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Published March 2014 | Submitted
Journal Article Open

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization


We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L^∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L^2 norm of the source terms; its (pre-)computation involves minimizing O(H^(-d)) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator -div(a∇.) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H)) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

Additional Information

© 2014 EDP Sciences. Received: 3 August 2013. The work of H. Owhadi is partially supported by the National Science Foundation under Award Number CMMI-092600, the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613, the Air Force Office of Scientific Research under Award Number FA9550-12-1-0389 and a contract from the DOE Exascale Co-Design Center for Materials in Extreme Environments. The work of L. Zhang is supported by the Young Thousand Talents Program of China. The work of L. Berlyand is supported by DOE under Award Number DE-FG02-08ER25862. H. Owhadi thanks M. Desbrun and F. de Goes for stimulating discussions. H. Owhadi also thanks François Murat for helpful discussions on Lemma 3.1. L. Berlyand thanks M. Potomkin for useful comments and suggestions. We also thank an anonymous referee for carefully reading the manuscript and detailed comments and suggestions.

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