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 Birth-and-Growth Model ⇔ Temperature Field ⇑ ? 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 (t0, x0) grows with some specified velocity into outward normal direction yielding the grain at time t: Θ(t, t0, x0) The union of all grains at time t is the crystallization phase Θ(t) =

• i

Θ(t, ti, xi)

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 E(δΘ) := E

• lim

r→0

1 |Br|

• Br

δΘ

• 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(δΘ) 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)

• ∂C(x,t)

A(y, s)d(y, s)., 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: ∂ ∂t (ˆ ρˆ cT ) = div (ˆ κ∇T ) + ∂ ∂t (hˆ ρδΘ(t)), in E × R+,(1) ∂T ∂n = ˆ β(T − Tout),

• n ∂E × R+,

(2) T (x, 0) = T 0(x) x ∈ E. (3) ˆ c, ˆ κ, ˆ ρ, h material parameters β heat transfer coefficient on boundary T 0 initial temperature δΘ(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 |Bmeso|

• Bmeso

δΘ(t)(x) ≃ E[δΘ(t)(x)] = V (x, t) ”law of large numbers” on a mesoscale scale

Crystal Growth and Inverse Problems

”law of large numbers” 1 |A|

• A

δΘ(t)(x) = 1 n2

n2

• i=1

1 |A′

i|

• A′

i

δΘ(t)(x) ∼ E 1 |A′

i|

• A′

i

δΘ(t)(x) ∼ V (x, t)

A A’

Crystal Growth and Inverse Problems

Simplified model: Mesoscale Approximation

Equation for averaged temperature on mesoscale ∂ ∂t (cρT) − div.(κ∇T) = ∂ ∂t (hρV (x, t)) in E × [0, tf ](5) ∂T ∂n = β(T − Tout)

• n ∂E × [0, tf ] (6)

T(x, 0) = T0(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 ∈ L2([0, t], H2(E)) for all 0 ≤ t ≤ tf 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) P

• 1

|Bl|

• Bl(x)

δΘ(x)dx − 1 |Bl|

• Bl(x)

V (x, t)dx

• > τ
• ∼ 1

τ ǫ l p ǫ crystallization scale l length scale of mesoscale Full model approximation: Not known

Crystal Growth and Inverse Problems

So far..

Random Crystal Birth-and-Growth Model V , C ⇔ α, γ ⇔ Temperature Field T

Crystal Growth and Inverse Problems

So far..

Random Crystal Birth-and-Growth Model V , C ⇔ α, γ ⇔ Temperature Field T ⇑ 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) − yδ2 + λα − α∗2

s

(9) Noisy data with noise level δ y − yδ ≤ 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 δ2/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 αλ converges to true one with rate O( √ δ)

Crystal Growth and Inverse Problems

So far

✬ ✫ ✩ ✪

Random Crystal Birth-and-Growth Model ⇔ Temperature Field ⇑ α Parameter Identification

Crystal Growth and Inverse Problems

So far

Parameter Id. in Random Environment ⇓

✬ ✫ ✩ ✪

Random Crystal Birth-and-Growth Model ⇔ Temperature Field ⇑ 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(., ωk) → F(α(., ωk)) 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)) = yδ(., 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

Tikhonov Regularization for Stochastic problems

Tikhonov regularization Regularized solution αα(., ω) = argminαF(α) − yδ(., ω)2 + λα − α∗2

s

Due to randomness of data, regularized solution is random variable

Crystal Growth and Inverse Problems

Tikhonov Regularization for Stochastic problems

How to measure error of random variables (resp. distribution): Prokhorov metric, Ky-Fan metric Ky-Fan metric: Distance between two random variables ρk(ξ1, ξ2) := inf{ǫ > 0 | P{ξ1 − ξ2 > ǫ} < ǫ}. If ρk(ξ1, ξ2) is small than the probability that ξ1 − ξ2 is large, is also small

Crystal Growth and Inverse Problems

Tikhonov Regularization for Stochastic problems

Convergence analysis of deterministic case can be ”lifted” to stochastic case Convergence theorem: Theorem Let ρk(y, yδ) ≤ δ, λ(δ) such that λ(δ) → 0 and δ2/λ(δ) → 0. Furthermore let the α∗-minimum norm α†

ω solution be unique in

D(F) ∩ Hs(R), for almost all ω, with s > 3

• 2. Then

ρk(α†

ω, αλ(δ),δ) → 0

as δ → 0 Also: convergence rates possible Regularization parameter is related to noise level

Crystal Growth and Inverse Problems

Special Case: Model error

Include modelling error as noise The mesoscale model is only an approximation to real model True model Mesoscale model Stochastic Temperature Deterministic Temperature Refinement for mesoscale model F(α) = y + modelling error Modelling error is a random variable !

Crystal Growth and Inverse Problems

Special Case: Model error

Stochastic framework of Tikhonov regularization allows to find bounds in the case of random modelling error. e.g. with simplifying assumption ρk(modelling error) ∼ ρk(δΘ(t), V ) Modelling error is determined by approximation error of mesoscale approximation Approximation depends on crystallization scale and on mesoscale (observation scale); e.g. for simplest model ρk(δΘ(t), V ) ∼ ǫ l p

Crystal Growth and Inverse Problems

Scale dependence of regularization parameter

Apply convergence theory for Tikhonov regularization in stochastic framework regularized solution converges to true one as mesoscale approximation error tends to 0. Consequence ⇒ λ has to be chosen depending on scale!

Crystal Growth and Inverse Problems

Summary

Mesoscale approximation allows a useful mathematical treatment

• f direct and inverse problem

Tikhonov Regularization in a stochastic framework is a flexible tool for inverse problems in a random enviroment Analysis can possibly be applied to other fields (?):

Crystal Growth and Inverse Problems

Future Goal: Medical Application

Randomness over individuals ⇓

✬ ✫ ✩ ✪

Angiogenesis

• f

tumors ⇔ Concentration

• f

chemicals, drugs ⇑ ? Coupling

Crystal Growth and Inverse Problems

References

• V. Capasso, H. W. Engl and S. Kindermann

Parameter identification in a random environment exemplified by a multiscale model of crystal growth, SIAM Mulit. Mod. Sim. 7, Nr. 2, pp 814-841 (2008) ...and the references therein

Crystal Growth and Inverse Problems