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 language | English (US) |
|---|---|
| Pages (from-to) | 517-552 |
| Number of pages | 36 |
| Journal | ESAIM: Mathematical Modelling and Numerical Analysis |
| Volume | 48 |
| Issue number | 2 |
| DOIs | |
| State | Published - Mar 11 2014 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Numerical Analysis
- Modeling and Simulation
- Computational Mathematics
- Applied Mathematics
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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 journal › Article
TY - JOUR
T1 - Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
AU - Owhadi, Houman
AU - Zhang, Lei
AU - Berlyand, Leonid
PY - 2014/3/11
Y1 - 2014/3/11
N2 - 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.
AB - 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.
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U2 - 10.1051/m2an/2013118
DO - 10.1051/m2an/2013118
M3 - Article
AN - SCOPUS:84907189656
VL - 48
SP - 517
EP - 552
JO - Mathematical Modelling and Numerical Analysis
JF - Mathematical Modelling and Numerical Analysis
SN - 0764-583X
IS - 2
ER -