Journal
MULTISCALE MODELING & SIMULATION
Volume 13, Issue 3, Pages 812-828Publisher
SIAM PUBLICATIONS
DOI: 10.1137/140974596
Keywords
numerical homogenization; Bayesian inference; Bayesian numerical analysis; coarse graining; polyharmonic splines; Gaussian filtering
Funding
- Air Force Office of Scientific Research [FA9550-12-1-0389]
- 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) [DE-AC52-06NA25396, 273448]
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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.
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