Numerical stochastic homogenization by quasilocal effective diffusion tensors

Dietmar Gallistl; Daniel Peterseim

doi:10.4310/CMS.2019.v17.n3.a3

Numerical stochastic homogenization by quasilocal effective diffusion tensors

Dietmar Gallistl
, Daniel Peterseim

Research output: Contribution to journal › Article › Academic › peer-review

7 Citations (Scopus)

1 Downloads (Pure)

Abstract

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

Original language	English
Pages (from-to)	637-651
Number of pages	15
Journal	Communications in Mathematical Sciences
Volume	17
Issue number	3
DOIs	https://doi.org/10.4310/CMS.2019.v17.n3.a3
Publication status	Published - 2019

Keywords

A posteriori error estimates
A priori error estimates
Model reduction
Modeling error estimate
Multiscale method
Numerical homogenization
Uncertainty
Upscaling
n/a OA procedure

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10.4310/CMS.2019.v17.n3.a3

Cite this

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title = "Numerical stochastic homogenization by quasilocal effective diffusion tensors",

abstract = "This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.",

keywords = "A posteriori error estimates, A priori error estimates, Model reduction, Modeling error estimate, Multiscale method, Numerical homogenization, Uncertainty, Upscaling, n/a OA procedure",

author = "Dietmar Gallistl and Daniel Peterseim",

note = "Funding Information: ∗Received: April 13, 2017; Accepted (in revised form): January 22, 2019. Communicated by Greg Pavliotis. D. Gallistl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Main parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program on multiscale problems. Funding Information: D. Gallistl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Main parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program on multiscale problems. Publisher Copyright: {\textcopyright} 2019 International Press.",

year = "2019",

doi = "10.4310/CMS.2019.v17.n3.a3",

language = "English",

volume = "17",

pages = "637--651",

journal = "Communications in Mathematical Sciences",

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AU - Peterseim, Daniel

N1 - Funding Information: ∗Received: April 13, 2017; Accepted (in revised form): January 22, 2019. Communicated by Greg Pavliotis. D. Gallistl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Main parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program on multiscale problems. Funding Information: D. Gallistl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Main parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program on multiscale problems. Publisher Copyright: © 2019 International Press.

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N2 - This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

AB - This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

KW - A posteriori error estimates

KW - A priori error estimates

KW - Model reduction

KW - Modeling error estimate

KW - Multiscale method

KW - Numerical homogenization

KW - Uncertainty

KW - Upscaling

KW - n/a OA procedure

UR - https://www.scopus.com/pages/publications/85077463980

U2 - 10.4310/CMS.2019.v17.n3.a3

DO - 10.4310/CMS.2019.v17.n3.a3

M3 - Article

AN - SCOPUS:85077463980

SN - 1539-6746

VL - 17

SP - 637

EP - 651

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 3

ER -

Numerical stochastic homogenization by quasilocal effective diffusion tensors

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Keywords

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