Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Houman Owhadi, Lei Zhang, Leonid Berlyand

Research output: Contribution to journalArticle

40 Citations (Scopus)

Abstract

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 L2norm of the source terms; its (pre-)computation involves minimizing ?(H-d) quadratic (cell) problems on (super-)localized sub-domains of size ?(Hln(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 (?(H)in energy norm and independent fromaspect 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.

Original languageEnglish (US)
Pages (from-to)517-552
Number of pages36
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume48
Issue number2
DOIs
StatePublished - Mar 11 2014

Fingerprint

Homogenization
Splines
Spline
Rough
Interpolation
Elliptic Equations
Divergence
Inverse problems
Mesh
Linear systems
Sparse Linear Systems
Biharmonic
Approximation Space
Interpolation Function
Variational Approach
Arbitrary
Coefficient
Ergodicity
Source Terms
Hyperbolic Equations

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Modeling and Simulation
  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "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 L2norm of the source terms; its (pre-)computation involves minimizing ?(H-d) quadratic (cell) problems on (super-)localized sub-domains of size ?(Hln(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 (?(H)in energy norm and independent fromaspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar{\'e} 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.",
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Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. / Owhadi, Houman; Zhang, Lei; Berlyand, Leonid.

In: ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, No. 2, 11.03.2014, p. 517-552.

Research output: Contribution to journalArticle

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