homogenization with stochastic differential equationsshottovy/presentationmp11f.pdfmodeling with sde...
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Homogenization with stochastic differential equations
Scott [email protected]
University of Arizona Program in Applied Mathematics
October 12, 2011
Modeling with SDE
• Use SDE to model system (e.g. particle displacesmentxt ∈ Rn at time t in viscous fluid)
dxt = b(xt) dt + σ(xt) dαWt , x0 = x.
• Wt , m-dim. Wiener Process, b drift σ noise. This assumeszero correlation.∫ t
0σ(xt) dαWt = lim
N→∞
N∑i=1
σ(xtαi , ω)(Wti −Wti−1),
tαi = αti + (1− α)ti−1,
• Integral varies with α. Special cases α = 0 Ito, α = 1/2Stratonovich, α = 1 anti-Ito.∫ t
0Ws dαWs ,=
1
2W2
t −(
1
2− α
)t
Modeling with SDE
• Use SDE to model system (e.g. particle displacesmentxt ∈ Rn at time t in viscous fluid)
dxt = b(xt) dt + σ(xt) dαWt , x0 = x.
• Wt , m-dim. Wiener Process, b drift σ noise. This assumeszero correlation.∫ t
0σ(xt) dαWt = lim
N→∞
N∑i=1
σ(xtαi , ω)(Wti −Wti−1),
tαi = αti + (1− α)ti−1,
• Integral varies with α. Special cases α = 0 Ito, α = 1/2Stratonovich, α = 1 anti-Ito.∫ t
0Ws dαWs ,=
1
2W2
t −(
1
2− α
)t
Experiment
(courtesy Giovanni Volpe)Measure forces
General Model
dxmt = vmt dt
dvmt =
(F (xmt )
m− γ(xmt )
mvmt
)dt +
σ(xmt )
mdWt .
• For σ, γ positive and Lipschitz, then use property ofstochastic integral:
• For f smooth (E [|f (s, ω)− f (t, ω)|2] ≤ k |s − t|1+ε),∫ t
0f (s, ω) dαWs =
∫ t
0f (s, ω) dWs , for all α.
• Approximated by the Smoluchowski-Kramers approximation asm→ 0,
dxt =F (xt)
γ(xt)dt +
σ(xt)
γ(xt)dαWt ,
• Lost smoothness of xt . Stochastic integral varies with α
dxt =
[F (xt)
γ(xt)+ α
σ(xt)
γ(xt)
d
dxt
(σ(xt)
γ(xt)
)]dt +
σ(xt)
γ(xt)dWt ,
• The second drift term is called spurious (or noise induced)drift.
• What is α?
• Approximated by the Smoluchowski-Kramers approximation asm→ 0,
dxt =F (xt)
γ(xt)dt +
σ(xt)
γ(xt)dαWt ,
• Lost smoothness of xt . Stochastic integral varies with α
dxt =
[F (xt)
γ(xt)+ α
σ(xt)
γ(xt)
d
dxt
(σ(xt)
γ(xt)
)]dt +
σ(xt)
γ(xt)dWt ,
• The second drift term is called spurious (or noise induced)drift.
• What is α?
Connection to PDE
dxmt = vmt dt
mdvmt = F (xmt )− γ(xmt )vt dt + σ(xmt ) dWt ,
Density pm(x ′, v ′, t ′|x , v , t) satisfies the backward Kolmogorovequation
∂pm∂t
=σ(x)2
2m
∂2pm∂v2
+ v∂pm∂x
+
((F (x)− γ(x)v)
m
)∂pm∂v
= Lx ,vpm,
Also, for all (x ′, v ′) ∈ R2, pm satisfies the forward Kolmogorov (orFokker-Planck) equation
∂pm∂t ′
= L∗x ′,v ′pm.
Homogenization
• “Homogenization theory studies the effects of high-frequencyoscillations in the coefficients upon solutions of PDE.” [Evans98]
• Ex: Conductor composed of several materials with differentconductivities.
http://www.sciencephoto.com/media/8839
Homogenization
• “Homogenization theory studies the effects of high-frequencyoscillations in the coefficients upon solutions of PDE.” [Evans98]
• Ex: Conductor composed of several materials with differentconductivities.
http://www.sciencephoto.com/media/8839
• Conductivity is a periodic function a(x).The temperature, uε satisfies,
Lu(x) = −n∑
i ,j=1
(aij
(xε
)uεxi (x)
)xj
= f (x) in U,
uε(x) = 0, in ∂U.
• Assume the uniform ellipticity condition
n∑i ,j=1
aij(y)ξiξj ≥ θ|ξ|2,
for constant θ > 0, and all y , ξ ∈ Rn.
• Two length scales, macro O(1), and micro O(ε).
• Assume solution uε(x)→ u(x) as ε→ 0.
• Goal is to find the PDE that u(x) satisfies.
uε(x) = u0(x , x/ε) + εu1(x , x/ε) + ε2(x , x/ε) + ...
• u : U × Q → R, where Q is the unit cube in Rn.
• Plug this solution into the original PDE
• result
Luε =1
ε2L1u0+
1
ε(L1u1+L2u0)+(L1u2+L2u1+L3u0)+O(ε) = f .
• Then Solve the system of PDEO(1ε2
)L1u0 = 0,
O(1ε
)L1u1 + L2u0 = 0,
O (1) L1u2 + L2u1 + L3u0 = f ,
Colored Noise Example
• For a physical process xt that satisfies
dxt = b(xt) + σ(xt) dWt ,
• xt is continuous, but not differentiable.
• Physically more realistic to replace Wt by differentiableprocess
ξεt =1
ε
∫k
((t − s)
ε2
)Ws ds,
• for smooth kernel k . ξεt is called colored noise.
• Now xεt is smooth (dependent on smoothness of k) andsatisfies
dxεt =
(b(xεt ) + σ(xεt )
dξεtdt
)dt
Colored Noise Example
• For a physical process xt that satisfies
dxt = b(xt) + σ(xt) dWt ,
• xt is continuous, but not differentiable.
• Physically more realistic to replace Wt by differentiableprocess
ξεt =1
ε
∫k
((t − s)
ε2
)Ws ds,
• for smooth kernel k . ξεt is called colored noise.
• Now xεt is smooth (dependent on smoothness of k) andsatisfies
dxεt =
(b(xεt ) + σ(xεt )
dξεtdt
)dt
• Since ξεt →Wt expect
dxεt =
(b(xεt ) + σ(xεt )
dξεtdt
)dt
• Since ξεt →Wt expect
dxt = b(xt) dt + σ(xt) dWt
Wong, Zakai (1965)
The solution to
dxεt =
(b(xεt ) + σ(xεt )
dξεtdt
)dt
converges to the solution of
dxt = b(xt) dt + σ(xt)dα=1/2Wt
= b(xt) +1
2σ(xt)
dσ(xt)
dxdt + σ(xt)d0Wt .
• Since ξεt →Wt expect
dxt = b(xt) dt + σ(xt) dWt
Wong, Zakai (1965)
The solution to
dxεt =
(b(xεt ) + σ(xεt )
dξεtdt
)dt
converges to the solution of
dxt = b(xt) dt + σ(xt)dα=1/2Wt
= b(xt) +1
2σ(xt)
dσ(xt)
dxdt + σ(xt)d0Wt .
Colored Noise Approximation
• Ornstein-Ulhenbeck SP to model colored noise
dyt = −aytε2
dt +
√2a
ε2dWt , y0 = 0.
yt =
√2a
ε2
∫ t
0exp
{ a
ε2(s − t)
}dW (s)
• Mean zero Gaussian random variable with covariance
E [yt1yt2 ] = exp{− a
ε2|t1 − t2|
}.
Colored Noise Approximation
• Ornstein-Ulhenbeck SP to model colored noise
dyt = −aytε2
dt +
√2a
ε2dWt , y0 = 0.
yt =
√2a
ε2
∫ t
0exp
{ a
ε2(s − t)
}dW (s)
• Mean zero Gaussian random variable with covariance
E [yt1yt2 ] = exp{− a
ε2|t1 − t2|
}.
• Use homogenization theory to derive Wong and Zakai result.{dxεt = b(xεt ) + σ(xεt )yt
ε dt
dyt = −aytε2
dt +√
2aε2dWt ,
• The generator for (xεt , yt)
Lε = b(x)∂
∂x+σ(x)y
ε
∂
∂x− ay
ε2∂
∂y+
a
ε2∂2
∂y2
= L2 +1
εL1 +
1
ε2Ly .
• Ansatz: Solution to the backward Kolmogorov equationpm(x , y) = p0(x , y) + εp1(x , y) + ε2p2(x , y) + ....
• BK equation
∂p0∂t
= (Lyp2 + L1p1 + L2p0) +1
ε2Lyp0 +
1
ε(Lyp1 + L1p0)
• O(ε−2),
−ay
ε2∂p0(x , y)
∂y+
a
ε2∂2p0(x , y)
∂y2= 0
.Only solution is p0(x , y) = p0(x).
• BK equation
∂p0∂t
= (Lyp2 + L1p1 + L2p0) +1
ε2Lyp0 +
1
ε(Lyp1 + L1p0)
• O(ε−2),
−ay
ε2∂p0(x , y)
∂y+
a
ε2∂2p0(x , y)
∂y2= 0
.Only solution is p0(x , y) = p0(x).
• BK equation
∂p0∂t
= (Lyp2 + L1p1 + L2p0) +1
ε2Lyp0 +
1
ε(Lyp1 + L1p0)
• O(ε−1),
−ay
ε2∂p1(x , y)
∂y+
a
ε2∂2p1(x , y)
∂y2= −σ(x)y
∂p0(x)
∂x.Set p1(x , y) = Φ(x , y)∂p0∂x + Φ1(x). (Cell Problem)
• Set Φ1(x) = 0.
Φ(x , y) =1
aσ(x)y
.
• Cell problem always solvable?
• O(1)
∂p0∂t
= (Lyp2 + L1p1 + L2p0)
• Cell problem always solvable?
• O(1)
−Lyp2 = −∂p0∂t
+ L1
(Φ(x , y)
∂p0∂x
)+ L2p0
• Use the Fredholm alternative to find a solvability condition.
• The r.h.s. integrated against the solution of
L∗yρ∞(x , y) = 0
,must hold.
• This is the stationary Fokker-Planck (forward Kolmogorovequation) for the OU process,
dyt = −ayt dt +√
2adWt
,
• Stationary density,
ρ∞(y) =1
2πexp
(−y2
2
).
• Solvability condition,
∫ ∞−∞
(−∂p0∂t
+ L1
(Φ(x , y)
∂p0∂x
)+ L2p0
)ρ∞(y) dy = 0
• Diffusion process,
∂p0∂t
=
(b(x) +
1
aσ′(x)σ(x)
)∂p0∂x
+1
aσ(x)
∂2p0∂x2
,
• Yields the SDE,
dxt =
(b(xt) +
1
aσ′(xt)σ(xt)
)dt +
√2
aσ(x) dWt
• xεt → xt in distribution, as ε→ 0 (with much more work).
• For the Newton equation with ut =√mvt ,
dxt =ut√m
dt
dut =F (xt)√
m− γ(xt)vt
mdt +
σ(xt)√m
dWt .
• Solve the system for p0
O(1m
)Lup0 =
(σ(x)2
2∂2
∂u2− γ(x)u ∂
∂u
)p0 = 0,
O(
1√m
)Lup1 = −L1p0 = −
(u ∂∂x + F (x) ∂
∂u
)p0,
O(1) ∂p0∂t = Lup2 + L1p1,
• For the Newton equation with ut =√mvt ,
dxt =ut√m
dt
dut =F (xt)√
m− γ(xt)vt
mdt +
σ(xt)√m
dWt .
• Solve the system for p0
O(1m
)Lup0 =
(σ(x)2
2∂2
∂u2− γ(x)u ∂
∂u
)p0 = 0,
O(
1√m
)Lup1 = −L1p0 = −
(u ∂∂x + F (x) ∂
∂u
)p0,
O(1) ∂p0∂t = Lup2 + L1p1,
• For Fredholm alternative arg. need solution to the stationaryFP of,
dut = −γ(x)ut dt + σ(x) dWt .
• Need σ, γ independent of u.
• OU process with parameter x .
• Solution is a Gaussian density.
ρ(u; x) = C (x) exp
{−γ(x)u2
σ(x)2
}.
• the BK equation satisfied by p0.
σ(x)2
2γ(x)2∂2p0∂x2
+
(F (x)
γ(x)− σ(x)2
2γ(x)3d(γ(x))
dx
)∂p0∂x
=∂p0∂t
.
• Compare to the PDE of SK
σ(x)2
2γ(x)2∂2p
∂x2+
(F (x)
γ(x)+ α
[− σ(x)
γ(x)3γ′(x)− σ(x)2
γ(x)2σ′(x)
])∂p
∂x=∂p
∂t.
• Effective SDE for xt ,
dxt =
(F (xt)
γ(xt)− σ(xt)
2
2γ(xt)3γ′(xt)
)dt +
σ(xt)
γ(xt)dWt ,
• Wt is an arbitrary Wiener process. (From studying the inf.operator)
• the BK equation satisfied by p0.
σ(x)2
2γ(x)2∂2p0∂x2
+
(F (x)
γ(x)− σ(x)2
2γ(x)3d(γ(x))
dx
)∂p0∂x
=∂p0∂t
.
• Compare to the PDE of SK
σ(x)2
2γ(x)2∂2p
∂x2+
(F (x)
γ(x)+ α
[− σ(x)
γ(x)3γ′(x)− σ(x)2
γ(x)2σ′(x)
])∂p
∂x=∂p
∂t.
• Effective SDE for xt ,
dxt =
(F (xt)
γ(xt)− σ(xt)
2
2γ(xt)3γ′(xt)
)dt +
σ(xt)
γ(xt)dWt ,
• Wt is an arbitrary Wiener process. (From studying the inf.operator)
Equation for α
α = α(x) =γ′(x)σ(x)
2(γ′(x)σ(x)− γ(x)σ′(x)).
α constant iff γ(x) = cσ(x)λ.
α =λ
2(λ− 1)
λ =α
α− 12
α vs λ
• Next Austin studies SDE systems with colored noise and delay.
α vs λ
• Next Austin studies SDE systems with colored noise and delay.