Crystal Growth and Inverse Problems in Random Environments Stefan - PowerPoint PPT Presentation

Crystal Growth and Inverse Problems in Random Environments Stefan Kindermann, Industrial Mathematics Institute Johannes Kepler University Linz, Austria joint work with Vincent Capasso, University of Milano Heinz W. Engl, University of Linz, (RICAM) Crystal Growth and Inverse Problems

Outline Parameter Id. in Random Environment ✬ ✩ ⇓ Random Crystal ⇔ Temperature Field Birth-and-Growth Model ⇑ ? ✫ ✪ Parameter Identification Crystal Growth and Inverse Problems

Crystal Growth on Microscale Birth-and-Growth Model Birth: At some random time t and some random location x ∈ Ω an infinitesimal small nucleus is born Growth: Each nucleus born at ( t 0 , x 0 ) grows with some specified velocity into outward normal direction yielding the grain at time t : Θ( t , t 0 , x 0 ) The union of all grains at time t is the crystallization phase � Θ( t ) = Θ( t , t i , x i ) i Crystal Growth and Inverse Problems

Crystal Growth: Model Poisson Birth Model P { Nucleus is born in region [ x + dx ] × [ t + dt ] } = A ( x , t ) dxdt Growth Model Each grain grows with normal velocity G ( x , t ) Crystal Growth and Inverse Problems

RACS, mean geometric density Θ( t ) is a ”set valued random variable” Theory of Choquet-Matheron allows to define a probability measure for such objects ⇒ Random closed sets (RACS). Mean geometric density � 1 � � E ( δ Θ ) := E lim δ Θ | B r | r → 0 B r is well defined for sufficiently regular RACS δ is characteristic function Crystal Growth and Inverse Problems

Mean geometric quantities If Θ is the crystalline phase, then the corresponding mean geometric density is the local degree of crystallinity V ( x , t ) := E ( δ Θ ) Expected value of relative volume of phase (locally) V ( x , t ) is a deterministic function Crystal Growth and Inverse Problems

Final PDE Model for Birth and Growth Final equations for evolution of mean geometric densities ∂ � ∂ t V ( x , t ) = (1 − V ( x , t )) G ( x , t ) A ( y , s ) d ( y , s ) ., ∂ C ( x , t ) Causal cone C ( x , t ) is determined by arrival time equation ∂ ∂ t ψ = G |∇ ψ | Under assumptions on G boundary of C ( x , t ) is well defined, Solution to equations exists in a viscosity sense and is unique. Analysis [Burger, Capasso, Pizzochero 06] Birth and growth PDE: Input G , A Output: V ( x , t ) Crystal Growth and Inverse Problems

Coupling with Temperature Crystallization affects temperature due to the release of latent heat. Weak coupling Crystallization ⇒ Temperature Equation for temperature: ∂ κ ∇T ) + ∂ in E × R + , (1) ∂ t (ˆ ρ ˆ c T ) = div (ˆ ∂ t ( h ˆ ρδ Θ( t ) ) , ∂ T ˆ on ∂ E × R + , = β ( T − T out ) , (2) ∂ n T 0 ( x ) T ( x , 0) = x ∈ E . (3) ˆ c , ˆ κ, ˆ ρ, h material parameters β heat transfer coefficient on boundary T 0 initial temperature δ Θ( t ) indicator function of the phase (heat source) Crystal Growth and Inverse Problems

More Difficulties - Strong Coupling Strong Coupling: Temperature also affects crystallization Crystallization ⇒ Temperature Crystallization ⇐ Temperature A ( x , t ) = α ( T ( x , t )) G ( x , t ) = γ ( T ( x , t )) , (4) Problems: G is random ⇒ mean geometric quantities are not well defined ?! ⇒ Urgent need for a simplied model Crystal Growth and Inverse Problems

Simplified model: Mesoscale Approximation Mesoscale: 1 large compared to crystal size 2 small compared to scale of temperature fluctuation (1) ⇒ Mesoscale approximation: Replace phase by local degree of crystallinity 1 � δ Θ( t ) ( x ) ≃ E [ δ Θ( t ) ( x )] = V ( x , t ) | B meso | B meso ”law of large numbers” on a mesoscale scale Crystal Growth and Inverse Problems

”law of large numbers” n 2 1 1 1 � � � δ Θ( t ) ( x ) = δ Θ( t ) ( x ) n 2 | A ′ | A | i | A A ′ i =1 i E 1 � ∼ δ Θ( t ) ( x ) ∼ V ( x , t ) | A ′ i | A ′ i A’ A Crystal Growth and Inverse Problems

Simplified model: Mesoscale Approximation Equation for averaged temperature on mesoscale ∂ ∂ ∂ t ( c ρ T ) − div . ( κ ∇ T ) = ∂ t ( h ρ V ( x , t )) in E × [0 , t f ](5) ∂ T = β ( T − T out ) on ∂ E × [0 , t f ] (6) ∂ n T ( x , 0) = T 0 ( x ) x in E (7) deterministic equation with deterministic coefficients Crystal Growth and Inverse Problems

Simplified model: Mesoscale Approximation (2) ⇒ (on mesoscale temperature is almost constant) (2) ⇒ Strong coupling model on mesoscale A ( x , t ) = α ( T ( x , t )) G ( x , t ) = γ ( T ( x , t ))) , (8) All equations only involve deterministic quantities T is averaged temperature on mesoscale Crystal Growth and Inverse Problems

Existence and Uniqueness for Mesoscale Approximation Assumptions: Coefficients c , ρ, κ, β, h and G ( x , t ) independent of T , V smooth geometry and parameters + standard positivity assumptions on c , ρβ Dimension n ≤ 3, α ∈ W 1 , ∞ ( R ) Theorem Under these assumptions there exists a unique T ∈ L 2 ([0 , t ] , H 2 ( E )) for all 0 ≤ t ≤ t f to coupled system. Proof by a fixed point argument by contraction Crystal Growth and Inverse Problems

Estimates for Mesoscale Approximation Does mesoscale model approximates true one ? Only partial answers [Burger, Capasso, Pizzochero 06]: Without coupling (and simplifying assumptions) �� � � 1 � 1 � ∼ 1 � ǫ � p � � δ Θ ( x ) dx − V ( x , t ) dx � > τ P � � � | B l | | B l | � τ l B l ( x ) B l ( x ) � ǫ crystallization scale l length scale of mesoscale Full model approximation: Not known Crystal Growth and Inverse Problems

So far.. Random Crystal Temperature Field Birth-and-Growth ⇔ α, γ ⇔ T Model V , C Crystal Growth and Inverse Problems

So far.. Random Crystal Temperature Field Birth-and-Growth ⇔ α, γ ⇔ T Model V , C ⇑ α , ( γ ) Parameter Identification Crystal Growth and Inverse Problems

Inverse Problems Forward Problem: Given the coupling functions, α , γ , find Temperature for mesoscale equation (solve coupled system of PDEs). Inverse Problem: Given partial information on temperature, find coupling function α , in mesoscale equations Crystal Growth and Inverse Problems

Inverse Problems as operator equation Solve F ( α ) = y F is forward operator F : α → solution of coupled mesoscale equation for T on boundary y are data y = T | ∂ E × [0 , T ] + error Deterministic inverse problem Crystal Growth and Inverse Problems

Tikhonov Regularization Approximate solution by minimizing Tikhonov functional α → J ( α ) := � F ( α ) − y δ � 2 + λ � α − α ∗ � 2 (9) s Noisy data with noise level δ � y − y δ � ≤ δ, (10) Crystal Growth and Inverse Problems

Result for Tikhonov Regularization If Regularization norm s > 3 2 then J ( α ) has a global minimizer Theorem If s > 3 2 and standard assumptions hold, then Tikhonov regularization is a a convergent regularization method if regularization parameter is chose appropriately: λ → 0 and δ 2 /λ → 0 , i.e regularized solution converges to true one. Theorem If s > 5 2 F is Frechet-differentiable and if a α satisfies source condition, and λ ∼ δ then the approximate solution α λ converges √ to true one with rate O ( δ ) Crystal Growth and Inverse Problems

So far ✬ ✩ Random Crystal ⇔ Temperature Field Birth-and-Growth Model ⇑ α ✫ ✪ Parameter Identification Crystal Growth and Inverse Problems

So far Parameter Id. in Random Environment ✬ ✩ ⇓ Random Crystal ⇔ Temperature Field Birth-and-Growth Model ⇑ α ✫ ✪ Parameter Identification Crystal Growth and Inverse Problems

Parameter Identification in Random Environments Reintroducing stochasticity due to environments: Random fluctuation in parameter due to random environments aka double stochasticity Idea: parameters of crystallization and/or coupling are not fixed functions but depend on external and experimental settings. Model: parameters are (Hilbert-space valued) random variables α = α ( ., ω ) A ( x , t , ω ) = α ( T ( x , t ) , ω ) ω randomness due to uncontrollable factors Crystal Growth and Inverse Problems

Parameter Identification in Random Environments Analysis of strongly coupled mesoscale model can be extended to doubly stochastic case: If α ( ., ω ) is sufficiently smooth almost surely, ⇒ birth-and-growth process conditioned on a realization of α = α ( ., ω ) ⇒ Well defined operator F ( α ( ., ω )) Crystal Growth and Inverse Problems

Parameter Identification in Random Environments Experimental setting: Parameter for experiment k → Outcome α ( ., ω k ) → F ( α ( ., ω k )) k = 1 , . . . n Data are samples out of a distribution Inverse Problem in a stochastic framework : Wanted Given Random Distribution of Parameter ⇐ Distribution of Data Crystal Growth and Inverse Problems

Parameter Identification in Random Environments Abstract formulation: F ( α ( ., ω )) = y δ ( ., ω ) Operator equation for random variables in Hilbert spaces Tikhonov regularization in a stochastic setup: [Engl, Kindermann, Hofinger 05] [Hofinger (PhD-Thesis) 06] Crystal Growth and Inverse Problems