TY - JOUR
T1 - Generalized rough polyharmonic splines for multiscale PDEs with rough coefficients
AU - Liu, Xinliang
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
UR - http://www.scopus.com/inward/record.url?scp=85115783459&partnerID=8YFLogxK
U2 - 10.4208/NMTMA.OA-2021-0100
DO - 10.4208/NMTMA.OA-2021-0100
M3 - Article
AN - SCOPUS:85115783459
SN - 1004-8979
VL - 14
SP - 862
EP - 892
JO - Numerical Mathematics
JF - Numerical Mathematics
IS - 4
ER -