Generalized rough polyharmonic splines for multiscale PDEs with rough coefficients

Xinliang Liu; Lei Zhang; Shengxin Zhu

doi:10.4208/NMTMA.OA-2021-0100

Generalized rough polyharmonic splines for multiscale PDEs with rough coefficients

Xinliang Liu, Lei Zhang^*, Shengxin Zhu^*

^*Corresponding author for this work

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

1 Scopus citations

Abstract

We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.

Original language	English (US)
Pages (from-to)	862-892
Number of pages	31
Journal	Numerical Mathematics
Volume	14
Issue number	4
DOIs	https://doi.org/10.4208/NMTMA.OA-2021-0100
State	Published - Nov 2021

Bibliographical note

Funding Information:
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871339, 11861131004). Dr Zhu research is also supported by the Foundation of LCP (Grant No. 6142A05180501), by the BNU-HKBU United International College Start-up Research Fund (Grant No. R72021114) and in part by the NSFC (Grant Nos. 11771002, 11671049, 11671051, 6162003, 11871339), by the joint project of Guangdong-Hong Kong-Macau Applied Mathematics Centre (Grant No. 2020B1515310022), and Guangdong Higher Education Research Platform and Research Project (Grant No. 2020ZDZX3076).

Publisher Copyright:
© 2021 Global Science Press. All rights reserved.

Keywords

Bayesian numerical homogenization
Derivative measurement
Edge measurement
Generalized Rough Polyharmonic Splines
Multiscale elliptic equation

ASJC Scopus subject areas

Modeling and Simulation
Control and Optimization
Computational Mathematics
Applied Mathematics

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10.4208/NMTMA.OA-2021-0100

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title = "Generalized rough polyharmonic splines for multiscale PDEs with rough coefficients",

abstract = "We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.",

keywords = "Bayesian numerical homogenization, Derivative measurement, Edge measurement, Generalized Rough Polyharmonic Splines, Multiscale elliptic equation",

author = "Xinliang Liu and Lei Zhang and Shengxin Zhu",

note = "Funding Information: This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871339, 11861131004). Dr Zhu research is also supported by the Foundation of LCP (Grant No. 6142A05180501), by the BNU-HKBU United International College Start-up Research Fund (Grant No. R72021114) and in part by the NSFC (Grant Nos. 11771002, 11671049, 11671051, 6162003, 11871339), by the joint project of Guangdong-Hong Kong-Macau Applied Mathematics Centre (Grant No. 2020B1515310022), and Guangdong Higher Education Research Platform and Research Project (Grant No. 2020ZDZX3076). Publisher Copyright: {\textcopyright} 2021 Global Science Press. All rights reserved.",

year = "2021",

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doi = "10.4208/NMTMA.OA-2021-0100",

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AU - Zhang, Lei

AU - Zhu, Shengxin

N1 - Funding Information: This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871339, 11861131004). Dr Zhu research is also supported by the Foundation of LCP (Grant No. 6142A05180501), by the BNU-HKBU United International College Start-up Research Fund (Grant No. R72021114) and in part by the NSFC (Grant Nos. 11771002, 11671049, 11671051, 6162003, 11871339), by the joint project of Guangdong-Hong Kong-Macau Applied Mathematics Centre (Grant No. 2020B1515310022), and Guangdong Higher Education Research Platform and Research Project (Grant No. 2020ZDZX3076). Publisher Copyright: © 2021 Global Science Press. All rights reserved.

PY - 2021/11

Y1 - 2021/11

N2 - We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.

AB - We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.

KW - Bayesian numerical homogenization

KW - Derivative measurement

KW - Edge measurement

KW - Generalized Rough Polyharmonic Splines

KW - Multiscale elliptic equation

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SN - 1004-8979

VL - 14

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JO - Numerical Mathematics

JF - Numerical Mathematics

IS - 4

ER -

Generalized rough polyharmonic splines for multiscale PDEs with rough coefficients

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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