icosahom 2014 conference in utah

1. Adaptive Wick-Malliavin approximation to nonlinear SPDEs
with discrete random variables
Mengdi Zheng, Boris Rozovsky
and George Em Karniadakis
(Brown University)
ICOSAHOM 2014 in Utah
June 24, 2014

2. Contents
General polynomial chaos (gPC) and stochastic partial differential
equations (SPDEs) (gPC order P)
Wick-Malliavin approximation (WM) to gPC (WM order Q)
Burgers equation with discrete random input by WM
P-Q convergence of error (exponential convergence when Q ≥ P − 1)
P-Q refinements with respect to time (adaptive)
Computational complexity comparison between gPC and WM
Introduce the WM diagram
Comparison on stochastic Burgers equation with multiple random
variables (RVs)
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3. GPC on SPDEs: spectral method on sample space
The random input of the SPDE is modeled by a random vector X
over a probabilistic space (Ω, F, P) (assuming independent
components of X)
The response random vector (the solution of the SPDE)
Y = M(X) is considered as an element of L2(Ω, F, P)
A basis of multivariate orthogonal polynomials is built up with
respect to the input PDF of X
Y =
α∈NM
yαΨα(X) =
α∈NM
yα1,...,αM
ψ(1)
α1
(X1)...ψ(M)
αM
(XM) (1)
and yα1,...,αM
is to be computed by taking the inner product of Y
w.r.t. each basis function Ω dP(X)Ψα(X)Y =< Ψα(X)Y >.
1
1
D. Xiu and G.E. Karniadakis, The Wiener–Askey polynomial chaos for
stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644.3 of 15

4. gPC propagator for 1D stochastic Burgers equation
As an example of gPC, we consider
ut + uux = νuxx + σc1(ξ; λ), x ∈ [−π, π], (2)
with deterministic initial condition, where ξ is a discrete RV
(Pois(λ)) and ck (Charlier polynomial) is the k-th polynomial that
is orthogonal w.r.t. the measure of ξ.
We expand the solution in a finite dimensional series as (up to gPC
order P)
u(x, t; ξ) ≈
P
k=0
ˆuk(x, t)ck(ξ; λ). (3)
The gPC propagator for this problem is: (motivation of WM)
∂ˆuk
∂t
+
P
m,n=0
ˆum
∂ˆun
∂x
< cmcnck > = ν
∂2ˆuk
∂x2
+ σδ1k, k = 0, 1, ..., P,
(4)4 of 15

5. Wick-Malliavin series expansion (Poisson RV)
Consider ξ ∼ Pois(λ) with measure Γ(x) = k∈S
e−λλk
k! δ(x − k),
on the support S = {0, 1, 2, ...}
With monic Charlier polynomials associated with Pois(λ):
k∈S
e−λλk
k!
cm(k; λ)cn(k; λ) =
n!λnδmn if m = n
0 if m = n
(5)
Define the Wick product ’ ’ as
cm(x; λ) cn(x; λ) = cm+n(x; λ), m, n = 0, 1, 2, ... (6)
Define the Malliavin derivative ’D’ as
Dp
ci (x; λ) =
i!
(i − p)!
ci−p(x; λ), i = 0, 1, 2, ..., p = 0, 1/2, 1, ..., i.
(7)
2
2
G.C. Wick, The evaluation of the collision matrix, Phys. Rev. 80(2), (1950),
pp. 268–272.5 of 15

6. Wick-Malliavin series expansion (continued)
The product of two polynomials can be expanded as
cm(x)cn(x) =
m+n
k=0
a(k, m, n)ck(x) =
m+n
2
p=0
Kmnpcm+n−2p(x; λ) (8)
where Kmnp = a(m + n − 2p, m, n)
Define the weighted Wick product ’ p’ in terms of the Wick
product as
cm p cn =
p!m!n!
(m + p)!(n + p)!
Km+p,n+p,pcm cn, (9)
Therefore
cm(x; λ)cn(x; λ) =
m+n
2
p=0
Dpcm p Dpcn
p!
(10)
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7. Wick-Malliavin series expansion (continued)
Given two random fields u and v on the same probability space
(S, B(S), Γ), with u = ∞
i=0 ui ci and v = ∞
i=0 vi ci
If we define
Dp
u =
∞
i=0
ui Dp
ci (11)
We can expand uv by
uv =
∞
p=0
Dpu p Dpv
p!
≈
Q
p=0
Dpu p Dpv
p!
(12)
We define the non-negative half integer Q ∈ {0, 1/2, 1, ...} as the
Wick-Malliavin order
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8. WM approximation for stochastic Burgers equation
We consider
ut + uux = νuxx + σ
d
j=1
c1(ξj )ψj (x, t), x ∈ [−π, π], (13)
with initial condition u(x, 0) = 1 − sin(x) and periodic boundary
conditions, where ξ1,...,d ∼ Pois(λ) are i.i.d. RVs.
The WM approximation to the equation is
ut +
Q1,...,Qd
p1,...,pd =0
1
p1!...pd !
Dp1...pd
u p Dp1...pd
ux
≈ νuxx + σ
d
j=1
c1(ξj )ψj (x, t)
(14)
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9. WM propagator for stochastic Burgers equation
We expand the solution in a finite dimensional series as
u(x, t; ξ1, ..., ξd ) ≈
P1,...,Pd
k1,...,kd =0
˜uk1,...,kd
(x, t)ck1 (ξ1; λ)...ckd
(ξd ; λ),
(15)
The WM propagator is (IMEX:RK2/CN)



∂˜uk1...kd
∂t + Q1...Qd
p1...pd =0
P1...Pd
m1...md =0(˜um1...md
∂
∂x ˜uk1+2p1−m1,...,kd +2pd −md
Km1,k1+2p1−m1,p1 ...Kmd ,kd +2pd −md ,pd
)
= ν
∂2˜uk1...kd
(x,t)
∂x2 + σ(δ1,k1 δ0,k2 ...δ0,kd
ψ1 + ... + δ0,k1 δ0,k2 ...δ1,kd
ψd )
˜u0,0,...,0(x, 0) = u(x, 0) = 1 − sin(x)
˜uk1,...,kd
(x, 0) = 0, (k1, ..., kd ) = (0, ..., 0)
Periodic B.C. on [−π, π]
,
(16)
, where 0 ≤ ki + 2pi − mi ≤ Pi .
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10. Spectral convergence when Q ≥ P − 1 (1RV)
Figure : Error l2u2(T) =
||E[u2
num(x,T;ξ)]−E[u2
ex (x,T;ξ)]||L2([−π,π])
||E[u2
ex (x,T;ξ)]||L2([−π,π])
for
ut + uux = νuxx + σc1(ξ; λ), x ∈ [−π, π], periodic BC, u(x, 0) = 1 − sin(x),
ξ ∼ Pois(λ), σ = 1, ν = 1, λ = 1, T = 0.5.
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11. PQ refinement w.r.t. time (1RV)
Figure : Error l2u2(T) for ut + uux = νuxx + σc1(ξ; λ), x ∈ [−π, π], periodic
BC, u(x, 0) = 1 − sin(x), ξ ∼ Pois(λ), σ = 1, ν = 1, λ = 1, T = 0.5.
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12. Burgers equation with 3RVs
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
T
l2u2(T)
Q
1
=Q
2
=Q
3
=0
Q
1
=1,Q
2
=Q
3
=0
Q
1
=Q
2
=1,Q
3
=0
Q1
=Q2
=Q3
=1
Figure : l2u2(T) for ut + uux = νuxx + σ
3
j=1 c1(ξj )cos(0.1jt),
x ∈ [−π, π], periodic BC, u(x, 0) = 1 − sin(x), ξ1,2,3 ∼ Pois(λ), λ = 0.1,
σ = 0.1, y0 = 1, ν = 1/100, P = 2.
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13. Computational complexity (1D): WM V.s. gPC
Figure : For ut + uux = νuxx + σc1(ξ; λ)
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14. Computational complexity (higher dimensions): WM
V.s. gPC
Table : Computational complexity ratio to evaluate u ∂u
∂x term in Burgers
equation with d RVs between WM and gPC, as C(P,Q)d
(P+1)3d : here we take the
WM order as Q = P − 1, and gPC with order P, in different dimensions
d = 2, 3, and 50. The higher the dimension, the less WM costs than gPC.
C(P, Q) is the number of terms as ˜ui
∂˜uj
∂x in the WM propagator for each RV.
C(P,Q)d
(P+1)3d P = 3, Q = 2 P = 4, Q = 3 P = 5, Q = 4
d=2 2500
46 ≈ 61.0% 10201
56 ≈ 65.3% 31329
66 ≈ 67.2%
d=3 12500
49 ≈ 47.7% 1030301
59 ≈ 52.8% 5545233
69 ≈ 55.0%
d=50 8.89e+84
4150
≈ 0.000436% 1.64e+100
5150
≈ 0.0023% 2.5042e+112
6150
≈ 0.0047%
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15. Thanks and references
D. Bell, The Malliavin calculus, Dover, (2007).
S. Kaligotla and S.V. Lototsky, Wick product in the
stochastic Burgers equation: a curse or a cure?, Asymptotic
Analysis 75, (2011), pp. 145–168.
S.V. Lototsky, B.L. Rozovskii, and D. Selesi, On
generalized Malliavin calculus, Stochastic Processes and their
Applications 122(3), (2012), pp. 808–843.
D. Venturi, X. Wan, R. Mikulevicius, B.L. Rozovskii,
G.E. Karniadakis, Wick-Malliavin approximation to nonlinear
stochastic PDEs: analysis and simulations, Proceedings of the
Royal Society, vol.469, no.2158, (2013).
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