Multi Scale problems

Below is a list of useful literature on the subject, along with comments.

Literature

  • Tutorials in multi scale problems by Runborg et al. Some tutorials on Multi Scale Simulations (FMM, Wavelet analysis.) The one on Wavelet analysis is interesting (Runborg.)

  • Introductory book
    Homogenization recap in Appendix B: Typically a problem of the form

    $ \nabla \cdot (a(x/\epsilon) \nabla u(x)) + a_0(x/\epsilon) u(x) = f(x), \implies L_\epsilon u = f $

    where $a$ and $a_0$ are 1-periodic functions. Goal is to study the limit as $\epsilon \to 0$. Homogenisation theory is not what we are interested in. Instead, we want to minimize error of a low resolution approximation, at a nonzero $\epsilon$. Of course we still want convergence to homogenized solution as $\epsilon \to 0$. The 1-d homogenization consists of taking the harmonic average.

    The standard assumption seems to be periodic multi scale:

    $ u(x) = \tilde{u}(x) + \phi(x/\epsilon), $

    where $\tilde{u}$ and $\phi$ are periodic, and $\tilde{u}$ is band limited to some large scale $»1/\epsilon$.