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