Quantitative estimates in stochastic homogenization Stefan Neukamm - PowerPoint PPT Presentation

Quantitative estimates in stochastic homogenization Stefan Neukamm Max Planck Institute for Mathematics in the Sciences joint work with Antoine Gloria and Felix Otto RDS 2012 – Bielefeld

Motivation: Effective large scale behavior of random media – description by statistics – effective large scale behavior � stochastic homogenization – qualitative theory � well-established � formula for effective properties In practice: Evaluation of formula requires approximation – only few results; non-optimal estimates for approximation error – lack of understanding on very basic level 1/32

Motivation: Effective large scale behavior of random media – description by statistics – effective large scale behavior � stochastic homogenization – qualitative theory � well-established � formula for effective properties In practice: Evaluation of formula requires approximation – only few results; non-optimal estimates for approximation error – lack of understanding on very basic level Our motivation: Quantitative methods leading to optimal estimates ... model problem: linear, elliptic, scalar, on Z d 1/32

Summary ◮ Framework: discrete elliptic equation with random coefficients ◮ Qualitative homogenization ◮ Homogenization formula and corrector – periodic case ◮ Corrector equation in probality space ◮ Main results ◮ A decay estimate for a diffusion semigroup

Discrete elliptic equation with random coefficients

b b b b b b b b b b b b b b b b b b b b b b b b b Discrete elliptic equation with random coefficients Z d ∇ ∗ a ( x ) ∇ u ( x ) = f ( x ) Coefficient field a : Z d → R d × d diag , λ 0 < λ � a ( x ) � 1 (uniform ellipticity) x 2/32

b b b b b b b b b b b b b b b b b b b b b b b b b Discrete elliptic equation with random coefficients Z d ∇ ∗ a ( x ) ∇ u ( x ) = f ( x ) Coefficient field a : Z d → R d × d diag , λ 0 < λ � a ( x ) � 1 (uniform ellipticity) x Lattice Z d sites x , y , coord. directions e 1 , . . . , e d Gradient ∇ ∇ u = ( ∇ 1 u , . . . , ∇ d u ) , ∇ i u ( x ) = u ( x + e i ) − u ( x ) (negative) Divergence ∇ ∗ ( = ℓ 2 -adjoint of ∇ ) ∇ ∗ g = ∇ ∗ 1 g 1 + . . . + ∇ ∗ d g d , ∇ ∗ i g i ( x ) = g ( x − e i ) − g ( x ) . 2/32

b b b b b b b b b b b b b b b b b b b b b b b b b Discrete elliptic equation with random coefficients Z d ∇ ∗ a ( x ) ∇ u ( x ) = f ( x ) Coefficient field a : Z d → R d × d diag , λ 0 < λ � a ( x ) � 1 (uniform ellipticity) x Random coefficients diag , λ ) ( Z d ) ( R d × d := Ω = space of coefficient fields �·� = probability measure on Ω = ”the ensemble” Behavior in the large � stochastic homogenization 3/32

Simplest setting: { a ( x ) } x ∈ Z d are i ndependent and i dentically d istributed according to a random variable A Most general setting: �·� is stationary and ergodic Stationarity: ∀ z ∈ Z d : a ( · ) and a ( · + z ) have same distribution a ( · + z ) a shift by z z Z d Z d Ergodicity: If ∀ z ∈ Z d F ( a ( · + z )) = F ( a ) then F = � F � a. s. 4/32

Qualitative homogenization

Numerical simulation - 1d, Dirichlet problem ∇ ∗ a ( x ) ∇ u ( x ) = 1 , x ∈ ( 0 , L ) ∩ Z , L ≫ 1 u ( 0 ) = u ( L ) = 0 statistics of a i ndependent, i dentically, d istributed uniformly in ( 0 . 2 , 1 ) L = 50 a a 5/32

Numerical simulation - 1d, Dirichlet problem ∇ ∗ a ( x ) ∇ u ( x ) = 1 , x ∈ ( 0 , L ) ∩ Z , L ≫ 1 u ( 0 ) = u ( L ) = 0 statistics of a i ndependent, i dentically, d istributed uniformly in ( 0 . 2 , 1 ) L = 50 a a a 5/32

Numerical simulation - 1d, Dirichlet problem ∇ ∗ a ( x ) ∇ u ( x ) = 1 , x ∈ ( 0 , L ) ∩ Z , L ≫ 1 u ( 0 ) = u ( L ) = 0 statistics of a i ndependent, i dentically, d istributed uniformly in ( 0 . 2 , 1 ) L = 50 a a a a 5/32

L = 100 L = 500 L = 2000 6/32

L = 100 L = 500 a hom = � a − 1 � − 1 − ∇ · a hom ∇ u = 1 L = 2000 u ( 0 ) = u ( 1 ) = 0 6/32

Qualitative homogenization result Kozlov [’79], Papanicolaou & Varadhan [’79] Suppose �·� is stationary & ergodic . Then: ∃ unique a hom ∈ R d × d sym such that: x ) consider right-hand side f L ( x ) = L − 2 f 0 ( x L ) , x ∈ Z d Given f 0 ( ˆ � ∇ ∗ a ( x ) ∇ u L = f L in x ∈ ([− 0 , L ) ∩ Z ) d Solve discrete : outside ([ 0 , L ) ∩ Z ) d Dirichlet problem u L = 0 � in x ∈ [ 0 , 1 ) d − ∇ · a hom ∇ u 0 = f 0 Solve continuum : outside [ 0 , 1 ) d Dirichlet problem u 0 = 0 Then lim L ↑ ∞ u L ( L ˆ x ) = u 0 ( ˆ x ) almost surely. 7/32

Motivation of this talk: approximation of a hom ...requires quantitative estimates for corrector problem 8/32

Motivation of this talk: approximation of a hom ...requires quantitative estimates for corrector problem Related, but different: – homogenization error , i.e. for | u L ( L · ) − u 0 ( · ) | (Naddaf et al., Conlon et al., ...) – correlation function in Euclidean field theory (Naddaf/Spencer, Giacomin/Olla/Spohn,...) 8/32

Formula for a hom

Formula for a hom — the periodic case — Let �·� L be stationary and concentrated on L -periodic coefficients : ∀ z ∈ Z d a ( · + Lz ) = a ( · ) a. s.

Formula for a hom — the periodic case — Let �·� L be stationary and concentrated on L -periodic coefficients : ∀ z ∈ Z d a ( · + Lz ) = a ( · ) a. s. We may think about the L -periodic ensemble �·� L as a periodic approximation of the stationary and ergodic ensemble �·� .

Definition of a hom , L = a hom , L ( a ) a hom , L e := L − d � ∀ e ∈ R d : x ∈ [ 0 , L ) d a ( x )( e + ∇ ϕ ( x )) where ϕ ( · ) = ϕ ( a , · ) is the L -periodic (mean-free) solution to ∇ ∗ a ( x )( e + ∇ ϕ ( x )) = 0 x ∈ [ 0 , L ) d 10/32

Definition of a hom , L = a hom , L ( a ) a hom , L e := L − d � ∀ e ∈ R d : x ∈ [ 0 , L ) d a ( x )( e + ∇ ϕ ( x )) where ϕ ( · ) = ϕ ( a , · ) is the L -periodic (mean-free) solution to ∇ ∗ a ( x )( e + ∇ ϕ ( x )) = 0 x ∈ [ 0 , L ) d ϕ is called the corrector associated with a and e 10/32

Definition of a hom , L = a hom , L ( a ) a hom , L e := L − d � ∀ e ∈ R d : x ∈ [ 0 , L ) d a ( x )( e + ∇ ϕ ( x )) where ϕ ( · ) = ϕ ( a , · ) is the L -periodic (mean-free) solution to ∇ ∗ a ( x )( e + ∇ ϕ ( x )) = 0 x ∈ [ 0 , L ) d ϕ is called the corrector associated with a and e ◮ existence and uniqueness by Poincar´ e’s inequality : � x ∈ [ 0 , L ) d | ϕ ( x ) | 2 � L 2 � x ∈ [ 0 , L ) d | ∇ ϕ ( x ) | 2 ◮ stationarity : ϕ ( a ( · + z ) , · ) = ϕ ( a , · + z ) for all z ∈ Z d a.s. 10/32

b b b b b b b b b b b b b b b b b b b b b b b b b Intuition of a hom , L : Given e ∈ R d and associated ϕ , e a hom e consider u L ( x ) := e · x + ϕ ( x ) . Then ∇ ∗ a ∇ u L = 0 � average gradient = L − d ∇ u L ( x ) = e x ∈ [ 0 , L ) d � average flux = L − d a ( x ) ∇ u L ( x ) = a hom , L e x ∈ [ 0 , L ) d 11/32

Formal passage L ↑ ∞ yields: Def. for stationary corrector ϕ = ϕ ( a , x ) for �·� defined by (i) corrector equation for all x ∈ Z d a.e. a ∈ Ω ∇ ∗ a ( x )( e + ∇ ϕ ( a , x )) = 0 (ii) sublinear growth on average � 2 = 0 . L ↑ ∞ L − d � � � L − 1 ϕ ( a , x ) � lim [ 0 , L ) d (iii) stationarity 12/32

Formal passage L ↑ ∞ yields: Def. for stationary corrector ϕ = ϕ ( a , x ) for �·� defined by (i) corrector equation for all x ∈ Z d a.e. a ∈ Ω ∇ ∗ a ( x )( e + ∇ ϕ ( a , x )) = 0 (ii) sublinear growth on average � 2 = 0 . L ↑ ∞ L − d � � L − 1 ϕ ( a , x ) � � lim [ 0 , L ) d (iii) stationarity Def. for homogenized coefficient matrix L ↑ ∞ L − d � ergodicity a hom e = lim [ 0 , L ) d a ( e + ∇ ϕ ) = � a ( e + ∇ ϕ ) � 12/32

Can we directly get existence of stationary corrector for �·� from existence of periodic corrector by limit L ↑ ∞ ? 13/32

Can we directly get existence of stationary corrector for �·� from existence of periodic corrector by limit L ↑ ∞ ? No , since Poincar´ e’s inequality degenerates for L ↑ ∞ : � x ∈ [ 0 , L ) d | ϕ ( x ) | 2 � L 2 � x ∈ [ 0 , L ) d | ∇ ϕ ( x ) | 2 13/32

Can we directly get existence of stationary corrector for �·� from existence of periodic corrector by limit L ↑ ∞ ? No , since Poincar´ e’s inequality degenerates for L ↑ ∞ : � x ∈ [ 0 , L ) d | ϕ ( x ) | 2 � L 2 � x ∈ [ 0 , L ) d | ∇ ϕ ( x ) | 2 In fact, for d � 2 stationary correctors in general do not exist ! 13/32

The corrector equation in L 2 �·� D ∗ a ( 0 )( e + D φ ) = 0 .

From Z d to Ω by stationarity Def.: A random field f ( a , x ) is called stationary , if ∀ x , z , a f ( a ( · + z ) , x ) = f ( a , x + z ) . Def.: The stationary extension of a random variable F ( a ) is defined by F ( a , x ) := F ( a ( · + x )) . a ( · + z ) a shift by z Z d Z d z 14/32