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Volume 14, Issue 4
Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients

Xinliang Liu, Lei Zhang & Shengxin Zhu

DOI: 10.4208/nmtma.OA-2021-0100

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 862-892.

Published online: 2021-09

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  • 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

Generalized Rough Polyharmonic Splines, multiscale elliptic equation, Bayesian numerical homogenization, edge measurement, derivative measurement.

  • AMS Subject Headings

41A15, 34E13, 35B27

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-862, author = {Liu , XinliangZhang , Lei and Zhu , Shengxin}, title = {Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {4}, pages = {862--892}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0100}, url = {http://global-sci.org/intro/article_detail/nmtma/19522.html} }
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 -

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.

Liu , XinliangZhang , Lei and Zhu , Shengxin. (2021). Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients. Numerical Mathematics: Theory, Methods and Applications. 14 (4). 862-892. doi:10.4208/nmtma.OA-2021-0100
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