y
chicago: Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery
with Applications to Phase Retrieval.” Foundations of Computational Mathematics.
Springer, 2019. https://doi.org/10.1007/s10208-018-9395-y.
ieee: M. Mondelli and A. Montanari, “Fundamental limits of weak recovery with applications
to phase retrieval,” Foundations of Computational Mathematics, vol. 19,
no. 3. Springer, pp. 703–773, 2019.
ista: Mondelli M, Montanari A. 2019. Fundamental limits of weak recovery with applications
to phase retrieval. Foundations of Computational Mathematics. 19(3), 703–773.
mla: Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery
with Applications to Phase Retrieval.” Foundations of Computational Mathematics,
vol. 19, no. 3, Springer, 2019, pp. 703–73, doi:10.1007/s10208-018-9395-y.
short: M. Mondelli, A. Montanari, Foundations of Computational Mathematics 19 (2019)
703–773.
date_created: 2019-07-22T13:23:48Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2021-01-12T08:08:28Z
day: '01'
doi: 10.1007/s10208-018-9395-y
extern: '1'
external_id:
arxiv:
- '1708.05932'
intvolume: ' 19'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1708.05932
month: '06'
oa: 1
oa_version: Preprint
page: 703-773
publication: Foundations of Computational Mathematics
publication_identifier:
eissn:
- 1615-3383
publication_status: published
publisher: Springer
quality_controlled: '1'
status: public
title: Fundamental limits of weak recovery with applications to phase retrieval
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2019'
...