On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
i give some indeas on how to use asymptotic series and expansion to prove Riemann Hypothesis, solve integral equations and even define a regularized integral of powers
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
i give some indeas on how to use asymptotic series and expansion to prove Riemann Hypothesis, solve integral equations and even define a regularized integral of powers
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
"The Metropolis adjusted Langevin Algorithm
for log-concave probability measures in high
dimensions", talk by Andreas Elberle at the BigMC seminar, 9th June 2011, Paris
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Probabilistic Modular Embedding for Stochastic Coordinated SystemsStefano Mariani
Embedding and modular embedding are two well-known tech- niques for measuring and comparing the expressiveness of languages— sequential and concurrent programming languages, respectively. The emer- gence of new classes of computational systems featuring stochastic be- haviours – such as pervasive, adaptive, self-organising systems – requires new tools for probabilistic languages. In this paper, we recall and refine the notion of probabilistic modular embedding (PME) as an extension to modular embedding meant to capture the expressiveness of stochas- tic systems, and show its application to different coordination languages providing probabilistic mechanisms for stochastic systems.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
"The Metropolis adjusted Langevin Algorithm
for log-concave probability measures in high
dimensions", talk by Andreas Elberle at the BigMC seminar, 9th June 2011, Paris
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Probabilistic Modular Embedding for Stochastic Coordinated SystemsStefano Mariani
Embedding and modular embedding are two well-known tech- niques for measuring and comparing the expressiveness of languages— sequential and concurrent programming languages, respectively. The emer- gence of new classes of computational systems featuring stochastic be- haviours – such as pervasive, adaptive, self-organising systems – requires new tools for probabilistic languages. In this paper, we recall and refine the notion of probabilistic modular embedding (PME) as an extension to modular embedding meant to capture the expressiveness of stochas- tic systems, and show its application to different coordination languages providing probabilistic mechanisms for stochastic systems.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 14.
More info at http://summerschool.ssa.org.ua
Stochastic Analysis of Resource Plays: Maximizing Portfolio Value and Mitiga...Portfolio Decisions
This paper outlines a method for applying stochastic representations of key macro-economic parameters and risks in evaluating the portfolio value of resource plays. Stochastic pricing, costs forecasts, and regulatory uncertainties are often neglected or im-properly applied when evaluating what are typically considered to be ‘low risk’ resource opportunities. Portfolio allocation and project development timing may be critically dependent upon these macro-economic variables, particularly given the long production life and operationally intensive nature of many resource opportunities. These techniques will allow corporate planners and E&P executives to better leverage their opportunity inventory and ensure resource play development plans are in alignment with stochastic pricing or other macro-economic forecasts.
Research internship on optimal stochastic theory with financial application u...Asma Ben Slimene
This is a presntation of my second year intership on optimal stochastic theory and how we can apply it on some financial application then how we can solve such problems using finite differences methods!
Enjoy it !
Stochastic Approximation and Simulated AnnealingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 8.
More info at http://summerschool.ssa.org.ua
My presentation of the "Hierarchical Stochastic Neighbor Embedding" algorithm at EuroVIS 2016.
HSNE is a non-linear dimensionlity reduction technique for the exploration of large high-dimensional datasets.
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Volatility
1) Volatility modelling
2) Local stochastic volatility models: stochastic volatility, jumps, local volatility
3) Calibration of parametric local volatility models using partial differential equation (PDE) methods
4) Calibration of non-parametric local volatility volatility models with jumps and stochastic volatility using PDE methods
5) Numerical methods for PDEs
6) Illustrations using SPX and VIX data
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This is a talk I gave at the University of Sussex in June 2011. It outlines the newly released numerical code Pyflation and the results published in arXiv:1103.0912.
Here we give an overview of the causes of pinning in multiphase lattice Boltzmann models and propose a stochastic sharpening approach to overcome this spurious phenomenon.
The widespread popularity of Bayesian tree-structured regression methods has raised considerable interest in theoretical understanding of their empirical success. However, theoretical literature on methods such as Bayesian CART and BART is still in its infancy. This paper affords new insights about Bayesian CART in the context of structured wavelet shrinkage under the white noise model. We exhibit precise connections between tree-shaped sparsity priors and unstructured spike-and-slab priors, which are regarded as ideal but are rather theoretical in nature. We show that the more practical Bayesian CART priors lead to adaptive rate-minimax posterior concentration in the l∞sense, performing nearly as well as the theoretical ideal (up to a log term). To further explore the benefits of structured shrinkage, we propose the g-prior for trees, which departs from the typical wavelet product priors by harnessing correlation structure induced by the tree topology. While the majority of wavelet type theoretical results for CART focus on dyadic trees, here we do not require that splits are at dyadic locations. We introduce the library of weakly balanced Haar wavelets and show that Bayesian CART is equivalent to Bayesian basis selection from this library. To illustrate that l∞adaptation is an intricate phenomenon, where internal sparsity plays a key role, we show that dense trees are incapable of adaptation. While one of the major appeals of BART is uncertainty quantification via credible sets, asymptotic normality justifications have thus far been unavailable. Building on the l∞adaptation property, we provide new fully non-parametric and adaptive Bernstein-von Mises statements for Bayesian CART using multiscale techniques.
(Joint work with Ismael Castillo)
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
1. Some computational aspects of stochastic phase-field
models
Omar Lakkis
Mathematics – University of Sussex – Brighton, England UK
based on joint work with
M.Katsoulakis, G.Kossioris & M.Romito
17 November 2009
3rd
Workshop
On
Random
Dynamical
Systems
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 1 / 62
3. Motivation
Dendrites
A real-life dendrite lab picture: A phase-field simulation:
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 3 / 62
4. Dendrites
Computed dendrites without ther- A computed dendrite with thermal
mal noise [Nestler et al., 2005] noise [Nestler et al., 2005]
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 4 / 62
5. Deterministic Phase Field
following [Chen, 1994]
Phase transition in solidification process
α ∂t u − ∆u + (u3 − u)/ = σw (“interface” motion)
c∂t w − ∆w = −∂t u (diffusion in bulk) ,
where
≈ 1
solid phase,
u : order parameter/phase field ∈ (−1 + δ , 1 − δ ) , interface (region)
≈ −1 liquid phase.
w : temperature in liquid/solid bulk
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 5 / 62
6. The Stochastic (or Noisy) Allen-Cahn Problem
is the following semilinear parabolic stochastic PDE with additive white
noise
∂t u(x, t) − ∆u(x, t) + f (u(x, t)) = γ
∂xt W (x, t), x ∈ (0, 1), t ∈ R+ .
u(x, 0) = u0 (x), x ∈ (0, 1)
∂x u(0, t) = ∂x u(1, t) = 0, t ∈ R+ .
1
u f (ξ) = 2 ξ 3 − ξ : first
f (u) 0
f
derivative of double well
potential
−1 0 1 u ∈ R+ : “diffuse inter-
face” parameter
γ ∈ R: noise intensity pa-
rameter
∂xt W : space-time white
noise
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 6 / 62
7. Why do we care?
about the (deterministic) Allen–Cahn
1
∂t u − ∆u + 2
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];
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 7 / 62
8. Why do we care?
about the stochastic Allen–Cahn
1
∂t u − ∆u + 2
u3 − u = γ
∂xt W
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 noises
possible [Souganidis and Yip, 2004], [Funaki, 1999],
[Dirr et al., 2001].
Rigorous mathematical setting to noise-induced dendrites.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 8 / 62
9. Why do we care?
Modeling and computation with stochastic Allen–Cahn
Stochastic Allen–Cahn (aka Ginzburg–Landau) with noise in materials
science:
Modeling in phenomenological/lattice approximation, noise = unknown
meso/micro-scopic fluctuation with known statistics effect in
macroscopic 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 ?
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 9 / 62
10. Deterministic Allen–Cahn in 1 dimension
metastability and profiles
unstable eq
y
1
PDE: ∂t u−∆u+ f (u) = 0, 0< 1, y = f (u)
2
1 1
potential: 2
f (ξ) = 2
ξ3 − ξ ,
1
1 −1 0 u
ODE (no diffusion): u = −
˙ 2
f (u)
equilibriums: f (±1) = 0, f (0) = 0,
±1 stable − f (±1) < 0,
linearize:
0 unstable − f (0) > 0.
stable eqs
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 10 / 62
11. Deterministic Allen–Cahn in 1 dimension
metastability and profiles
1
PDE: ∂t u−∆u+ 2
f (u) = 0, 0< 1, resolved profile
u
1 1 3
potential: 2
f (ξ) = 2
ξ −ξ ,
1 ξ1
ODE (no diffusion): u = −
˙ 2
f (u) ξ2 x
equilibriums: f (±1) = 0, f (0) = 0, nonresolved profile
±1 stable − f (±1) < 0,
linearize:
0 unstable − f (0) > 0.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 10 / 62
12. Deterministic Allen–Cahn in 1 dimension
resolved profiles terminology
diffuse interface
u O( ) thick
profile’s center ξ
x
profile √
{(x,u): u≈tanh((x−ξ)/ 2)}
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 11 / 62
13. Deterministic Allen–Cahn in dimension d > 1
close relation to Mean Curvature Flow (MCF)
Γ = {u = 0}
v
Level set Γ = {u = 0}, moves, as →0
a mean curvature flow:
∂t u|Γ − ∆ u|Γ + 1 (u3 − u) Γ = 0
↓ ↓ ↓ H
−v −H +0 = 0
normal −mean
velocity curvature
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 12 / 62
14. The Stochastic (or Noisy) Allen-Cahn Problem
is the following semilinear parabolic stochastic PDE with additive white
noise
∂t u(x, t) − ∆u(x, t) + f (u(x, t)) = γ
∂xt W (x, t), x ∈ (0, 1), t ∈ R+ .
u(x, 0) = u0 (x), x ∈ (0, 1)
∂x u(0, t) = ∂x u(1, t) = 0, t ∈ R+ .
1
u f (ξ) = 2 ξ 3 − ξ : first
f (u) 0
f
derivative of double well
potential
−1 0 1 u ∈ R+ : “diffuse inter-
face” parameter
γ ∈ R: noise intensity pa-
rameter
∂xt W : space-time white
noise
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 13 / 62
15. White noise ∂xt W : 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 to
multi-dim, built on Ω.
Solutions of PDE understood as mild solution
Notation for ∂xt W
∞ 1 ∞ 1
f (x, t) dW (x, t) := f (x, t)∂xt W (x, t) dx dt,
0 0 0 0
stochastic integral [Walsh, 1986].
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 14 / 62
16. White noise 101
∀A ∈ Borel(R2 ) dW (x, t) =: W (A) ∈ N (0, |A|) , I.e, W (A) is a
A
0-mean Gaussian random variable with variance = |A|.
A ∩ B = ∅ ⇒ W (A),W (B) independent and
W (A ∪ B) = W (A) + W (B).
t x
Brownian sheet: Wx,t = W ([0, t] × [0, x]) = dW (x, t).
0 0
Basic yet crucial property:
2
E f (x, t) dW (x, t) =E f (x, t)2 dx dt , ∀f ∈ L2 .
I D I D
√
(Aka dW = dx dt.)
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 15 / 62
17. Solutions of the stochastic linear heat equation
∂t w − ∆w = ∂xt W, in D × R+
0
w(0) = 0, on D
∂x w(1, t) = ∂x w(0, t) = 0, ∀t ∈ [0, ∞).
Solution is defined as Gaussian process produced by the stochastic integral
t
Zt (x) = Z(x, t) := Gt−s (x, y) dW (x, y).
0 D
where G is the heat kernel:
∞
Gt (x, y) = 2 cos(πkx) cos(πky) exp(−π 2 k 2 t).
k=0
(An “explicit semigroup” approach.)
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 16 / 62
18. Solutions of the stochastic Allen–Cahn problem
γ
∂t u(x, t) − ∆u(x, t) + f (u(x, t)) = ∂xt W (x, t), for x ∈ D = [0, 1], t ∈ [0, ∞
u(x, 0) = u0 (x), ∀x ∈ D
∂x u(0, t) = ∂x u(1, t) = 0, ∀t ∈ R+ .
Defined as the continuous solution of integral equation
t
u(x, t) = − Gt−s (x, y)f (u(y, s)) dy ds
0 D
γ
+ Gt (x, y)u0 (y) dy + Zt (x).
D
Unique continuous integral solution exists in 1d provided the initial
condition u0 fulfills boundary conditions.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 17 / 62
19. Remarks on solution process
The solution u of Stochastic Allen-Cahn, is adapted continuous
Gaussian process (it is H¨lder of exponents 1/2,1/4).
o
The process u has continuous, but nowhere differentiable sample
paths.
This approach works only in 1d+time, for higher dimensions; when
defined solutions are extremely singular distributions (let alone
continuous).
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.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 18 / 62
20. Discretization strategy
In two main steps:
1. Replace white noise ∂xt W by a smoother object: the approximate
¯
white noise ∂xt W .
¯
2. Discretize the approximate problem with ∂xt W via a finite element
scheme for the Allen-Cahn equation.
Inspired by [Allen et al., 1998], [Yan, 2005].
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 19 / 62
21. 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:
1
ηm,n :=
¯ χm (x)ψn (t) dW (x, t).
σρ I D
N N
¯
∂xt W (x, t) := ηm,n χm (x)ϕn (t).
¯
n=1 m=1
χm = 1Dm , ϕn = 1In (characteristic functions).
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 20 / 62
22. Regularity of AWN
N N
¯
∂xt W (x, t) := ηm,n χm (x)ϕn (t).
¯
n=1 m=1
ηm,n are independent N (0, 1/(σρ)) variables and
¯
2 2
E ¯
f (x, t) dW (x, t) ≤E f (x, t) dW (x, t) .
I D I D
2
E ¯
dW (x, t) =T
I D
¯ 2 1
E ∂xt W (x, t) dx dt ≤
I D σρ
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 21 / 62
23. The regularized problem
∂t u − ∆¯ + f (¯) =
¯ u u γ ¯
∂xt W ,
∂x u(t, 0) = ∂x u(t, 1) = 0,
¯ ¯
u(0) = u0 ,
¯
Admits a “classical” solution. ∀ω ∈ Ω ⇒ corresponding realization of the
¯
AWN ∂xt W (ω) ∈ L∞ ([0, T ] × D) (parabolic regularity) ⇒
∂t u(ω) ∈ L2 ([0, T ]; L2 (D)).
¯
⇒ variational formulation and thus FEM (or other standard methods) now
possible.
Error estimate schedule:
1. Compare u with u.
¯
2. Approximate u by U in a finite element space and compare.
¯
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 22 / 62
24. The Finite Element Method
Parameters (numerical mesh) τ, h > 0 (approximation mesh) ρ, σ.
Not necessary, but “natural” coupling: τ = ρ = h2 = σ 2 .
Linearized Semi-Implicit Euler
U n − U n−1
, Φ + ∂x U n , ∂x Φ + f (U n−1 )U n , Φ
τ
= f (U n−1 )U n−1 + f (U n−1 ), Φ + γ ∂xt W , Φ , ∀ Φ ∈ Vh .
¯
Vh is a finite element space of piecewise linear
[Kessler et al., 2004, Feng and Wu, 2005].
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 23 / 62
25. Implementation
1 1 1 1
M + A + 2 N (un−1 ) un = 2 g(un−1 ) + M un−1 + γ
w.
ρ ρ
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: for
each node i we have
γ h
wi = (ηi−1 + ηi ) ηi ∈ N (0, 1) pseudo-random.
2 ρ
Simple Monte Carlo method on w.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 24 / 62
26. Convergence of regularized to stochastic solution
Theorem (quasi-strong (least squares) convergence)
Let γ > −1/2, T > 0. For some c1 , c2 (← ), and for each ∈ (0, 1) there
correspond (i) an event Ω ∞ , (ii) constants C , C1 , C2 s.t.
P (Ω ∞ ) ≥ 1 − 2c1 exp(−c2 / 1+2γ
)
r
σ2
|¯ − u|2 dx dt dP ≤ C
u C1 ρ1/2 + C2 , ∀σ, ρ > 0.
Ω∞ 0 D ρ1/2
Remark
1 Event Ω ∞ = {|u, u| < 3} (measured by maximum principle &
¯
exponential decay of “chi-squared” distribution).
2 C grows exponentially with 1/ .
3 Constant improves for γ 1 (exploiting spectral gap).
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 25 / 62
27. 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
1+2γ
P sup (u, u)(t)
¯ L∞ (D) >3 ≤ c1 exp(−c2 / ).
t∈[0,T ]
Remark
1 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 to
hold.
3 Because, nonlinearity is not globally Lipschitz.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 26 / 62
28. Small noise resolution (for γ big: weak intensity)
Theorem (small noise)
Let q solution of deterministic the Allen–Cahn problem
∂t q − ∆q + f (q) = 0, q(0) = u0 , on D.
Then ∀ T > 0 : ∃ K1 (T ) > 0, 0 (T ) >0
3
P sup u − q
¯ L2 (D) ≤
[0,T ]
T /(σρ)−1
K1 (T ) 6−2γ K1 (T ) 6−2γ
≥1− 1+ exp −
2 2
for all ∈ (0, 0 ), γ > 3 and ρ, σ > 0.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 27 / 62
29. Spectrum estimate
Theorem ([Chen, 1994], [de Mottoni and Schatzman, 1995])
Let q be the (classical) solution of the problem
∂t q − ∆q + f (q) = 0, q(0) = u0 , on D
Key to argument in convergence for weak noise is the use of spectral
estimate for q: There exists a constant λ0 > 0 independent of such that
for any ∈ (0, 1] we have
2 2
∂x φ L2 (D) + f (q)φ, φ ≥ −λ0 φ L2 (D) , ∀φ H1 (D).
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 28 / 62
30. X(0, T ; L2 (D)) bounds for AWN
Lemma
∀K > 0
P sup ¯
∂xt W (t) ≤K
L2 (D)
t∈[0,T ]
1/(2σ)−1 +
T K2 K2
≥ 1− 1+ ρ exp − ρ
ρ 2 2
T
¯ 2
and P ∂xt W (t) L2 (D)
≤ K2
0
1/(2ρσ)−1
K2 K2
≥1− 1+ exp −
2 2
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 29 / 62
31. Convergence of FEM
Taking τ = ρ = h2 = σ 2 to simplify, we have
|¯(tn ) − U n |2 dx dP ≤ C(T, )h.
u
Ω∞ D
Remark
1 compare w.r.t. deterministic case,
2 impossible to take E, but only Ω∞ , due to rare events.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 30 / 62
32. Regularity of u
¯
Theorem (Regularity of u)
¯
|∂t u|2 dx dt ≤ c1 ρ−1/2 + c2 σ −1
¯
Ω∞ Im D
|∆¯|2 dx ≤ c3 ρ−3/2 + c4 σ −1 ρ−1 .
u
Ω∞ D
For Ω∞ an event of high probability.
Remark
Expected loss of regularity, as σ, ρ → 0.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 31 / 62
33. Numerical experiments
1 Computations for = 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 (2500
samples).
3 Meshsize h < .
4 Timestep τ = c0 h2 ≈ 2.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 32 / 62
34. Sample path with = 0.01, γ = 0
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 33 / 62
35. Sample path with = 0.01, γ = −0.2
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 34 / 62
36. Sample path with = 0.01, γ = 1.0
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 35 / 62
37. 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 → 0, a (1d) Brownian motion,
lim P sup ˜ ˜
ut − q(x − ξt ) >δ = 0,
→0 t≤ −1−2γ T L2 (D)
for δ > 0.
Remark
q is an instanton for the deterministic Allen-Cahn equation
˜
ξt solves a SODE representing the motion of the interface
ut (x) is an appropriate rescaling of u(x, t).
˜
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 36 / 62
38. Benchmarking
Scaling
In our scaling, the interface position ξt performs a process that looks
asymptotically (as → 0) like
√
3
ξt ≈ √ γ+1/2 Bt ,
2 2
where Bt is the 1d Brownian Motion.
We use statistics on the interface (discarding all cases of
nucleation/coalescence).
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 37 / 62
39. Numerically computed average and variance
γ = 0.5 and = 0.08
computed interface position vs. time for = 0.5and ε = 0.08
γ
0.1
lev=6; smp=450
lev=7; smp=451
0.05 lev=8; smp=55
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.05 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.1
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
1.5
computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.08 Sample number smp
lev=6; smp=450
lev=7; smp=451 differs with lev. Samples
lev=8; smp=55
1 lev=9; smp=0 are rejected if annihila-
tion/nucleation occurs.
0.5
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 38 / 62
40. Numerically computed average and variance
γ = 0.5 and = 0.04
computed interface position vs. time for = 0.5and ε = 0.04
γ
0.03
lev=6; smp=920
0.025 lev=7; smp=924
0.02 lev=8; smp=918
0.015
lev=9; smp=918 Average of Monte Carlo
0.01
samples against time for
0.005
0 various levels of refine-
−0.005
ment lev = 6, 7, 8, 9,
−0.01
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
4
computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.04 Sample number smp
lev=6; smp=920
3.5
lev=7; smp=924 differs with lev. Samples
3 lev=8; smp=918
2.5
lev=9; smp=918 are rejected if annihila-
2 tion/nucleation occurs.
1.5
1 Variance against time
0.5
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 39 / 62
41. Numerically computed average and variance
γ = 0.5 and = 0.02
computed interface position vs. time for = 0.5and ε = 0.02
γ
0.01
lev=6; smp=949
lev=7; smp=949
0.005 lev=8; smp=949
lev=9; smp=948 Average of Monte Carlo
0
samples against time for
−0.005 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.01
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
5
computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.02 Sample number smp
lev=6; smp=949
4
lev=7; smp=949 differs with lev. Samples
lev=8; smp=949
3
lev=9; smp=948 are rejected if annihila-
2
tion/nucleation occurs.
1
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 40 / 62
42. Numerically computed average and variance
γ = 0.5 and = 0.01
−3
x 10 computed interface position vs. time for = 0.5and ε = 0.01
γ
4
lev=6; smp=948
lev=7; smp=949
2
lev=8; smp=949
0
lev=9; smp=949 Average of Monte Carlo
−2
samples against time for
−4
various levels of refine-
ment lev = 6, 7, 8, 9,
−6
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
4
computed variance/20ε1+2γ vs. time forγ = 0.5and ε = 0.01 Sample number smp
lev=6; smp=948
3.5
lev=7; smp=949 differs with lev. Samples
3 lev=8; smp=949
2.5
lev=9; smp=949 are rejected if annihila-
2 tion/nucleation occurs.
1.5
1 Variance against time
0.5
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 41 / 62
43. Simulations vs. theory
log(var[Uh ] /ti ) vs. log plot for γ = 0.5 at various times ti = t0 10i
i
log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0.5
−2
ref. slope=1+2*γ
time 0.00152587
−3 time 0.0152587
time 0.152587
time 1.52587
time 15.2587
−4
−5 Theoretical slope
(predicted by
−6
[Funaki, 1995] and
−7 [Brassesco et al., 1995])
is 1 + 2γ.
−8
Plotted in orange refer-
−9 ence line
As → 0 slope im-
−10
−5 −4.5 −4 −3.5 −3 −2.5
proves.
Meshsize h = 1/512.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 42 / 62
44. Numerically computed average and variance
γ = 0.2 and = 0.08
computed interface position vs. time for = 0.2and ε = 0.08
γ
0.4
lev=6; smp=19
0.3
lev=7; smp=0
0.2 lev=8; smp=0
0.1
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.1
−0.2 various levels of refine-
−0.3
ment lev = 6, 7, 8, 9,
−0.4
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
0.5
computed variance/20ε1+2γ vs. time forγ = 0.2and ε = 0.08 Sample number smp
lev=6; smp=19
0.4
lev=7; smp=0 differs with lev. Samples
lev=8; smp=0
0.3
lev=9; smp=0 are rejected if annihila-
0.2
tion/nucleation occurs.
0.1
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 43 / 62
45. Numerically computed average and variance
γ = 0.2 and = 0.04
computed interface position vs. time for = 0.2and ε = 0.04
γ
0.1
lev=6; smp=351
lev=7; smp=303
0.05 lev=8; smp=163
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.05 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.1
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
1
computed variance/20ε1+2γ vs. time forγ = 0.2and ε = 0.04 Sample number smp
lev=6; smp=351
0.8
lev=7; smp=303 differs with lev. Samples
lev=8; smp=163
0.6
lev=9; smp=0 are rejected if annihila-
0.4
tion/nucleation occurs.
0.2
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 44 / 62
46. Numerically computed average and variance
γ = 0.2 and = 0.02
computed interface position vs. time for = 0.2and ε = 0.02
γ
0.15
lev=6; smp=907
lev=7; smp=755
0.1 lev=8; smp=727
lev=9; smp=732 Average of Monte Carlo
0.05
samples against time for
0 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.05
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
2.5
computed variance/20ε1+2γ vs. time forγ = 0.2and ε = 0.02 Sample number smp
lev=6; smp=907
2
lev=7; smp=755 differs with lev. Samples
lev=8; smp=727
1.5
lev=9; smp=732 are rejected if annihila-
1
tion/nucleation occurs.
0.5
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 45 / 62
47. Numerically computed average and variance
γ = 0.2 and = 0.01
computed interface position vs. time for = 0.2and ε = 0.01
γ
0.02
lev=6; smp=949
0.015 lev=7; smp=949
lev=8; smp=944
0.01 lev=9; smp=929 Average of Monte Carlo
0.005
samples against time for
0
various levels of refine-
−0.005
ment lev = 6, 7, 8, 9,
−0.01
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
5
computed variance/20ε1+2γ vs. time forγ = 0.2and ε = 0.01 Sample number smp
lev=6; smp=949
4
lev=7; smp=949 differs with lev. Samples
lev=8; smp=944
3
lev=9; smp=929 are rejected if annihila-
2
tion/nucleation occurs.
1
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 46 / 62
48. Simulations vs. theory
log(var[Uh ] /ti ) vs. log plot for γ = 0.2 at various times ti = t0 10i
i
log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0.2
0
ref. slope=1+2*γ
time 0.00152587
time 0.0152587
−1 time 0.152587
time 1.52587
time 15.2587
−2
Theoretical slope
−3
(predicted by
−4
[Funaki, 1995] and
[Brassesco et al., 1995])
−5 is 1 + 2γ.
Plotted in orange refer-
−6
ence line
As → 0 slope im-
−7
−5 −4.5 −4 −3.5 −3 −2.5
proves.
Meshsize h = 1/512.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 47 / 62
49. Numerically computed average and variance
γ = 0.0 and = 0.08
computed interface position vs. time for = 0and ε = 0.08
γ
1
lev=6; smp=0
lev=7; smp=0
0.5 lev=8; smp=0
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.5 various levels of refine-
ment lev = 6, 7, 8, 9,
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
and meshsize h = 1/2lev .
1
computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.08 Sample number smp
lev=6; smp=0
lev=7; smp=0 differs with lev. Samples
0.5 lev=8; smp=0
lev=9; smp=0 are rejected if annihila-
0 tion/nucleation occurs.
−0.5 Variance against time
for the various levels. As
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 48 / 62
50. Numerically computed average and variance
γ = 0.0 and = 0.04
computed interface position vs. time for = 0and ε = 0.04
γ
0.2
lev=6; smp=120
0.15
lev=7; smp=0
0.1 lev=8; smp=0
0.05
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.05
−0.1 various levels of refine-
−0.15
ment lev = 6, 7, 8, 9,
−0.2
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
0.25
computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.04 Sample number smp
lev=6; smp=120
0.2
lev=7; smp=0 differs with lev. Samples
lev=8; smp=0
0.15
lev=9; smp=0 are rejected if annihila-
0.1
tion/nucleation occurs.
0.05
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 49 / 62
51. Numerically computed average and variance
γ = 0.0 and = 0.02
computed interface position vs. time for = 0and ε = 0.02
γ
0.1
lev=6; smp=792
lev=7; smp=481
0.05 lev=8; smp=307
lev=9; smp=0 Average of Monte Carlo
0
samples against time for
−0.05 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.1
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
0.5
computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.02 Sample number smp
lev=6; smp=792
0.4
lev=7; smp=481 differs with lev. Samples
lev=8; smp=307
0.3
lev=9; smp=0 are rejected if annihila-
0.2
tion/nucleation occurs.
0.1
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 50 / 62
52. Numerically computed average and variance
γ = 0.0 and = 0.01
computed interface position vs. time for = 0and ε = 0.01
γ
0.15
lev=6; smp=2497
lev=7; smp=1646
0.1 lev=8; smp=1148
lev=9; smp=1116 Average of Monte Carlo
0.05
samples against time for
0 various levels of refine-
ment lev = 6, 7, 8, 9,
−0.05
0 2 4 6 8 10 12 14 16 18 20
and meshsize h = 1/2lev .
1
computed variance/20ε1+2γ vs. time forγ = 0and ε = 0.01 Sample number smp
lev=6; smp=2497
0.8
lev=7; smp=1646 differs with lev. Samples
lev=8; smp=1148
0.6
lev=9; smp=1116 are rejected if annihila-
0.4
tion/nucleation occurs.
0.2
Variance against time
for the various levels. As
0
0 2 4 6 8 10 12 14 16 18 20
time grows, variance ac-
curacy deteriorates.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 51 / 62
53. Simulations vs. theory
log(var[Uh ] /ti ) vs. log plot for γ = 0.0 at various times ti = t0 10i
i
log(ε)−log(σ(t)2/t) and log(ε)*(1+2*γ) for γ=0
0
ref. slope=1+2*γ
time 0.00152587
−0.5
time 0.0152587
time 0.152587
−1 time 1.52587
time 15.2587
−1.5
−2
Theoretical slope
(predicted by
−2.5
[Funaki, 1995] and
−3
[Brassesco et al., 1995])
−3.5
is 1 + 2γ.
−4 Plotted in orange refer-
−4.5
ence line
As → 0 slope im-
−5
−5 −4.5 −4 −3.5 −3 −2.5
proves.
Meshsize h = 1/512.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 52 / 62
54. What we learned
Stability and convergence numerical method for the stochastic
Allen-Cahn in 1d.
Monte-Carlo type simulations.
Benchmarking via statistical comparison.
Adaptivity mandatory (but different from “standard” phase-field
approach) for higher dimensional study.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 53 / 62
55. What we’re trying to learn
Current investigation [Kossioris et al., 2009]
finite “exact” solutions
multiple interface convergence
structure of the stochastic solution
colored noise in d > 1
adaptivity
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 54 / 62
56. 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
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 55 / 62
57. References I
Allen, E. J., Novosel, S. J., and Zhang, Z. (1998).
Finite element and difference approximation of some linear stochastic
partial 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 its
application 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-Landau
equation with noise.
Ann. Inst. H. Poincar´ Probab. Statist., 31(1):81–118.
e
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 56 / 62
58. 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.
Carr, J. and Pego, R. L. (1989).
Metastable patterns in solutions of ut = 2 uxx − f (u).
Comm. Pure Appl. Math., 42(5):523–576.
Chen, X. (1994).
Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations
for generic interfaces.
Comm. Partial Differential Equations, 19(7-8):1371–1395.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 57 / 62
59. References III
de Mottoni, P. and Schatzman, M. (1995).
Geometrical evolution of developed interfaces.
Trans. Amer. Math. Soc., 347(5):1533–1589.
Dirr, N., Luckhaus, S., and Novaga, M. (2001).
A stochastic selection principle in case of fattening for curvature flow.
Calc. Var. Partial Differential Equations, 13(4):405–425.
Evans, L. C., Soner, H. M., and Souganidis, P. E. (1992).
Phase transitions and generalized motion by mean curvature.
Comm. Pure Appl. Math., 45(9):1097–1123.
Faris, W. G. and Jona-Lasinio, G. (1982).
Large fluctuations for a nonlinear heat equation with noise.
J. Phys. A, 15:3025–3055.
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60. References IV
Fatkullin, I. and Vanden-Eijnden, E. (2004).
Coarsening by di usion-annihilation in a bistable system driven by
noise.
Preprint, Courant Institute, New York University, New York, NY
10012.
Feng, X. and Wu, H.-j. (2005).
A posteriori error estimates and an adaptive finite element method for
the Allen-Cahn equation and the mean curvature flow.
Journal of Scientific Computing, 24(2):121–146.
Funaki, T. (1995).
The scaling limit for a stochastic PDE and the separation of phases.
Probab. Theory Relat. Fields, 102:221–288.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 59 / 62
61. References V
Funaki, T. (1999).
Singular limit for stochastic reaction-diffusion equation and generation
of random interfaces.
Acta Math. Sin. (Engl. Ser.), 15(3):407–438.
Fusco, G. and Hale, J. K. (1989).
Slow-motion manifolds, dormant instability, and singular
perturbations.
J. Dynam. Differential Equations, 1(1):75–94.
Katsoulakis, M. A. and Szepessy, A. (2006).
Stochastic hydrodynamical limits of particle systems.
Commun. Math. Sci., 4(3):513–549.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 60 / 62
62. References VI
Kessler, D., Nochetto, R. H., and Schmidt, A. (2004).
A posteriori error control for the Allen-Cahn problem: Circumventing
Gronwall’s inequality.
M2AN Math. Model. Numer. Anal., 38(1):129–142.
Kossioris, G., Lakkis, O., and Romito, M. (In preparation 2009).
Interface dynamics for the stochastic Allen–Cahn model: asymptotics
and computations.
Technical report.
Liu, D. (2003).
Convergence of the spectral method for stochastic Ginzburg-Landau
equation driven by space-time white noise.
Commun. Math. Sci., 1(2):361–375.
O Lakkis (Sussex) computational stochastic phase-field Bielefeld, 18 November 2009 61 / 62
63. References VII
Lythe, G. (1998).
Stochastic PDEs: domain formation in dynamic transitions.
In Proceedings of the VIII Reuni´n de F´
o ısica Estad´
ıstica, FISES ’97,
volume 4, pages 55—63. Anales de F´ ısica, Monograf´ RSEF.
ıas
Nestler, B., Danilov, D., and Galenko, P. (2005).
Crystal growth of pure substances: phase-field simulations in
comparison with analytical and experimental results.
J. Comput. Phys., 207(1):221–239.
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