le¸cons de math´ematiques jacques-louis lions effective
TRANSCRIPT
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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
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Effective behavior of media
under statistical/incomplete information:
Early explicit approximate treatment,
recent numerical applications
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Maxwell: Effective resistance of a composite
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Einstein: Effective viscosity of a suspension
→ F → F
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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
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A crucial influence of J.-L. Lions’ school ...
source: Conference 60th birthday
source: Academie des Sciences,
Institut de France
... on “homogenization”
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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
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Stochastic homogenization:
The Representative Volume method
and its numerical analysis
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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
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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 ?
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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
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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
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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
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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
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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
)
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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
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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
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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?
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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
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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
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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
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Stochastic homogenization:
A regularizing effect of randomness on large scales
captured by Liouville properties of all order
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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?
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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
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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 ?
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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+.
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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
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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