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.
Bibliographical noteFunding 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).
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- 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