TY - JOUR
T1 - Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients
AU - Liu , Xinliang
AU - Zhang , Lei
AU - Zhu , Shengxin
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 862
EP - 892
PY - 2021
DA - 2021/09
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2021-0100
UR - https://global-sci.org/intro/article_detail/nmtma/19522.html
KW - Generalized Rough Polyharmonic Splines, multiscale elliptic equation, Bayesian numerical homogenization, edge measurement, derivative measurement.
AB - <p style="text-align: justify;">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.</p>