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Quantitative stochastic homogenization Jean-Christophe Mourrat with S. Armstrong and T. Kuusi CNRS ENS Paris July 27, 2018 Jc Mourrat Quantitative stochastic homogenization Elliptic equations We consider { t u ( a u )

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Quantitative stochastic homogenization

Jean-Christophe Mourrat with S. Armstrong and T. Kuusi

CNRS – ENS Paris

July 27, 2018

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { ∂tu − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

a ∶ Rd → Rd×d

sym

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

a ∶ Rd → Rd×d

sym

Λ−1 ⩽ a(x) ⩽ Λ

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

a ∶ Rd → Rd×d

sym

random Λ−1 ⩽ a(x) ⩽ Λ

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

a ∶ Rd → Rd×d

sym

random Λ−1 ⩽ a(x) ⩽ Λ translation-invariant law

Jc Mourrat Quantitative stochastic homogenization

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Elliptic equations

We consider { − ∇ ⋅ (a∇u) = 0 in U, u = f

  • n ∂U.

a ∶ Rd → Rd×d

sym

random Λ−1 ⩽ a(x) ⩽ Λ translation-invariant law finite range of dependence

Jc Mourrat Quantitative stochastic homogenization

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Coefficients

Jc Mourrat Quantitative stochastic homogenization

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Coefficients

Jc Mourrat Quantitative stochastic homogenization

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Scaling

{ −∇ ⋅ (a(ε−1⋅)∇uε) = 0 in U, uε = f

  • n ∂U.

Jc Mourrat Quantitative stochastic homogenization

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Homogenization

{ −∇ ⋅ (a(ε−1⋅)∇uε) = 0 in U, uε = f

  • n ∂U.

There exists a matrix a s.t. uε

L2

  • →

ε→0 ¯

u, { −∇ ⋅ (a∇¯ u) = 0 in U, ¯ u = f

  • n ∂U.

Jc Mourrat Quantitative stochastic homogenization

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Homogenization

{ −∇ ⋅ (a(ε−1⋅)∇uε) = 0 in U, uε = f

  • n ∂U.

There exists a matrix a s.t. uε

L2

  • →

ε→0 ¯

u, { −∇ ⋅ (a∇¯ u) = 0 in U, ¯ u = f

  • n ∂U.

∇uε ⇀ ∇¯ u, a(ε−1⋅)∇uε ⇀ a∇¯ u.

Jc Mourrat Quantitative stochastic homogenization

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Law of large numbers

A law of large numbers. . .

Jc Mourrat Quantitative stochastic homogenization

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Law of large numbers

A law of large numbers. . . But a ≠ E[a] !

Jc Mourrat Quantitative stochastic homogenization

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Law of large numbers

A law of large numbers. . . But a ≠ E[a] !

Jc Mourrat Quantitative stochastic homogenization

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Numerical approximations

Very interesting result from a computational point of view.

Jc Mourrat Quantitative stochastic homogenization

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Numerical approximations

Very interesting result from a computational point of view. Computation of a and then of ¯ u.

Jc Mourrat Quantitative stochastic homogenization

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Numerical approximations

Very interesting result from a computational point of view. Computation of a and then of ¯ u. Higher-order approximations; approximations in law; CLT.

Jc Mourrat Quantitative stochastic homogenization

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Numerical approximations

Very interesting result from a computational point of view. Computation of a and then of ¯ u. Higher-order approximations; approximations in law; CLT. Efficient algorithms for exact computation at fixed ε.

Jc Mourrat Quantitative stochastic homogenization

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Numerical approximations

Very interesting result from a computational point of view. Computation of a and then of ¯ u. Higher-order approximations; approximations in law; CLT. Efficient algorithms for exact computation at fixed ε. Goal: estimate rates of convergence.

Jc Mourrat Quantitative stochastic homogenization

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Approach

Difficulty:

Jc Mourrat Quantitative stochastic homogenization

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Approach

Difficulty: solutions are non-local, non-linear functions of the coefficient field.

Jc Mourrat Quantitative stochastic homogenization

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Approach

Difficulty: solutions are non-local, non-linear functions of the coefficient field. 1st approach (Gloria, Neukamm, Otto, . . . ): “non-linear” concentration inequalities (cf. also Naddaf-Spencer).

Jc Mourrat Quantitative stochastic homogenization

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Approach

Difficulty: solutions are non-local, non-linear functions of the coefficient field. 1st approach (Gloria, Neukamm, Otto, . . . ): “non-linear” concentration inequalities (cf. also Naddaf-Spencer). 2nd approach (Armstrong, Kuusi, M., Smart, . . . ): renormalization, focus on energy quantities.

Jc Mourrat Quantitative stochastic homogenization

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Motivations

Jc Mourrat Quantitative stochastic homogenization

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Motivations

Prove stronger results

Jc Mourrat Quantitative stochastic homogenization

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Motivations

Prove stronger results Renormalization: very inspiring, broad and powerful idea, with still a lot of potential as a mathematical technique

Jc Mourrat Quantitative stochastic homogenization

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Motivations

Prove stronger results Renormalization: very inspiring, broad and powerful idea, with still a lot of potential as a mathematical technique Develop tools that will hopefully shed light on variety of

  • ther problems: other equations, Gibbs measures,

interacting particle systems, etc.

Jc Mourrat Quantitative stochastic homogenization

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Motivations

Prove stronger results Renormalization: very inspiring, broad and powerful idea, with still a lot of potential as a mathematical technique Develop tools that will hopefully shed light on variety of

  • ther problems: other equations, Gibbs measures,

interacting particle systems, etc. Suggests new numerical algorithms

Jc Mourrat Quantitative stochastic homogenization

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Problem reduction

Jc Mourrat Quantitative stochastic homogenization

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Problem reduction

Jc Mourrat Quantitative stochastic homogenization

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Problem reduction

For p ∈ Rd, write a-harmonic function with slope p as x ↦ p ⋅ x + φp(x), that is, −∇ ⋅ a(p + ∇φp) = 0.

Jc Mourrat Quantitative stochastic homogenization

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Problem reduction

For p ∈ Rd, write a-harmonic function with slope p as x ↦ p ⋅ x + φp(x), that is, −∇ ⋅ a(p + ∇φp) = 0. ∣φp(x)∣ ≪ ∣x∣ ?

Jc Mourrat Quantitative stochastic homogenization

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Problem reduction

For p ∈ Rd, write a-harmonic function with slope p as x ↦ p ⋅ x + φp(x), that is, −∇ ⋅ a(p + ∇φp) = 0. ∣φp(x)∣ ≪ ∣x∣ ? Quantify

  • Spat. av. ∇φp → 0
  • Spat. av. a(p + ∇φp) → ap.

Jc Mourrat Quantitative stochastic homogenization

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Gradual homogenization

If 1 − δ ⩽ a(x) ⩽ 1 + δ, then ∣a − E[a]∣ ⩽ Cδ2.

Jc Mourrat Quantitative stochastic homogenization

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Gradual homogenization

If 1 − δ ⩽ a(x) ⩽ 1 + δ, then ∣a − E[a]∣ ⩽ Cδ2. Gradual homogenization a(x) ↝ ar(x) ↝ a

Jc Mourrat Quantitative stochastic homogenization

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Gradual homogenization

If 1 − δ ⩽ a(x) ⩽ 1 + δ, then ∣a − E[a]∣ ⩽ Cδ2. Gradual homogenization a(x) ↝ ar(x) ↝ a Linearization for r ≫ 1.

Jc Mourrat Quantitative stochastic homogenization

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Energies

Dal Maso, Modica 1986: ν(U,p) ∶= inf

v∈ℓp+H1

0(U)

1 2 ⨏U ∇v ⋅ a∇v.

Jc Mourrat Quantitative stochastic homogenization

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Energies

Dal Maso, Modica 1986: ν(U,p) ∶= inf

v∈ℓp+H1

0(U)

1 2 ⨏U ∇v ⋅ a∇v. U ↦ ν(U,p) is sub-additive.

Jc Mourrat Quantitative stochastic homogenization

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Energies

Dal Maso, Modica 1986: ν(U,p) ∶= inf

v∈ℓp+H1

0(U)

1 2 ⨏U ∇v ⋅ a∇v. U ↦ ν(U,p) is sub-additive. ν(U,p) =∶ 1 2p ⋅ a(U)p.

Jc Mourrat Quantitative stochastic homogenization

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Energies

Dal Maso, Modica 1986: ν(U,p) ∶= inf

v∈ℓp+H1

0(U)

1 2 ⨏U ∇v ⋅ a∇v. U ↦ ν(U,p) is sub-additive. ν(U,p) =∶ 1 2p ⋅ a(U)p. ν(◻,p)

a.s.

  • →

∣◻∣→∞

1 2p ⋅ ap.

Jc Mourrat Quantitative stochastic homogenization

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Coarse-grained coefficients

vp ∶= minimizer for ν(U,p)

Jc Mourrat Quantitative stochastic homogenization

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Coarse-grained coefficients

vp ∶= minimizer for ν(U,p) ⨏U ∇vp = p

Jc Mourrat Quantitative stochastic homogenization

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Coarse-grained coefficients

vp ∶= minimizer for ν(U,p) ⨏U ∇vp = p q ⋅ a(U)p = ⨏U ∇vq ⋅ a∇vp

Jc Mourrat Quantitative stochastic homogenization

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Coarse-grained coefficients

vp ∶= minimizer for ν(U,p) ⨏U ∇vp = p q ⋅ a(U)p = ⨏U ∇vq ⋅ a∇vp a(U)p = ⨏U a∇vp.

Jc Mourrat Quantitative stochastic homogenization

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Strategy

Jc Mourrat Quantitative stochastic homogenization

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Strategy

Get a small rate of convergence: ∃α > 0 s.t. ∣ν(◻,p) − 1 2p ⋅ ap∣ ≲ ∣◻∣−α.

Jc Mourrat Quantitative stochastic homogenization

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Strategy

Get a small rate of convergence: ∃α > 0 s.t. ∣ν(◻,p) − 1 2p ⋅ ap∣ ≲ ∣◻∣−α. Coarse-grained coefficients vary by ±∣◻∣−α, so ∣ν(◻,p) − 2−d ∑

z

ν(z + ◻′,p)∣ ≲ ∣◻∣−2α.

Jc Mourrat Quantitative stochastic homogenization

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Strategy

Get a small rate of convergence: ∃α > 0 s.t. ∣ν(◻,p) − 1 2p ⋅ ap∣ ≲ ∣◻∣−α. Coarse-grained coefficients vary by ±∣◻∣−α, so ∣ν(◻,p) − 2−d ∑

z

ν(z + ◻′,p)∣ ≲ ∣◻∣−2α. Control of fluctuations ∣ν(◻,p) − E[ν(◻,p)]∣ ≲ ∣◻∣−(2α)∧ 1

2. Jc Mourrat Quantitative stochastic homogenization

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Strategy

Get a small rate of convergence: ∃α > 0 s.t. ∣ν(◻,p) − 1 2p ⋅ ap∣ ≲ ∣◻∣−α. Coarse-grained coefficients vary by ±∣◻∣−α, so ∣ν(◻,p) − 2−d ∑

z

ν(z + ◻′,p)∣ ≲ ∣◻∣−2α. Control of fluctuations ∣ν(◻,p) − E[ν(◻,p)]∣ ≲ ∣◻∣−(2α)∧ 1

2.

  • ⇒

Exponent improvement α → (2α) ∧ 1

2.

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

For ◻r ∶= ]− r

2; r 2[ d,

a(x + ◻r) ≃ a + Wr(x), where Wr(x) ≃ N(0,r −d).

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

For ◻r ∶= ]− r

2; r 2[ d,

a(x + ◻r) ≃ a + Wr(x), where Wr(x) ≃ N(0,r −d). Recall that −∇ ⋅ a(p + ∇φp) = 0. For φp,r ∶= φp ⋆ 1◻r ∣◻r∣,

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

For ◻r ∶= ]− r

2; r 2[ d,

a(x + ◻r) ≃ a + Wr(x), where Wr(x) ≃ N(0,r −d). Recall that −∇ ⋅ a(p + ∇φp) = 0. For φp,r ∶= φp ⋆ 1◻r ∣◻r∣, we expect −∇ ⋅ (a + Wr(x))(p + ∇φp,r) ≃ 0.

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

−∇ ⋅ (a + Wr)(p + ∇φp,r) ≃ 0.

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

−∇ ⋅ (a + Wr)(p + ∇φp,r) ≃ 0. −∇ ⋅ (a + Wr)∇φp,r ≃ ∇ ⋅ (Wrp).

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

−∇ ⋅ (a + Wr)(p + ∇φp,r) ≃ 0. −∇ ⋅ (a + Wr)∇φp,r ≃ ∇ ⋅ (Wrp).

  • ⇒

∣∇φp,r∣ ≲ r − d

2 . Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

−∇ ⋅ (a + Wr)(p + ∇φp,r) ≃ 0. −∇ ⋅ (a + Wr)∇φp,r ≃ ∇ ⋅ (Wrp).

  • ⇒

∣∇φp,r∣ ≲ r − d

2 .

−∇ ⋅ a∇φp,r ≃ ∇ ⋅ (Wrp).

Jc Mourrat Quantitative stochastic homogenization

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Corrector estimates

−∇ ⋅ (a + Wr)(p + ∇φp,r) ≃ 0. −∇ ⋅ (a + Wr)∇φp,r ≃ ∇ ⋅ (Wrp).

  • ⇒

∣∇φp,r∣ ≲ r − d

2 .

−∇ ⋅ a∇φp,r ≃ ∇ ⋅ (Wrp).

  • ⇒

r

d 2 (∇φp)(r ⋅)

law

  • →

r→∞ ∇(GFF).

Jc Mourrat Quantitative stochastic homogenization

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Correctors

Jc Mourrat Quantitative stochastic homogenization

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Correctors

Jc Mourrat Quantitative stochastic homogenization

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GFF −∇ ⋅ a∇Φ = ∇ ⋅ W

Jc Mourrat Quantitative stochastic homogenization

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GFF −∇ ⋅ a∇Φ = ∇ ⋅ W

Jc Mourrat Quantitative stochastic homogenization

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Perspectives and dreams

Optimal error estimates, next-order information, new numerical algorithms

Jc Mourrat Quantitative stochastic homogenization

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Perspectives and dreams

Optimal error estimates, next-order information, new numerical algorithms Expand the reach of rigorous renormalization techniques

Jc Mourrat Quantitative stochastic homogenization

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Perspectives and dreams

Optimal error estimates, next-order information, new numerical algorithms Expand the reach of rigorous renormalization techniques New tools to attack other models, e.g. other equations, gradient Gibbs measures, interacting particle systems, . . .

Jc Mourrat Quantitative stochastic homogenization

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Perspectives and dreams

Optimal error estimates, next-order information, new numerical algorithms Expand the reach of rigorous renormalization techniques New tools to attack other models, e.g. other equations, gradient Gibbs measures, interacting particle systems, . . .

Jc Mourrat Quantitative stochastic homogenization

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Perspectives and dreams

Optimal error estimates, next-order information, new numerical algorithms Expand the reach of rigorous renormalization techniques New tools to attack other models, e.g. other equations, gradient Gibbs measures, interacting particle systems, . . . Check out our book!

Jc Mourrat Quantitative stochastic homogenization

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Thank you!

Jc Mourrat Quantitative stochastic homogenization