An introduction to homogenization Setting Least square formulation A parameter identification problem in stochastic homogenization William Minvielle CERMICS, École des Ponts ParisTech, Matherials research-team, INRIA Rocquencourt william.minvielle@cermics.enpc.fr École Nationale des Ponts et Chaussées – 2014, October 2nd. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Multiscale materials often leads to very expensive computations, and practical difficulties. We consider a simple (linear) problem for a complex materials: � − div [ A ε ( x ) ∇ u ε ( x )] = f ( x ) x ∈ D ⊂⊂ R d , u ε = 0 ∂ D . Airplane wing. Courtesy M. Thomas and EADS William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation u ε = 0 − div ( A ε ( x ) ∇ u ε ) = f in D , on ∂ D Application A ε u ε f Elasticity elastic moduli displacement mechanical load Thermal conductivity thermal conductivity temperature heat source Electrostatics permittivity electric potential charge density Darcy flow flow conductivity pressure sources William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y ) a Z d -periodic matrix field. � x � � ∇ u ε � u ε = 0 − div = f in D , on ∂ D (1) A ε William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y ) a Z d -periodic matrix field. � x � � ∇ u ε � u ε = 0 − div = f in D , on ∂ D (1) A ε This difficult oscillatory problem homogenizes to: u ⋆ = 0 − div ( A ⋆ ∇ u ⋆ ) = f in D , on ∂ D , (2) William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y ) a Z d -periodic matrix field. � x � � ∇ u ε � u ε = 0 − div = f in D , on ∂ D (1) A ε This difficult oscillatory problem homogenizes to: u ⋆ = 0 − div ( A ⋆ ∇ u ⋆ ) = f in D , on ∂ D , (2) The homogenized matrix A ⋆ is defined by an average in the unit cell Q = (0 , 1) d involving so-called correctors functions w : � A ⋆ e j = A ( x ) ( ∇ w e j ( x ) + e j ) dx , (3) Q and the (easy) corrector equation reads: on R d , − div [ A ( ∇ w p + p )] = 0 (4) � ∇ w p periodic, ∇ w p = 0 . Q William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Courtesy M. Thomas and EADS William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y , ω ) a stationary matrix field. � x � � ∇ u ε � u ε = 0 − div A ε , ω = f in D , on ∂ D . William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y , ω ) a stationary matrix field. � x � � ∇ u ε � u ε = 0 − div A ε , ω = f in D , on ∂ D . This difficult oscillatory problem homogenizes to: u ⋆ = 0 − div ( A ⋆ ∇ u ⋆ ) = f in D , on ∂ D , William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y , ω ) a stationary matrix field. � x � � ∇ u ε � u ε = 0 − div A ε , ω = f in D , on ∂ D . This difficult oscillatory problem homogenizes to: u ⋆ = 0 − div ( A ⋆ ∇ u ⋆ ) = f in D , on ∂ D , where A ⋆ is defined by: � E � A ( y , · ) ( ∇ w e j ( y , · ) + e j ) � A ⋆ e j = dy , Q William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation Consider A ( y , ω ) a stationary matrix field. � x � � ∇ u ε � u ε = 0 − div A ε , ω = f in D , on ∂ D . This difficult oscillatory problem homogenizes to: u ⋆ = 0 − div ( A ⋆ ∇ u ⋆ ) = f in D , on ∂ D , where A ⋆ is defined by: � E � A ( y , · ) ( ∇ w e j ( y , · ) + e j ) � A ⋆ e j = dy , Q and the corrector equation, in R d , reads, for any p ∈ R d : in R d a.s. , − div [ A ( ∇ w p + p )] = 0 � ∇ w p stationary, E [ ∇ w p ] = 0 . Q Note that A ⋆ (and hence u ⋆ ) is deterministic. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation In practice, truncate over Q N := (0 , N ) d : − div � A ( ∇ w N p + p ) � w N = 0 in Q N a.s. , Q N − periodic . p � 1 A ( y , ω )( e j + ∇ w N A ⋆ N ( ω ) e j := e j ( y , ω )) dy . | Q N | Q N For that reason alone, randomness comes again in the picture. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation In practice, truncate over Q N := (0 , N ) d : − div � A ( ∇ w N p + p ) � w N = 0 in Q N a.s. , Q N − periodic . p � 1 A ( y , ω )( e j + ∇ w N A ⋆ N ( ω ) e j := e j ( y , ω )) dy . | Q N | Q N For that reason alone, randomness comes again in the picture. In the sequel, we focus on computing E [ A ⋆ N ]. M := 1 � Introduce the estimator I MC A ⋆ N ( ω m ), where ( ω m ) are i.i.d. M M m =1 William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Multiscale materials Setting Truncation Least square formulation In practice, truncate over Q N := (0 , N ) d : − div � A ( ∇ w N p + p ) � w N = 0 in Q N a.s. , Q N − periodic . p � 1 A ( y , ω )( e j + ∇ w N A ⋆ N ( ω ) e j := e j ( y , ω )) dy . | Q N | Q N For that reason alone, randomness comes again in the picture. In the sequel, we focus on computing E [ A ⋆ N ]. M := 1 � Introduce the estimator I MC A ⋆ N ( ω m ), where ( ω m ) are i.i.d. M M m =1 A ⋆ − I MC = A ⋆ − E [ A ⋆ N ] − I MC N ] + E [ A ⋆ (5) M M The bias error is often small. The statistical error is controlled by the variance . Variance reduction approaches are useful to reduce the error. � V ar[ A ⋆ N ] � N ] − I MC � ≤ 1 . 96 � � E [ A ⋆ √ M M F. Legoll and WM A control variate approach based on a defect-type theory for variance reduction in stochastic homogenization , 2014, Submitted. ArXiv 1407.8029 William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation An inverse problem in stochastic homogenization joint work with F. Legoll, A. Obliger, M. Simon. F. Legoll, W.M., A. Obliger, M. Simon. A parameter identification problem in stochastic homogenization, 2014, arXiv 1402.0982. Accepted in ESAIM:ProcS. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation Subsurface modeling (Courtesy PECSA, Paris VI) Diffusion in clay modeled by the so-called Pore Network Model. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation Subsurface modeling (Courtesy PECSA, Paris VI) Diffusion in clay modeled by the so-called Pore Network Model. y x e e 2 e 1 Discrete elliptic equation − div [ A ( x ε , ω ) ∇ u ε ] = f William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation Can we recover some microscopic quantities on the basis of a few macroscopic quantities? William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation Modelling: ◮ Diameters of channel: Weibull law d e ∼ W ( λ, k ) i.i.d. ◮ Conductance: A ( x , ω ) = diag (( d 4 x , x + e j ( ω )) j ∈{ 1 ,..., d } ). Figure 1 : Weibull distributions. William Minvielle An inverse problem in stochastic homogenization
An introduction to homogenization Physics Setting Forward problem Least square formulation Modelling: ◮ Diameters of channel: Weibull law d e ∼ W ( λ, k ) i.i.d. ◮ Conductance: A ( x , ω ) = diag (( d 4 x , x + e j ( ω )) j ∈{ 1 ,..., d } ). Forward problem: given A ( · , ω ), compute ◮ Macroscopic permeability A ⋆ N ( ω ). ◮ Macroscopic variance V ar[ A ⋆ N ]. William Minvielle An inverse problem in stochastic homogenization
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