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Bayesian Numerical Homogenization

Owhadi, Houman (2015) Bayesian Numerical Homogenization. Multiscale Modeling and Simulation, 13 (3). pp. 812-828. ISSN 1540-3459.

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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.

Item Type:Article
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URLURL TypeDescription DOIArticle Paper
Owhadi, Houman0000-0002-5677-1600
Additional Information:© 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.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-12-1-0389
Department of Energy (DOE)DE-AC52-06NA25396
Subject Keywords:numerical homogenization, Bayesian inference, Bayesian numerical analysis, coarse graining, polyharmonic splines, Gaussian filtering
Issue or Number:3
Classification Code:AMS subject classifications. 41A15, 34E13, 62C10, 60H30
Record Number:CaltechAUTHORS:20151023-105214463
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:61485
Deposited By: Tony Diaz
Deposited On:26 Oct 2015 20:55
Last Modified:03 Oct 2019 09:08

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