Bayesian Numerical Homogenization
Numerical homogenization, i.e., the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian inference problem in which a given PDE with rough coefficients (or multiscale operator) is excited with noise (random right-hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how rough polyharmonic splines can be rediscovered as the optimal solution of a Gaussian filtering problem.
© 2015 SIAM. Received by the editors June 26, 2014; accepted for publication (in revised form) May 5, 2015; published electronically July 14, 2015. The author gratefully acknowledges support from the Air Force Office of Scientific Research under award FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators) and the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, through the Exascale Co-Design Center for Materials in Extreme Environments (ExMatEx, LANL contract DE-AC52-06NA25396, Caltech subcontract 273448). The author thanks Dongbin Xiu, Lei Zhang, and Guillaume Bal for stimulating discussions, and Leonid Berlyand for comments on the manuscript. The author also thanks two anonymous referees for valuable comments and suggestions.
Submitted - 1406.6668v2.pdf
Published - 140974596.pdf