used to guide the discovery of new Selberg identities). We also refer to subsection \[subsecone3\] for a review of the correspondence between statistical inference and numerical approximation.

Outline of the paper
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The essential difficulty in generalizing the multigrid concept to PDEs with rough coefficients lies in the fact that the interpolation (downscaling) and restriction (upscaling) operators are, a priori, unknown. Indeed, in this situation, piecewise linear finite-elements can perform arbitrarily badly [@BaOs:2000] and the design of the interpolation operator requires the identification of accurate basis elements adapted to the microstructure $a(x)$.

This identification problem has also been the essential difficulty in numerical homogenization [@WhHo87; @BaOs:1983; @BaCaOs:1994; @Beylkin:1995; @HoWu:1997; @EEngquist:2003; @OwZh:2007a; @BraWu09]. Although inspired by classical homogenization ideas and concepts (such as oscillating test functions [@Mur78; @EfHo:2007; @EfGiHouEw:2006], cell problems/correctors and effective coefficients [@BeLiPa78; @PapanicolaouVaradhan:1981; @Abdulle:2004; @NoPaPi:2008; @EnSou08; @AnGlo06], harmonic coordinates [@Kozlov:1979; @BaOs:1983; @BaCaOs:1994; @Owhadi:2003; @BenarousOwhadi:2003; @AllBri05; @OwZh:2007a], compactness by compensation [@Spagnolo:1968; @Gio75; @Muratb:1978; @BeOw:2010]) an essential goal of numerical homogenization has been the numerical approximation of the solution space of with arbitrary rough coefficients [@OwZh:2007a], i.e., in particular, without the assumptions found in classical homogenization, such as scale separation, ergodicity at fine scales and $\epsilon$-sequences of operators (otherwise the 