6.1.
Other approaches
In this survey we have essentially focused on the so-called HMM (direct approach) and so-called MsFEM (dual approach), which are two popular numerical homogenization methods. The convergence analysis of these methods can be made using the same analytical framework — the difference between the two approaches arising during the discretization. Another more global approach has been introduced by Schwab and Hoang in [41], inspired by
• two-scale convergence [4] and periodic unfolding [14],
• sparse tensor-products approximation [40].
The consistency of this approach can be proved using our analytical framework as well. With the notation of Section 2.1, their starting point is indeed the fact that the equation foruρ,ε takes the equivalent form (provided
Aεis extended on a neigborhood ofD): Finduρ,ε ∈H01(D) andUρ,ε∈L2(D, H01(Q)) such that for allv∈H01(D)
and allV ∈L2(D, H01(Q)), ˆ D ˆ Q (∇xv(x) +∇yV(x, y))·Aε(x+ρy)(∇xuρ,ε(x) +∇yUρ,ε(x, y))dxdy = ˆ D f(x)v(x)dx.
This point of view allows them to look for a efficient approximation of the spaceL2(D, H1
0(Q)), and implement
sparse-tensor product strategies.
Another approach advocated by Owhadi and Zhang in [53], whose origin is not the homogenization theory properly speaking, is based on the notion of harmonic coordinates. The starting point of their work is that although solutionsu∈H1
0(D) to the equation
−∇ ·A∇u = f
are usually not more regular than H1 in the Euclidian coordinates, they are H2 in the so-called harmonic
coordinates provided f ∈ L2(D). These harmonic coordinates are defined as x 7→ x+ Γ(x)·x, where Γ =
(γ1, . . . , γd) andγk is the unique weak solution inH01(D) to
−∇ ·A(ek+∇γk) = 0
for all k ∈ {1, . . . , k}. The issue is then to compute (approximations of) these harmonic coordinates. An interesting link between harmonic coordinates and the MsFEM has been made by Allaire and Brizzi in [5], who consider as multiscale finite element space the functionsx7→vH(x+ ΓH(x)), where ΓH is a local approximation
of the harmonic coordinates Γ on each simplex of the macroscopic tesselation ofD. The combination of harmonic coordinates with the regularization method was recently studied by Owhadi and Zhang in [54].
6.2.
Beyond the linear case
The reader may wonder what remains of all this when we replace the linear equation by a nonlinear equation, say elliptic monotone. Then part of the results translates quite naturally to the nonlinear case, namely the analytical framework (see [20, 21] for the description of the MsFEM method for nonlinear problems and error estimates in the periodic case, and [27,28] for the extension of the analytical framework of Section 2 to monotone elliptic operators and nonconvex integral functionals). Yet most of the quantitative results break down. It is also not clear how the regularization method behaves in that case.
In addition, from a practical point of view, things get much worse. In the linear case, one could have the excuse that the multiscale finite element basis or the local homogenized coefficients and local correctors only had to be computed once, and could be used to solve the equation for a wide variety of boundary conditions and right hand sides. This is definitely an advantage when dealing with optimal control for instance. In the nonlinear case, this does not hold any longer, and the local basis or local values of the operator have to be computed on the fly. This is not doable in practice.
For nonlinear problems, even periodic homogenization is not that easy since the object to reconstruct is a homogenized nonlinear map (and not a single matrix). The same fact holds for stochastic homogenization (with the additional difficulty that the corrector equation is posed on the whole space). A possible strategy is to compute approximations of the nonlinear map at some sampling points, and try to reconstruct an analytical approximation of this map by solving an inverse problem. This analytical approximation should at least satisfy similar properties as the homogenized nonlinear map (say convexity, isotropy, etc.). This strategy has been used in [16] to approximate the homogenized energy density associated with a discrete model for rubber introduced in [33] and whose homogenization limit has been etablished in [3]. There, the homogenized energy density is known to be quasiconvex, isotropic, and minimal at identity. An important issue is then to identify a suitable (explicit) subset of such functions in order to approximate the inverse problem. The constraint being nonlinear and the error landscape being quite rough, deterministic optimization methods usually fail to converge. In [16], the use of an evolutionary algorithm has been quite efficient.
An intermediate case between nonlinear problems and the linear equations considered throughout this survey is the case of parametrized linear equations. Such an example naturally appears in [31], where a coupled system of elliptic/parabolic equations is studied (for application to nuclear waste storage). Indeed, although the two equations are linear and the coefficients periodic, the nonlinear coupling condition makes the homogenized coefficient of the parabolic equation depend locally on the gradient of the solution of the homogenized elliptic equation. We are thus lead to solve a family of corrector equations of the form: Find φζ ∈H1
#(Q) solution to
−∇ ·A(ζ)(ξ+∇φζ) = 0 in Q,
parametrized byζ∈Rd. This is an ideal framework for the reduced basis method [48], whose paradigm is that
the family{φζ, ζ∈Rd}maybe a thin subset ofH1
#(Q), so that it could be well approximated by a suitable low
dimensional subspaces ofH1
#(Q) (see [9, 15] for the case whenA(ζ) =A0+ζA1, withζin some compact set of
R). This method has already been used in the context of homogenization by Boyaval in [13]. As usual with the
reduced basis method, most of the work has to be done on the choice of the estimator, and on the fast-assembly method. In [31] we have used an estimator which is specific to homogenization problems, and appealed to the fast Fourier transform to assemble rigidity matrices efficiently.
6.3.
What next ?
In the nonlinear case, much remains to be done. The approach which consists in constructing analyti- cal proxies for the homogenized operators raise quite either unaddressed or unsolved challenging questions in approximation theory.
Even in the linear case, things are far from complete yet. In particular, as emphasized by Nolen, Papanicolaou, and Pironneau in [52], the numerical homogenization Grail is an adaptive method which would take the best of all worlds: use efficient domain decomposition techniques when needed, and exploit homogenization when possible.