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1 results

  • 4 - Numerical Homogenization

  • from Part I - The Sobolev Space Setting
  • Houman Owhadi, California Institute of Technology, Clint Scovel, California Institute of Technology
  • Book:
    Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
    Published online:
    10 October 2019
    Print publication:
    24 October 2019, pp 38-62

  • Summary

    This chapter reviews classical homogenizationconcepts such as the cell problem; correctors; compactness by compensation; oscillating test functions; H, G, and Gamma convergence; and periodic and stochastic homogenization. Numerical homogenization is presented as the problem of identifying basis functions that are both as accurate and as localized as possible. Optimal recovery splines constructed from simple measurement functions (Diracs, indicator functions, and local polynomials) provide a simple to solution to this problem: they achieve the Kolmogorov n-width optimal accuracy (up to a constant) and they are exponentially localized. Current numerical homogenization methods are reviewed. Gamblets, the LOD method, the variational multiscale method, andpolyharmonic splines are shown to have a common characterization as optimal recovery splines.