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Lecons de mathematiques Jacques-Louis Lions
Effective behavior of random media
From an error analysis to regularity theory
Julian Fischer*, Antoine Gloria, Stefan Neukamm, Felix
Otto*
* Max Planck Institute for Mathematics in the Sciences
Leipzig, Germany
Effective behavior of media
under statistical/incomplete information:
Early explicit approximate treatment,
recent numerical applications
Maxwell: Effective resistance of a composite
Einstein: Effective viscosity of a suspension
→ F → F
Recent: composite materials & porous media
Effective elasticity Effective permeability
Effective behavior by simulationof “Representative Volume”Rigorous mathematics/qualitative theory:Varadhan&Papanicolaou, Kozlov ’79, H-convergence by Murat&Tartar
A crucial influence of J.-L. Lions’ school ...
source: Conference 60th birthday
source: Academie des Sciences,
Institut de France
... on “homogenization”
Quantitative homogenization theory
and connections to other fields
Motivation from numerical analysis:
Representative Volume method
Connections with elliptic regularity theory:
generic regularity theory on large scales
Stochastic homogenization:
The Representative Volume method
and its numerical analysis
Random media ...
λ-uniformly coefficient fields a = a(x) on d-dim. space Rd
∃ λ > 0 ∀x ∈ Rd ∀ ξ ∈ R
d ξ · a(x)ξ ≥ λ|ξ|2, |a(x)ξ| ≤ |ξ|
Elliptic operator −∇ · a∇u
Ensemble 〈·〉 of (=probability measure on)
λ-uniformly elliptic coefficient fields a
Example of ensemble 〈·〉points Poisson distributed with density 1,
union of balls of radius 14 around points,
a = 1id on union, a = λid on complement,
... = elliptic operators with random coefficient fields
Effective behavior of random medium
= homogenized coefficient
Eg conductive medium:
function u electric potential, −∇u electric field,
coefficient field a conductivity,
q := −a∇u electric current, ∇ · q = 0 stationary current
For a-harmonic function u = u(x), that is, −∇ · a∇u = 0
homogenized coefficient ahom provides linear relation between
average field − limR↑∞
−∫
BR∇u to average current − lim
R↑∞−∫
BRa∇u
How to get from 〈·〉 to ahom ?
Representative Volume Method
Introduce artificial period L
Example of periodized ensemble 〈·〉L
points Poisson distributed with density 1,
on torus [0, L)d
union of balls of radius 14 around points,
a = 1id on union, a = λid on complement,
Given coordinate direction i = 1, · · · , d seek L-periodic φi with
−∇ · a(ei +∇φi) = 0 on [0, L)d.
Note ei +∇φi = ∇(xi + φi) ; since −∫
[0,L)d(ei +∇φi) = ei take
−∫
[0,L)da(ei+∇φi) as approximation to ahomei for L ≫ 1
Solving d elliptic equations −∇ · a(∇φi + ei) = 0 ...
direction e1
potential
x1 + φ1
current
a(e1 +∇φ1)
average current
−∫
a(e1 +∇φ1)
=(
0.49641−0.02137
)
≈ ahome1
direction e2
potential
x2 + φ2
current
a(e2 +∇φ1)
average current
−∫
a(e2 +∇φ2)
=(
−0.021370.53240
)
≈ ahome2
simulations by R. Kriemann (MPI)
... gives approximation to ahom
Two types of errors: Random error
ahom,L(a)ei := −∫
[0,L)da(ei+∇φi) where −∇·a(ei+∇φi) = 0, φi per
φi and thus ahom,Ldepend on realization aahom,L still random
fluctuations of ahom,Las measured by variance
〈(aL,hom−〈aL,hom〉L)2〉L decrease
as L increases
Random error: approx. depends on realization ...
realization 1
potential
current
ahom ≈(
0.49641 −0.02137−0.02137 0.53240
)
realization 2
potential
current
ahom ≈(
0.45101 0.011040.01104 0.45682
)
realization 3
potential
current
ahom ≈(
0.56213 0.008570.00857 0.60043
)
... and thus fluctuates
Fluctuations decrease with increasing L
ahom ≈
(
0.49641 −0.02137−0.02137 0.53240
)
ahom ≈
(
0.51837 0.003830.00383 0.51154
)
ahom ≈
(
0.45101 0.011040.01104 0.45682
)
ahom ≈
(
0.53204 0.004850.00485 0.52289
)
ahom ≈
(
0.56213 0.008570.00857 0.60043
)
ahom ≈
(
0.51538 −0.00052−0.00052 0.52152
)
Two types of errors: systematic error
ahom,L(a)ei := −∫
[0,L)da(ei+∇φi) where −∇·a(ei+∇φi) = 0, φi per
Expectation 〈ahom,L〉Lstill depends on L
since from 〈·〉 to 〈·〉Lstatistics are altered
by artificial long-range correlations
Also systematic error decreases with increasing L
〈ahom,L〉L = λhom,L
(
1 00 1
)
because of symmetry of 〈·〉 under rotation
L = 2 L = 5 L = 10 L = 20 L = 50
0.551 0.524 0.520 0.522 0.522
Scaling of both errors in L ...
Pick a according to 〈·〉L, solve for φ (period L),
compute spatial average ahom,Lei := −∫
[0,L)da(ei +∇φi)
Take random variable ahom,L as approximation to ahom
〈error2〉L = random2 + systematic2:
〈|ahom,L−ahom|2〉L = var〈·〉L[ahom,L]+ |〈ahom,L〉L − ahom|2
Qualitative theory yields:
limL↑∞
var〈·〉L[ahom,L] = 0, limL↑∞
〈ahom,L〉L = ahom
... why of interest?
Number of samples N vs. artificial period L
Take N samples, i. e. N independent picks a1, · · · , aN from 〈·〉L.
Compute empirical mean1N
N∑
n=1−∫
[0,L)dan(∇φi,n + ei)
〈total error2〉L = 1N random error2 + systematic error2
L ↑ reduces
systematic error and
random error
N ↑ reduces only
effect of random error
State of art in quantitative stochastic homogenization ...
Yurinskii ’86 : suboptimal rates in L for mixing 〈·〉
Naddaf & Spencer ’98, & Conlon ’00:optimal rates for small contrast 1− λ ≪ 1,for 〈·〉 with spectral gap
Gloria & Otto ’11, & Neukamm ’13, & Marahrens ’13:optimal rates for all λ > 0 for 〈·〉 with spectral gap,Logarithmic Sobolev (concentration of measure)
Armstrong & Smart ’14, & Mourrat ’14, & Kuusi ’15:towards optimal stochastic integrability (Gaussian),for mixing 〈·〉
... of linear equations in divergence form
An optimal result
Let 〈·〉L be ensemble of a’s with period L,
with 〈·〉L suitably coupled to 〈·〉
For a with period L
solve −∇ · (ei +∇φi) = 0 for φi of period L.
Set ahom,Lei = −∫
[0,L)da(ei +∇φi).
Theorem [Gloria&O.’13, G.&Neukamm&O. Inventiones’15]
Random error2 = var〈·〉L
[
ahom,L
]
≤ C(d, λ)L−d
Systematic error =∣
∣
∣〈ahom,L〉L − ahom∣
∣
∣ ≤ C(d, λ)L−d lnd2 L
Gloria&Nolen ’14: (random) error approximately Gaussian
Stochastic homogenization:
A regularizing effect of randomness on large scales
captured by Liouville properties of all order
a-harmonic functions of polynomial growth ...
Uniformly elliptic symm. coefficient field a = a(x) on Rd
λ|ξ|2 ≤ ξ · a(x)ξ ≤ |ξ|2 for some λ > 0
For any exponent α ≥ 0 consider
Xα :={
u a-harmonic∣
∣
∣
∣
limR↑∞1Rα
(
−∫
BRu2
)12 = 0
}
.
Under which conditions dimXα = dimXEuclα ?
... Liouville properties of all order?
For mere uniform ellipticity...
Always dimXα ≤ C(λ)αd−1 [Colding & Minicozzi, Li ’97]
(volume doubling, Poincare-Neumann inequality on all scales)
In general dimXα 6= dimXEuclα for any α > 0 and d > 1!
For (x1, x2) = (r cos θ, r sin θ)
consider metric cone
a = (dr)2 + r2
α2(dθ)2.
Then u = rα cos θ is a-harmonic,
hence u ∈ Xα+,
hence dimXα+ > 1.
Persist under smoothing out the vertex
... complete failure of Liouville
More on uniform ellipticity
∀ α dimXα ≤ C(λ)αd−1 [Colding & Minicozzi, Li ’97]
∀ α ∈ (0,1) ∃ a dimXα > 1 (d ≥ 2)
∃ 0 < α0(d, λ) ≪ 1 ∀ α ≤ α0 dimXα = 1
[Nash, De Giorgi ’57]
Systems: ∃ a dimXα=0 > 1 (d ≥ 3)
[De Giorgi ’68]
In which sense does ahave to be flat at infinity
for dimXα = dimXEuclα ?
A criterion for Liouville inspired by homogenization
In which sense does a have to be flat at infinity
for dimXα = dimXEuclα ?
Harmonic coordinates {ui}i=1,··· ,d : −∇ · a∇ui = 0
field ∇ui closed 1-form ↔ current a∇ui closed (d− 1)-form
“Flatness at infinity” means
∇ui − ei = dφi with 0-form φi of sublinear growth
a∇ui−ahomei = dσi with (d− 2)-form σi of sublinear growth
a∇ui − ahomei = ∇ · σi for σi = {σijk}jk=1,··· ,d skew
... scalar and vector potential of corrector
Liouville of order α < 2 ...
λ|ξ|2 ≤ ξ · aξ ≤ |ξ|2, Xα :={
u a-harm.∣
∣
∣ limR↑∞1Rα
(
−∫
BRu2)12 = 0
}
.
Proposition 1[Gloria, Neukamm, & O. ’14]
For i = 1, · · · , d let φi scalar and σi skew be related to a via
a(ei +∇φi) = ahomei +∇ · σi for some matrix ahom.
Suppose sub-linear growth in sense of
limR↑∞1R
(
−∫
BR|(φ, σ)|2
)12 = 0.
Then Xα = [{1}] for α ∈ (0,1)
and Xα = [{1, x1+φ1, · · · , xd+φd}] for α ∈ [1,2).
In particular dimXα = dimXEuclα for all α < 2.
... from sublinear growth of scalar & vector potential
Tool: decay of an intrinsic excess
Exc(u,BR) :=
infξ∈Rd
(
−∫
BR|∇(u− ξi(xi+φi))|
2)12
energy distance to a-linear
Proposition 1’[Gloria, Neukamm, & O. ’14]
∀ α < 2 ∃ δ = δ(d, λ, α) > 0 such that provided
1R
(
−∫
BR|(φ, σ)|2
)12 ≤ δ for R ≥ r
for some r < ∞, then for any a-harmonic u
Exc(u,Br) ≤ C(d, λ)(r
R)α−1Exc(u,BR) for R ≥ r
Classical proof in a picture
ahom-harmonic uhom with same boundary data on ∂BR
Correct (1 + φi∂i)uhom so that ≈ a-harmonic
Cut-off (1 + ηφi∂i)uhom so that = u on ∂BR
Taylor on small ball Br uhom(0) + ξi(xi+φi) with ξ = ∇uhom(0)
Campanato’s approach to Schauder theory
Classical proof: Campanato’s approach to Schauder theory
Compare u to ahom-harmonic function uhom on every BR:
−∇ · ahom∇uhom = 0 in BR, uhom = u on ∂BR
Write u = (1+ ηφi∂i)uhom + w then w = 0 on ∂BR and
−∇·a∇w = ∇·((φia−σi)∇η∂iuhom)+∇ · ((1− η)(a− ahom)∇uhom),
∇(u− ξi(xi+φi)) = (∂iuhom− ξi)(ei+∇φi)+ φi∇∂iu
hom+∇w
Energy estimate for w, inner & bdry regularity for uhom:
Exc(u,Br) ≤(
−∫
Br |∇(u− ξi(xi+φi))|2)12
.(
rR + (Rr )
d2∆(Rρ )
d2+1 + ρ
R
)(
−∫
B2R|∇u|2
)12
where ∆ := 1R
(
−∫
BR|(φ, σ)|2
)12. Campanato iteration.
... merit of vector potential σ
General form of randomness ...
Rd ∋ z acts on space of coefficient fields a via a(z + ·).
An ensemble 〈·〉 on space of coefficient fields a
1) is called stationary if for any measurable F = F(a),F(a(z + ·)) and F(a) have same expectation,
2) is called ergodic if for any measurable F = F(a),∀z F(a(z + ·)) = F(a) =⇒ ∀ 〈·〉-a.e. a F(a) = 〈F 〉
Theorem 1[Gloria, Neukamm, O’14]
Let 〈·〉 be stationary and ergodic ensemble
on space of λ-uniformly elliptic coefficient fields on Rd.
Then ∀ 〈·〉-a.e. a ∀ α < 2 dimXα = dimXEuclα .
... creates generic large-scale regularity
A stochastic construction
Proposition 1”[Gloria, Neukamm, O ’14]
Let 〈·〉 be stationary and ergodic ensemble
on space of λ-uniformly elliptic coefficient fields on Rd.
Then for i = 1, · · · , d ∃ φi(a, x) scalar and σi(a, x) skew s.t.
∇(φ, σ) is shift-equivariant ∇(φ, σ)(a(z + ·), x) = ∇(φ, σ)(a, z + x),
of vanishing expectation 〈∇(φ, σ)〉 = 0,
of bounded second moment 〈|∇(φ, σ)|2〉 ≤ C(λ),
and a(ei +∇φi) = 〈a(ei +∇φi)〉+∇ · σi.
This yields ∀ 〈·〉-a.e. a limR↑∞1R
(
−∫
BR|(φ, σ)|2
)12 = 0.
Proof a la Papanicolaou & Varadhan
after natural gauge for σi:−△σijk = ∂jσik − ∂kqij where qi = a(ei +∇φi)
Credits
Avellaneda&Lin’87 for periodic adimXα = dimXEucl
α for all α > 0
Benjamini&Copin&Kozma&Yadin’11 stat. & ergod. 〈·〉dimXα = dimXEucl
α for all α ≤ 1
Marahrens&O. ’13 for stationary & mixing 〈·〉:C0,α-theory for all α < 1
Armstrong&Smart ’14 for stationary & mixing 〈·〉:C0,1-theory
Liouville for all α ...
λ|ξ|2 ≤ ξ · aξ ≤ |ξ|2, Xα :={
u a-harm.∣
∣
∣ limR↑∞1Rα
(
−∫
BRu2)12 = 0
}
.
Proposition 2[Fischer & O. ’15]
For i = 1, · · · , d let φi scalar and σi skew be related to a via
a(ei +∇φi) = ahomei +∇ · σi for some matrix ahom.
Suppose∫ ∞
1
dRR
1R
(
−∫
BR|(φ, σ)|2
)12 < ∞.
Then dimXα = dimXEuclα for all α < ∞.
... under slightly quantified sublinear growth
Idea of proof: Xα large enough
Xhomk+
:={
u ahom-harmonic
∣
∣
∣
∣
lim supR↑∞
1Rk
(
−∫
BRu2
)12 < ∞
}
= XEuclk+
Step 1 Xhomk+
/Xhom(k−1)+ ⊂ Xk+/X(k−1)+
For k-homogeneous qhom ∈ Xhomk+
construct q ∈ Xk+ via
q = (1+φi∂i)qhom+
∑∞n=0(wn−p′n) where p′n ∈ X(k−1)+
−∇ · a∇wn = ∇ · (I(B2n/B2n−1)(φia− σi)∇∂iphom).
Goal:
sub-k-growth of w :=∑∞n=0(wn−p′n)
i.e. lim supR↑∞1
Rk−1
(
−∫
BR|∇w|2
)12 = 0
Idea of proof: Xα not too large
Step 2 Xk+/X(k−1)+ ⊂ Xhomk+
/Xhom(k−1)+
preliminary version of intrinsic excess of order k
Exck(u,BR) := infp′∈X(k−1)+,q
hom∈Xhomk+ ,k-hom
(
−∫
BR|∇(u−p′−q)|2
)12
Goal: Excess decay of order (k+1)−:
∀ α < k, a-harm. u Exck(u,Br) . ( rR)αExck(u,BR) for 1 ≪ r ≤ R
Step 3 Buckle under summability assumption:
Growth of w at order k− ↔ excess decay of order (k+1)−
Have canonical Xk+/X(k−1)+∼= Xhom
k+/Xhom
(k−1)+
but no canonical Xk+∼= Xhom
k+.
Slightly quantified sublinear growth ...
Proposition 2” [Fischer & O. ’15]
Let a be centered stationary Gaussian field s.t.
|〈a(x)⊗ a(0)〉| ≤ |x|−β for all x ∈ Rd for some 0 < β ≤ β0(d, λ).
Let Φ be 1-Lipschitz with image in λ-elliptic tensors;
set a(x) := Φ(a(x)).
Then ∃ r∗ = r∗(a) with 〈exp(rβ∗ )〉 ≤ C(d, λ, β) and
1R2−
∫
BR|(φ, σ)|2 ≤ C(d, λ, β)(r∗R)βlog(e+ log R
r∗) for R ≥ r∗.
Proof: estimates on sensitivity ∂∇(φ,σ)(x)∂a(y)
, concentration of measure
cf Armstrong& Mourrat ’14
... under mild decorrelation
Quantitative homogenization theory
from practitioner’s questions to geometric rigidity
Motivation from numerical analysis:
Representative Volume method
Connections with elliptic regularity theory:
generic large-scale regularity
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