Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients

TL;DR

This paper introduces a Bayesian framework for constructing generalized rough polyharmonic splines to efficiently solve multiscale PDEs with rough coefficients, providing theoretically justified localized bases and demonstrating their effectiveness through numerical experiments.

Contribution

It presents a novel Bayesian approach to automatically derive optimal coarse bases for multiscale PDEs with rough coefficients, with proven localization and approximation properties.

Findings

The constructed bases are quasi-optimal and localized.

Numerical experiments confirm theoretical approximation properties.

The method effectively handles multiscale PDEs with rough coefficients.

Abstract

In this paper, we demonstrate the construction of generalized Rough Polyhamronic 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 edge or derivative measurements. We prove the (quasi)-optimal localization and approximation properties of the obtained bases, and justify the theoretical results with numerical experiments.