computational stochastic phase-field

Some computational aspects of stochastic phase-fieldmodels

Omar Lakkis

Mathematics – University of Sussex – Brighton, England UK

based on joint work withM.Katsoulakis, G.Kossioris & M.Romito

17 November 2009

3rdWorkshopOnRandomDynamicalSystems

O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 1 / 62

Outline

1 Motivation

2 Background

3 White noise

4 Numerical method

5 Convergence

6 Benchmarking

7 Conclusions

8 References


Motivation

Dendrites

A real-life dendrite lab picture: A phase-field simulation:


Dendrites

Computed dendrites without ther-mal noise [Nestler et al., 2005]

A computed dendrite with thermalnoise [Nestler et al., 2005]


Deterministic Phase Fieldfollowing [Chen, 1994]

Phase transition in solidification process

αε∂tu− ε∆u+ (u3 − u)/ε = σw (“interface” motion)

c∂tw −∆w = −∂tu (diffusion in bulk) ,

where

u : order parameter/phase field

≈ 1 solid phase,

∈ (−1 + δε, 1− δε) , interface (region)

≈ −1 liquid phase.

w : temperature in liquid/solid bulk


The Stochastic (or Noisy) Allen-Cahn Problem

is the following semilinear parabolic stochastic PDE with additive whitenoise

∂tu(x, t)−∆u(x, t) + fε(u(x, t)) = εγ∂xtW (x, t), x ∈ (0, 1), t ∈ R+.

u(x, 0) = u0(x), x ∈ (0, 1)

∂xu(0, t) = ∂xu(1, t) = 0, t ∈ R+.

f(u)

10−1 u

∫ u

0f

fε(ξ) =1

ε2(ξ3 − ξ

): first

derivative of double wellpotentialε ∈ R

+: “diffuse inter-face” parameterγ ∈ R: noise intensity pa-rameter∂xtW : space-time whitenoise


Why do we care?about the (deterministic) Allen–Cahn

∂tu−∆u+1

ε2(u3 − u

)= 0

Phase-separation models in metallurgy [Allen and Cahn, 1979].

Simplest model of more complicated class [Cahn and Hilliard, 1958].

Cubic nonlinearity approximation of “harder” (logarithmic) potential.

Double obstacle can replace cubic by other.

Phase-field models of phase separation and geometric motions[Rubinstein et al., 1989], [Evans et al., 1992], [Chen, 1994],[de Mottoni and Schatzman, 1995], . . . ;

Metastability, exponentially slow motion in d = 1,[Carr and Pego, 1989], [Fusco and Hale, 1989],[Bronsard and Kohn, 1990];


Why do we care?about the stochastic Allen–Cahn

∂tu−∆u+1

ε2(u3 − u

)= εγ∂xtW

Noise = stabilizing/destabilizing mechanism [Brassesco et al., 1995],[Funaki, 1995].

Stochastic 1d: Basic existence theory [Faris and Jona-Lasinio, 1982].

Stochastic MCF of interfaces colored space-time/time-only noisespossible [Souganidis and Yip, 2004], [Funaki, 1999],[Dirr et al., 2001].

Rigorous mathematical setting to noise-induced dendrites.


Why do we care?Modeling and computation with stochastic Allen–Cahn

Stochastic Allen–Cahn (aka Ginzburg–Landau) with noise in materialsscience:

Modeling in phenomenological/lattice approximation, noise = unknownmeso/micro-scopic fluctuation with known statistics effect inmacroscopic scale [Halperin and Hoffman, 1977],[Katsoulakis and Szepessy, 2006, cf.];

Simulation noise as ad-hoc nucleation/instability inducing mechanism[Warren and Boettinger, 1995, Nestler et al., 2005, e.g.,];

Numerics for d = 1 SDE approach [Shardlow, 2000], spectral methods[Liu, 2003], interface nucleation/annihilation [Lythe, 1998,Fatkullin and Vanden-Eijnden, 2004].

Numerics for d ≥ 2 ?


Deterministic Allen–Cahn in 1 dimensionmetastability and profiles

PDE: ∂tu−∆u+1

ε2f(u) = 0, 0 < ε 1,

potential:1

ε2f(ξ) =

1

ε2(ξ3 − ξ

),

ODE (no diffusion): u = − 1

ε2f(u)

equilibriums: f(±1) = 0, f(0) = 0,

linearize:±1 stable − f ′(±1) < 0,

0 unstable − f ′(0) > 0.

y = f(u)

u

y

−1 0

1

stable eqs

unstable eq


Deterministic Allen–Cahn in 1 dimensionmetastability and profiles

PDE: ∂tu−∆u+1

ε2f(u) = 0, 0 < ε 1,

potential:1

ε2f(ξ) =

1

ε2(ξ3 − ξ

),

ODE (no diffusion): u = − 1

ε2f(u)

equilibriums: f(±1) = 0, f(0) = 0,

linearize:±1 stable − f ′(±1) < 0,

0 unstable − f ′(0) > 0.

resolved profile

nonresolved profile

x

u

ξ1

ξ2


Deterministic Allen–Cahn in 1 dimensionresolved profiles terminology

profile(x,u): u≈tanh((x−ξ)/ε

√2)

x

u

profile’s center ξ

diffuse interfaceO(ε) thick


Deterministic Allen–Cahn in dimension d > 1close relation to Mean Curvature Flow (MCF)

Level set Γ = u = 0, moves, as ε → 0a mean curvature flow:

ε ∂tu|Γ −ε∆ u|Γ +1ε (u3 − u)

∣∣Γ

= 0

↓ ↓ ↓−v −H +0 = 0

normal −meanvelocity curvature

Γ = u = 0

ε

H

v


The Stochastic (or Noisy) Allen-Cahn Problem

is the following semilinear parabolic stochastic PDE with additive whitenoise

∂tu(x, t)−∆u(x, t) + fε(u(x, t)) = εγ∂xtW (x, t), x ∈ (0, 1), t ∈ R+.

u(x, 0) = u0(x), x ∈ (0, 1)

∂xu(0, t) = ∂xu(1, t) = 0, t ∈ R+.

f(u)

10−1 u

∫ u

0f

fε(ξ) =1

ε2(ξ3 − ξ

): first

derivative of double wellpotentialε ∈ R

+: “diffuse inter-face” parameterγ ∈ R: noise intensity pa-rameter∂xtW : space-time whitenoise


White noise ∂xtW : an informal definition

Informally: white noise is mixed derivative of (2 dimensional)Brownian sheet W = Wx,t.

Brownian sheet W : extension of 1-dim Brownian motion tomulti-dim, built on Ω.

Solutions of PDE understood as mild solution

Notation for ∂xtW∫ ∞

0

∫ 1

0f(x, t) dW (x, t) :=

∫ ∞

0

∫ 1

0f(x, t)∂xtW (x, t) dx dt,

stochastic integral [Walsh, 1986].


White noise 101

∀A ∈ Borel(R2)

∫A

dW (x, t) =: W (A) ∈ N (0, |A|) , I.e, W (A) is a

0-mean Gaussian random variable with variance = |A|.A ∩B = ∅⇒ W (A),W (B) independent andW (A ∪B) = W (A) +W (B).

Brownian sheet: Wx,t = W ([0, t]× [0, x]) =

∫ t

0

∫ x

0dW (x, t).

Basic yet crucial property:

E

[(∫I

∫Df(x, t) dW (x, t)

)2]

= E

[∫I

∫Df(x, t)2 dx dt

], ∀f ∈ L2.

(Aka dW =√

dx dt.)


Solutions of the stochastic linear heat equation

∂tw −∆w = ∂xtW, in D × R+0

w(0) = 0, on D

∂xw(1, t) = ∂xw(0, t) = 0, ∀t ∈ [0,∞).

Solution is defined as Gaussian process produced by the stochastic integral

Zt(x) = Z(x, t) :=

∫ t

0

∫DGt−s(x, y) dW (x, y).

where G is the heat kernel:

Gt(x, y) = 2∞∑

k=0

cos(πkx) cos(πky) exp(−π2k2t).

(An “explicit semigroup” approach.)


Solutions of the stochastic Allen–Cahn problem

∂tu(x, t)−∆u(x, t) + fε(u(x, t)) = εγ∂xtW (x, t), for x ∈ D = [0, 1], t ∈ [0,∞)

u(x, 0) = u0(x), ∀x ∈ D∂xu(0, t) = ∂xu(1, t) = 0, ∀t ∈ R+.

Defined as the continuous solution of integral equation

u(x, t) =−∫ t

0

∫DGt−s(x, y)fε(u(y, s)) dy ds

+

∫DGt(x, y)u0(y) dy + εγZt(x).

Unique continuous integral solution exists in 1d provided the initialcondition u0 fulfills boundary conditions.


Remarks on solution process

The solution u of Stochastic Allen-Cahn, is adapted continuousGaussian process (it is Holder of exponents 1/2,1/4).

The process u has continuous, but nowhere differentiable samplepaths.

This approach works only in 1d+time, for higher dimensions; whendefined solutions are extremely singular distributions (let alonecontinuous).

If white noise (uncorrelated) is replaced by colored noise (correlated),then “regular” u can be sought in higher dimensions.

Direct numerical discretization of this problem not obvious.


Discretization strategy

In two main steps:

1. Replace white noise ∂xtW by a smoother object: the approximatewhite noise ∂xtW .

2. Discretize the approximate problem with ∂xtW via a finite elementscheme for the Allen-Cahn equation.

Inspired by [Allen et al., 1998], [Yan, 2005].


Approximate White Noise (AWN)

Fix a final time T > 0 and consider uniform space and time partitions

in space: D = [0, 1], Dm := (xm−1, xm), xm − xm−1 = σ, m ∈ [1 : M ] ,

in time: I = [0, T ], In := [tn−1, tn), tn − tn−1 = ρ, n ∈ [1 : N ] .

Regularization of the white noise is projection on piecewise constants:

ηm,n :=1

σρ

∫I

∫Dχm(x)ψn(t) dW (x, t).

∂xtW (x, t) :=N∑

n=1

N∑m=1

ηm,nχm(x)ϕn(t).

χm = 1Dm , ϕn = 1In (characteristic functions).


Regularity of AWN

∂xtW (x, t) :=N∑

n=1

N∑m=1

ηm,nχm(x)ϕn(t).

ηm,n are independent N (0, 1/(σρ)) variables and

E

[(∫I


)2]≤ E

[(∫I


)2].

E

[(∫I

∫D

dW (x, t)

)2]

= T

E

[∫I

∫D

∣∣∂xtW (x, t)∣∣2 dx dt

]≤ 1

σρ


The regularized problem

∂tu−∆u+ fε(u) = εγ∂xtW ,

∂xu(t, 0) = ∂xu(t, 1) = 0,

u(0) = u0,

Admits a “classical” solution. ∀ω ∈ Ω ⇒ corresponding realization of theAWN ∂xtW (ω) ∈ L∞([0, T ]×D) (parabolic regularity) ⇒∂tu(ω) ∈ L2([0, T ]; L2(D)).⇒ variational formulation and thus FEM (or other standard methods) nowpossible.Error estimate schedule:

1. Compare u with u.

2. Approximate u by U in a finite element space and compare.


The Finite Element Method

Parameters (numerical mesh) τ, h > 0 (approximation mesh) ρ, σ.

Not necessary, but “natural” coupling: τ = ρ = h2 = σ2.

Linearized Semi-Implicit Euler⟨Un − Un−1

τ,Φ

⟩+ 〈∂xU

n, ∂xΦ〉+⟨f ′ε(U

n−1)Un,Φ⟩

=⟨f ′ε(U

n−1)Un−1 + fε(Un−1),Φ

⟩+⟨εγ∂xtW ,Φ

⟩, ∀ Φ ∈ Vh.

Vh is a finite element space of piecewise linear

[Kessler et al., 2004, Feng and Wu, 2005].


Implementation

[1

ρM + A +

1

ε2N(un−1)

]un =

1

ε2g(un−1) +

1

ρMun−1 + εγw.

M , A usual mass and stiffness matrices,

N and g nonlinear mass matrix and load vector

w = (wi) a random load vector generated at each time-step: foreach node i we have

wi =εγ

2

√h

ρ(ηi−1 + ηi) ηi ∈ N (0, 1) pseudo-random.

Simple Monte Carlo method on w.


Convergence of regularized to stochastic solution

Theorem (quasi-strong (least squares) convergence)

Let γ > −1/2, T > 0. For some c1, c2 (6←[ ε), and for each ε ∈ (0, 1) therecorrespond (i) an event Ω∞, (ii) constants Cε, C1, C2 s.t.

P (Ω∞) ≥ 1− 2c1 exp(−c2/ε1+2γ)∫Ω∞

∫ r

0

∫D|u− u|2 dx dt dP ≤ Cε

(C1ρ

1/2 + C2σ2

ρ1/2

), ∀σ, ρ > 0.

Remark1 Event Ω∞ = |u, u| < 3 (measured by maximum principle &

exponential decay of “chi-squared” distribution).

2 Cε grows exponentially with 1/ε.

3 Constant improves for γ 1 (exploiting spectral gap).


Maximum principle in probability sense

Lemma

Suppose γ > −1/2. Given T , there exist c1, c2, δ0 > 0 such that if‖u0‖L∞(D) ≤ 1 + δ0 then

P

sup

t∈[0,T ]‖(u, u)(t)‖L∞(D) > 3

≤ c1 exp(−c2/ε1+2γ).

Remark1 The constant 3 is for convenience, can be replaced by 1 + δ1, if

needed, by adjusting c1, c2 and δ0.

2 This result is used to determine the event Ω∞ for convergence tohold.

3 Because, nonlinearity is not globally Lipschitz.


Small noise resolution (for γ big: weak intensity)

Theorem (small noise)

Let q solution of deterministic the Allen–Cahn problem

∂tq −∆q + fε(q) = 0, q(0) = u0, on D.

Then ∀ T > 0 : ∃K1(T ) > 0, ε0(T ) > 0

P

(sup[0,T ]‖u− q‖L2(D) ≤ ε

3

)

≥ 1−(

1 +K1(T )

2ε6−2γ

)T/(σρ)−1

exp

(−K1(T )

2ε6−2γ

)for all ε ∈ (0, ε0), γ > 3 and ρ, σ > 0.


Spectrum estimate

Theorem ([Chen, 1994], [de Mottoni and Schatzman, 1995])

Let q be the (classical) solution of the problem

∂tq −∆q + fε(q) = 0, q(0) = u0, on D

Key to argument in convergence for weak noise is the use of spectralestimate for q: There exists a constant λ0 > 0 independent of ε such thatfor any ε ∈ (0, 1] we have

‖∂xφ‖2L2(D) +⟨f ′ε(q)φ, φ

⟩≥ −λ0 ‖φ‖2L2(D) , ∀φH1(D).


X(0, T ; L2(D)) bounds for AWN

Lemma

∀K > 0

P

sup

t∈[0,T ]

∥∥∂xtW (t)∥∥

L2(D)≤ K

≥

[1− T

ρ

(1 +

K2

2ρ

)1/(2σ)−1

exp

(−K

2

2ρ

)]+

and P

∫ T

0

∥∥∂xtW (t)∥∥2

L2(D)≤ K2

≥ 1−

(1 +

K2

2

)1/(2ρσ)−1

exp

(−K

2

2

)O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 29 / 62

Convergence of FEM

Taking τ = ρ = h2 = σ2 to simplify, we have∫Ω∞

∫D|u(tn)− Un|2 dx dP ≤ C(T, ε)h.

Remark1 compare w.r.t. deterministic case,

2 impossible to take E, but only∫Ω∞ , due to rare events.


Regularity of u

Theorem (Regularity of u)∫Ω∞

∫Im

∫D|∂tu|2 dx dt ≤ c1ρ−1/2 + c2σ

−1∫Ω∞

∫D|∆u|2 dx ≤ c3ρ−3/2 + c4σ

−1ρ−1.

For Ω∞ an event of high probability.

Remark

Expected loss of regularity, as σ, ρ→ 0.


Numerical experiments

1 Computations for ε = 0.04, 0.02, 0.01 and γ = 1.0, 0.5, 0.2, 0.0,−0.2(γ < 0 though not discussed in theory can be interesting in practice).

2 Run a Monte-Carlo simulation for each set of parameters (2500samples).

3 Meshsize h < ε.

4 Timestep τ = c0h2 ≈ ε2.


Sample path with ε = 0.01, γ = 0


Sample path with ε = 0.01, γ = −0.2


Sample path with ε = 0.01, γ = 1.0


How to the benchmark our computations?

Benchmarking = “comparing with known solutions” Very few known“exact result”.

Theorem ( [Funaki, 1995, Brassesco et al., 1995])

The interface motion is asymptotically, as ε→ 0, a (1d) Brownian motion,

limε→0

P

sup

t≤ε−1−2γT

∥∥∥ut − q(x− ξεt )∥∥∥

L2(D)> δ

= 0,

for δ > 0.

Remark

q is an instanton for the deterministic Allen-Cahn equation

ξεt solves a SODE representing the motion of the interface

ut(x) is an appropriate rescaling of u(x, t).


BenchmarkingScaling

In our scaling, the interface position ξt performs a process that looksasymptotically (as ε→ 0) like

ξεt ≈

√3

2√

2εγ+1/2Bt,

where Bt is the 1d Brownian Motion.We use statistics on the interface (discarding all cases ofnucleation/coalescence).


Numerically computed average and variance

γ = 0.5 and ε = 0.08

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05

0.1computed interface position vs. time forγ = 0.5and ε = 0.08

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.08

lev=6; smp=450lev=7; smp=451lev=8; smp=55lev=9; smp=0


Average of Monte Carlo

samples against time for

various levels of refine-

ment lev = 6, 7, 8, 9,

and meshsize h = 1/2lev.

Sample number smp

differs with lev. Samples

are rejected if annihila-

tion/nucleation occurs.

Variance against time

for the various levels. As

time grows, variance ac-

curacy deteriorates.



γ = 0.5 and ε = 0.04

0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025


0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.04






ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.5 and ε = 0.02

0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4







ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.5 and ε = 0.01

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4 x 10−3 computed interface position vs. time forγ = 0.5and ε = 0.01

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5







ment lev = 6, 7, 8, 9,


Sample number smp









Simulations vs. theory

log(var[U ih] /ti) vs. log ε plot for γ = 0.5 at various times ti = t0 10i

−5 −4.5 −4 −3.5 −3 −2.5−10

−9

−8

−7

−6

−5

−4

−3

−2log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0.5

ref. slope=1+2*γtime 0.00152587time 0.0152587time 0.152587time 1.52587time 15.2587

Theoretical slope

(predicted by

[Funaki, 1995] and

[Brassesco et al., 1995])

is 1 + 2γ.

Plotted in orange refer-

ence line

As ε → 0 slope im-

proves.

Meshsize h = 1/512.O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 42 / 62


γ = 0.2 and ε = 0.08

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4







ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.2 and ε = 0.04

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8







ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.2 and ε = 0.02

0 2 4 6 8 10 12 14 16 18 20−0.05

0

0.05

0.1


0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2







ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.2 and ε = 0.01

0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005

0.01

0.015


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4







ment lev = 6, 7, 8, 9,


Sample number smp











−5 −4.5 −4 −3.5 −3 −2.5−7

−6

−5

−4

−3

−2

−1

0log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0.2


Theoretical slope

(predicted by

[Funaki, 1995] and


is 1 + 2γ.


ence line


proves.



γ = 0.0 and ε = 0.08

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1computed interface position vs. time forγ = 0and ε = 0.08

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.08






ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.0 and ε = 0.04

0 2 4 6 8 10 12 14 16 18 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2computed interface position vs. time forγ = 0and ε = 0.04

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.04






ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.0 and ε = 0.02

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.02






ment lev = 6, 7, 8, 9,


Sample number smp










γ = 0.0 and ε = 0.01

0 2 4 6 8 10 12 14 16 18 20−0.05

0

0.05

0.1


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.01






ment lev = 6, 7, 8, 9,


Sample number smp











−5 −4.5 −4 −3.5 −3 −2.5−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0


Theoretical slope

(predicted by

[Funaki, 1995] and


is 1 + 2γ.


ence line


proves.


What we learned

Stability and convergence numerical method for the stochasticAllen-Cahn in 1d.

Monte-Carlo type simulations.

Benchmarking via statistical comparison.

Adaptivity mandatory (but different from “standard” phase-fieldapproach) for higher dimensional study.


What we’re trying to learn

Current investigation [Kossioris et al., 2009]

finite ε “exact” solutions

multiple interface convergence

structure of the stochastic solution

colored noise in d > 1

adaptivity


What still needs to be learned

Extension to Cahn-Hilliard type equations, Phase-Field, Dendrites.

Other stochastic applications

Chaos Expansion

Kolmogorov (Fokker-Plank) equations

Sparse methods


References I

Allen, E. J., Novosel, S. J., and Zhang, Z. (1998).Finite element and difference approximation of some linear stochasticpartial differential equations.Stochastics Stochastics Rep., 64(1-2):117–142.

Allen, S. M. and Cahn, J. (1979).A macroscopic theory for antiphase boundary motion and itsapplication to antiphase domain coarsening.Acta Metal. Mater., 27(6):1085–1095.

Brassesco, S., De Masi, A., and Presutti, E. (1995).Brownian fluctuations of the interface in the D = 1 Ginzburg-Landauequation with noise.Ann. Inst. H. Poincare Probab. Statist., 31(1):81–118.


References II

Bronsard, L. and Kohn, R. V. (1990).On the slowness of phase boundary motion in one space dimension.Comm. Pure Appl. Math., 43(8):983–997.

Cahn, J. W. and Hilliard, J. E. (1958).Free energy of a nonuniform system. I. Interfacial free energy.Journal of Chemical Physics, 28(2):258–267.

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Yan, Y. (2005).Galerkin finite element methods for stochastic parabolic partialdifferential equations.SIAM J. Numer. Anal., 43(4):1363–1384 (electronic).


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