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 ) = 0 in U , u = f on ∂ U . Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . a ∶ R d → R d × d sym Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . a ∶ R d → R d × d sym Λ − 1 ⩽ a ( x ) ⩽ Λ Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . a ∶ R d → R d × d random sym Λ − 1 ⩽ a ( x ) ⩽ Λ Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . a ∶ R d → R d × d random sym Λ − 1 ⩽ a ( x ) ⩽ Λ translation-invariant law Jc Mourrat Quantitative stochastic homogenization
Elliptic equations We consider { − ∇ ⋅ ( a ∇ u ) = 0 in U , u = f on ∂ U . a ∶ R d → R d × d random sym Λ − 1 ⩽ a ( x ) ⩽ Λ translation-invariant law finite range of dependence Jc Mourrat Quantitative stochastic homogenization
Coefficients Jc Mourrat Quantitative stochastic homogenization
Coefficients Jc Mourrat Quantitative stochastic homogenization
Scaling { −∇ ⋅ ( a ( ε − 1 ⋅)∇ u ε ) = 0 in U , u ε = f on ∂ U . Jc Mourrat Quantitative stochastic homogenization
Homogenization { −∇ ⋅ ( a ( ε − 1 ⋅)∇ u ε ) = 0 in U , u ε = f on ∂ U . There exists a matrix a s.t. � � L 2 u ε ε → 0 ¯ u , → { −∇ ⋅ ( a ∇ ¯ u ) = 0 in U , u = f ¯ on ∂ U . Jc Mourrat Quantitative stochastic homogenization
Homogenization { −∇ ⋅ ( a ( ε − 1 ⋅)∇ u ε ) = 0 in U , u ε = f on ∂ U . There exists a matrix a s.t. � � L 2 u ε ε → 0 ¯ u , → { −∇ ⋅ ( a ∇ ¯ u ) = 0 in U , u = f ¯ on ∂ U . ∇ u ε ⇀ ∇ ¯ a ( ε − 1 ⋅)∇ u ε ⇀ a ∇ ¯ u , u . Jc Mourrat Quantitative stochastic homogenization
Law of large numbers A law of large numbers. . . Jc Mourrat Quantitative stochastic homogenization
Law of large numbers A law of large numbers. . . But a ≠ E [ a ] ! Jc Mourrat Quantitative stochastic homogenization
Law of large numbers A law of large numbers. . . But a ≠ E [ a ] ! Jc Mourrat Quantitative stochastic homogenization
Numerical approximations Very interesting result from a computational point of view. Jc Mourrat Quantitative stochastic homogenization
Numerical approximations Very interesting result from a computational point of view. Computation of a and then of ¯ u . Jc Mourrat Quantitative stochastic homogenization
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
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
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
Approach Difficulty: Jc Mourrat Quantitative stochastic homogenization
Approach Difficulty: solutions are non-local, non-linear functions of the coefficient field. Jc Mourrat Quantitative stochastic homogenization
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
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
Motivations Jc Mourrat Quantitative stochastic homogenization
Motivations Prove stronger results Jc Mourrat Quantitative stochastic homogenization
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
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 other problems: other equations, Gibbs measures, interacting particle systems, etc. Jc Mourrat Quantitative stochastic homogenization
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 other problems: other equations, Gibbs measures, interacting particle systems, etc. Suggests new numerical algorithms Jc Mourrat Quantitative stochastic homogenization
Problem reduction Jc Mourrat Quantitative stochastic homogenization
Problem reduction Jc Mourrat Quantitative stochastic homogenization
Problem reduction For p ∈ R d , write a -harmonic function with slope p as x ↦ p ⋅ x + φ p ( x ) , that is, −∇ ⋅ a ( p + ∇ φ p ) = 0 . Jc Mourrat Quantitative stochastic homogenization
Problem reduction For p ∈ R d , write a -harmonic function with slope p as x ↦ p ⋅ x + φ p ( x ) , that is, −∇ ⋅ a ( p + ∇ φ p ) = 0 . ∣ φ p ( x )∣ ≪ ∣ x ∣ ? Jc Mourrat Quantitative stochastic homogenization
Problem reduction For p ∈ R d , write a -harmonic function with slope p as x ↦ p ⋅ x + φ p ( x ) , that is, −∇ ⋅ a ( p + ∇ φ p ) = 0 . ∣ φ p ( x )∣ ≪ ∣ x ∣ ? Quantify Spat. av. ∇ φ p → 0 Spat. av. a ( p + ∇ φ p ) → a p . Jc Mourrat Quantitative stochastic homogenization
Gradual homogenization 1 − δ ⩽ a ( x ) ⩽ 1 + δ, If ∣ a − E [ a ]∣ ⩽ C δ 2 . then Jc Mourrat Quantitative stochastic homogenization
Gradual homogenization 1 − δ ⩽ a ( x ) ⩽ 1 + δ, If ∣ a − E [ a ]∣ ⩽ C δ 2 . then a ( x ) ↝ a r ( x ) ↝ a Gradual homogenization Jc Mourrat Quantitative stochastic homogenization
Gradual homogenization 1 − δ ⩽ a ( x ) ⩽ 1 + δ, If ∣ a − E [ a ]∣ ⩽ C δ 2 . then a ( x ) ↝ a r ( x ) ↝ a Gradual homogenization Linearization for r ≫ 1 . Jc Mourrat Quantitative stochastic homogenization
Energies Dal Maso, Modica 1986: ν ( U , p ) ∶= 2 ⨏ U ∇ v ⋅ a ∇ v . 1 inf v ∈ ℓ p + H 1 0 ( U ) Jc Mourrat Quantitative stochastic homogenization
Energies Dal Maso, Modica 1986: ν ( U , p ) ∶= 2 ⨏ U ∇ v ⋅ a ∇ v . 1 inf v ∈ ℓ p + H 1 0 ( U ) U ↦ ν ( U , p ) is sub-additive. Jc Mourrat Quantitative stochastic homogenization
Energies Dal Maso, Modica 1986: ν ( U , p ) ∶= 2 ⨏ U ∇ v ⋅ a ∇ v . 1 inf v ∈ ℓ p + H 1 0 ( U ) U ↦ ν ( U , p ) is sub-additive. ν ( U , p ) = ∶ 1 2 p ⋅ a ( U ) p . Jc Mourrat Quantitative stochastic homogenization
Energies Dal Maso, Modica 1986: ν ( U , p ) ∶= 2 ⨏ U ∇ v ⋅ a ∇ v . 1 inf v ∈ ℓ p + H 1 0 ( U ) U ↦ ν ( U , p ) is sub-additive. ν ( U , p ) = ∶ 1 2 p ⋅ a ( U ) p . ν ( ◻ , p ) � � � 2 p ⋅ a p . 1 a.s. → ∣◻∣→∞ Jc Mourrat Quantitative stochastic homogenization
Coarse-grained coefficients v p ∶ = minimizer for ν ( U , p ) Jc Mourrat Quantitative stochastic homogenization
Coarse-grained coefficients v p ∶ = minimizer for ν ( U , p ) ⨏ U ∇ v p = p Jc Mourrat Quantitative stochastic homogenization
Coarse-grained coefficients v p ∶ = minimizer for ν ( U , p ) ⨏ U ∇ v p = p q ⋅ a ( U ) p = ⨏ U ∇ v q ⋅ a ∇ v p Jc Mourrat Quantitative stochastic homogenization
Coarse-grained coefficients v p ∶ = minimizer for ν ( U , p ) ⨏ U ∇ v p = p q ⋅ a ( U ) p = ⨏ U ∇ v q ⋅ a ∇ v p a ( U ) p = ⨏ U a ∇ v p . Jc Mourrat Quantitative stochastic homogenization
Strategy Jc Mourrat Quantitative stochastic homogenization
Strategy Get a small rate of convergence: ∃ α > 0 s.t. ∣ ν ( ◻ , p ) − 1 2 p ⋅ a p ∣ ≲ ∣ ◻ ∣ − α . Jc Mourrat Quantitative stochastic homogenization
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