Adaptive Wick-Malliavin approximation to nonlinear SPDEswith discrete random variables

Mengdi Zheng, Boris Rozovskyand George Em Karniadakis

(Brown University)

ICOSAHOM 2014 in Utah

June 24, 2014

Contents

General polynomial chaos (gPC) and stochastic partial differentialequations (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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GPC on SPDEs: spectral method on sample space

The random input of the SPDE is modeled by a random vector Xover a probabilistic space (Ω,F ,P) (assuming independentcomponents 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 withrespect to the input PDF of X

Y =∑α∈NM

yαΨα(X ) =∑α∈NM

yα1,...,αMψ(1)α1

(X1)...ψ(M)αM

(XM) (1)

and yα1,...,αMis 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 >.

11D. 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

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 thatis orthogonal w.r.t. the measure of ξ.

We expand the solution in a finite dimensional series as (up to gPCorder P)

u(x , t; ξ) ≈P∑

k=0

uk(x , t)ck(ξ;λ). (3)

The gPC propagator for this problem is: (motivation of WM)

∂uk∂t

+P∑

m,n=0

um∂un∂x

< cmcnck > = ν∂2uk∂x2

+ σδ1k , k = 0, 1, ...,P,

(4)where < cmcnck >=

∫S dΓ(x)ck(ξ)cm(ξ)cn(ξ).

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Wick-Malliavin series expansion (Poisson RV)

Consider ξ ∼ Pois(λ) with measure Γ(x) =∑

k∈Se−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 = n0 if m 6= 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

Dpci (x ;λ) =i !

(i − p)!ci−p(x ;λ), i = 0, 1, 2, ..., p = 0, 1/2, 1, ..., i .

(7)2

2G.C. Wick, The evaluation of the collision matrix, Phys. Rev. 80(2), (1950),pp. 268–272.5 of 15

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+n2∑

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+n2∑

p=0

Dpcm p Dpcnp!

(10)

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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 uici and v =

∑∞i=0 vici

If we define

Dpu =∞∑i=0

uiDpci (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 theWick-Malliavin order

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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 boundaryconditions, 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...pdu p Dp1...pdux

≈ νuxx + σ

d∑j=1

c1(ξj)ψj(x , t)

(14)

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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...Qdp1...pd=0

∑P1...Pdm1...md=0(um1...md

∂∂x uk1+2p1−m1,...,kd+2pd−md

Km1,k1+2p1−m1,p1 ...Kmd ,kd+2pd−md ,pd )

= ν∂2uk1...kd

(x ,t)

∂x2 + σ(δ1,k1δ0,k2 ...δ0,kdψ1 + ...+ δ0,k1δ0,k2 ...δ1,kdψd)u0,0,...,0(x , 0) = u(x , 0) = 1− sin(x)uk1,...,kd (x , 0) = 0, (k1, ..., kd) 6= (0, ..., 0)Periodic B.C. on [−π, π]

,

(16), where 0 ≤ ki + 2pi −mi ≤ Pi .

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Spectral convergence when Q ≥ P − 1 (1RV)

Figure : Error l2u2(T ) =||E [u2

num(x,T ;ξ)]−E [u2ex (x,T ;ξ)]||L2([−π,π])

||E [u2ex (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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PQ refinement w.r.t. time (1RV)

Figure : Error l2u2(T ) for ut + uux = νuxx + σc1(ξ;λ), x ∈ [−π, π], periodicBC, u(x , 0) = 1− sin(x), ξ ∼ Pois(λ), σ = 1, ν = 1, λ = 1, T = 0.5.

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Burgers equation with 3RVs

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−7

10−6

10−5

10−4

10−3

10−2

T

l2u2(T)

Q1=Q2=Q3=0

Q1=1,Q2=Q3=0

Q1=Q2=1,Q3=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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Computational complexity (1D): WM V.s. gPC

Figure : For ut + uux = νuxx + σc1(ξ;λ)13 of 15

Computational complexity (higher dimensions): WMV.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 dimensionsd = 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 250046 ≈ 61.0% 10201

56 ≈ 65.3% 3132966 ≈ 67.2%

d=3 1250049 ≈ 47.7% 1030301

59 ≈ 52.8% 554523369 ≈ 55.0%

d=50 8.89e+844150

≈ 0.000436% 1.64e+1005150

≈ 0.0023% 2.5042e+1126150

≈ 0.0047%

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Thanks and references

D. Bell, The Malliavin calculus, Dover, (2007).

S. Kaligotla and S.V. Lototsky, Wick product in thestochastic Burgers equation: a curse or a cure?, AsymptoticAnalysis 75, (2011), pp. 145–168.

S.V. Lototsky, B.L. Rozovskii, and D. Selesi, Ongeneralized Malliavin calculus, Stochastic Processes and theirApplications 122(3), (2012), pp. 808–843.

D. Venturi, X. Wan, R. Mikulevicius, B.L. Rozovskii,G.E. Karniadakis, Wick-Malliavin approximation to nonlinearstochastic PDEs: analysis and simulations, Proceedings of theRoyal Society, vol.469, no.2158, (2013).

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