This document summarizes a thesis on numerical methods for stochastic systems subject to generalized Levy noise. It includes: 1) Motivation for studying such systems from both mathematical and applicational perspectives, such as in mathematical finance and chaotic flows. 2) An introduction to Levy processes and the probability collocation method (PCM) for uncertainty quantification (UQ). 3) Details on improving PCM through a multi-element approach and constructing orthogonal polynomials for discrete measures.

1. Numerical methods
for stochastic systems
subject to generalized Levy noise
by Mengdi Zheng!
Thesis committee: George Em Karniadakis (Ph.D., advisor)!
Hui Wang (Ph.D., reader, APMA, Brown)!
Xiaoliang Wan (Ph.D., reader, Mathematics, LSU)

2. Contents
Besides motivation and introduction,

3. Motivation
Mathematical
Reasons
Applicational Reasons
Mathematical
Finance
!
Levy flights
in Chaotic
flows

4. Probability collocation method (PCM) in UQ
Xt (ω) ≈ Xt (ξ1,ξ2,...,ξn ) ω ∈Ω Ε[um
(x,t;ω)] ≈ Ε[um
(x,t;ξ1,ξ2,...,ξn )]
ξ1
ξ2
ξ3
... ξn
O Ω
PCM
ξ1
ξ2
Ω
O
MEPCM
B1 B2
B3 B4
n = 2
I = dΓ(x) f (x) ≈
a
b
∫ dΓ(x) f (xi )hi (x)
i=1
d
∑ = f (xi ) dΓ(x)hi (x)
a
b
∫i=1
d
∑
a
b
∫
Gauss integration:
u(x,t;ξ1i )
i=1
d
∑ wi
n = 1
{Pi (x)}orthogonal to Γ(x)
Pd (x)zeros of
Lagrange
interpolation
at zeros{xi ,i = 1,..,d}

5. sample path of a
Poisson process
Jt
t
Introduction of Levy processes

6. Gauss
quadrature
and weights
generate
orthogonal to{Pj
k (ξ j
)}
µ j
(ξ j
)
moment
statistics
5
ways
tensor
product
or
sparse
grid
Ε[um
(x,t;ω)]
0
17.5
35
52.5
70
1 2 3 4
measure of
ξi
ξi
what if is a set of experimental data?ξi
subjective assumption of!
distribution shapes
?
ut + 6uux + uxxx = σiξi ,
i=1
n
∑ x ∈!
u(x,0) =
a
2
sech2
(
a
2
(x − x0 ))
Data-driven UQ for stochastic KdV equations
M. Zheng, X. Wan, G.E. Karniadakis, Adaptive multi-element polynomial chaos with discrete measure: Algorithms and
application to SPDEs, Applied Numerical Mathematics, 90 (2015), pp. 91–110.
A(k + n,n) = (−1)k+n−|i| n −1
k + n− | i |
⎛
⎝⎜
⎞
⎠⎟ (Ui1
⊗...⊗Uin
)
k+1≤|i|≤k+n
∑
Sparse grids in Smolyak algorithm: level k, dimension n
sparse!
grid
tensor!
product!
grid

7. Construct orthogonal polynomials to discrete measures
1. (Nowak) S. Oladyshkin, W. Nowak, Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion,
Reliability Engineering & System Safety, 106 (2012), pp. 179–190.
2. (Stieltjes, Modified Chebyshev) W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Stat. Comp., 3 (1982), no.3, pp.
289–317.
3. (Lanczos) D. Boley, G. H. Golub, A survey of matrix inverse eigenvalue problems, Inverse Problems, 3 (1987), pp. 595–622.
4. (Fischer) H. J. Fischer, On generating orthogonal polynomials for discrete measures, Z. Anal. Anwendungen, 17 (1998), pp. 183–
205.
f (k;N, p) =
N!
k!(N − k)!
pk
(1− p)N−k
,k = 0,1,...,N.
10 20 40 80 100
10
−4
10
−3
10
−2
10
−1
10
0
polynomial order i
CPUtimetoevaluateorth(i)
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
C*i2
n=100,p=1/2
polynomial order i
Bino(100, 1/2)
0 10 20 30 40 50 60 70 80 90 100
10
−20
10
−15
10
−10
10
−5
10
0
polynomial order i
orth(i)
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
N=100, p=1/2
polynomial order i
Bino(100, 1/2)
orthogonality ?
cost ?
Bino(N, p)Binomial distribution

8. | f (ξ)µ(ξ)− Qm
Bi
f
i=1
Nes
∑
Γ
∫ |≤ Chm+1
|| EΓ ||m+1,∞,Γ | f |m+1,∞,Γ
{Bi
}i=1
Nes
: elementsNes : number of elements
: number of elementsΓ
µ: discrete measure
Qm
Bi
Gauss quadrature + tensor product
with exactness m=2d-1
h: maximum size of Bi
f : test function in W m+1,∞
(Γ)
(when the measure is continuous) J. Foo, X. Wan, G. E. Karniadakis, A multi-element probabilistic collocation method for PDEs with
parametric uncertainty: error analysis and applications, Journal of Computational Physics 227 (2008), pp. 9572–9595.
10
0
10
1
10
−6
10
−5
10
−4
10
−3
10
−2
N
es
absoluteerror
c=0.1,w=1
GENZ1
d=2
m=3
bino(120,1/2)Bino(120, 1/2)
10
0
10
1
10
−13
10
−12
10
−11
10
−10
10
−9
N
es
absoluteerrors
c=0.1,w=1
GENZ4
d=2
m=3
bino(120,1/2)
Bino(120, 1/2)
0 20 40 60 80 100
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
GENZ1 function (oscillations)
w=1, c=0.01
w=1,c=0.1
w=1,c=1
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
GENZ4 function (Gaussian)
c=0.01,w=1
c=0.1,w=1
c=1,w=1
Multi-element Gauss integration over discrete measures

9. UQ of stochastic KdV equation with 1RVut + 6uux + uxxx = σiξi ,
i=1
n
∑ x ∈!
u(x,0) =
a
2
sech2
(
a
2
(x − x0 ))
2 3 4 5 6 7 8
10
−3
10
−2
d
error
l2u1aPC
l2u2
aPC
2 3 5 10 15 20 30
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
Nes
error
l2u1
l2u2
C*Nel−4
h-convergence!
of MEPCM
p-convergence of PCM
Poisson distribution
Binomial distribution
σi
2
local variance to the measure µ(dξ) / µ(dξ)
Bi
∫
adaptive !
integration!
mesh
2 3 4 5 6
10
−5
10
−4
10
−3
10
−2
Number of PCM points on each element
errors 2 el, even grid
2 el, uneven grid
4 el, even grid
4 el, uneven grid
5 el, even grid
5 el, uneven grid
(MEPCM)!
adaptive!
vs.!
non-
adaptive!
meshes
error of Ε[u2
]
Improved !

10. ut + 6uux + uxxx = σiξi ,
i=1
n
∑ x ∈!
u(x,0) =
a
2
sech2
(
a
2
(x − x0 ))
(sparse grid) D. Xiu, J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Scientific
Computing 27(3) (2005), pp. 1118– 1139.
17 153 256 969 4,845
10
−10
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
r(k)
errors
l2u1(sparse grid)
l2u2(sparse grid)
l2u1(tensor product grid)
l2u2(tensor product grid)
sparse grid vs. tensor product grid
Binomial distribution
n=8
Improved !
2D sparse grid in Smolyak algorithm
UQ of stochastic KdV equation with multiple RVs

11. Summary of contributions (1)
✰ convergence study of multi-element integration over
discrete measure
!
✰ comparison of 5 ways to construct orthogonal
polynomials w.r.t. discrete measure
!
✰ improvement of moment statistics by adaptive
integration mesh (on discrete measure)
!
✰ improvement of moment statistics by sparse grid (on
discrete measure)

12. gPC for 1D stochastic Burgers equation
M. Zheng, B. Rozovsky, G.E. Karniadakis, Adaptive Wick-Malliavin Approximation to Nonlinear SPDEs with Discrete Random
Variables, SIAM J. Sci. Comput., revised. (multiple discrete RVs)
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). (multiple Gaussian RVs)
ut + uux = υuxx +σc1(ξ;λ), x ∈[−π,π]
u(x,t;ξ) ≈ u!k (x,t)ck (ξ;λ)
k=0
P
∑Expand the solution:
∂u!k
∂t
+ u!m
∂u!n
∂t
< cmcnck >=
m,n=0
P
∑ υ
∂2
u!k
∂x2
+σδ1k ,
general Polynomial Chaos (gPC) propagator
k = 0,1,...,P.
ξ ∼ Pois(λ)
ck : Charlier polynomial
e−λ
λk
k!
cm (k;λ)cn (k;λ) =< cmcn >= n!λn
δmn
k∈!
∑
cm
by Galerkin projection : < uck >
nonlinear
How many numbers of
terms !
!
!
!
there are !
u!m
∂u!n
∂t
< cmcnck >
?
(motivation)
Let us simplify the gPC propagator !
(P +1)3

13. Wick-Malliavin (WM) approximation
G.C. Wick, The evaluation of the collision matrix, Phys. Rev. 80 (2), (1950), pp. 268-272.
◊
ξ ∼ Pois(λ)✰ consider with measure
!
✰ define Wick product as:
✰ define Malliavin derivative D as:
!
✰ the product of two polynomials can be approximated by:
!
!
!
✰ here
!
✰ define weighted Wick product :
!
✰ rewrite the product of two polynomials:
!
!
✰ approximate the product of uv as:
Γ(x) =
e−λ
λk
k!
δ(x − k)
k∈!
∑
cm (x;λ)◊cn (x;λ) = cm+n (x;λ),
Dp
ci (x;λ) =
i!
(i − p)!
ci−p (x;λ)
cm (x)cm (x) = a(k,m,n)ck (x) =
k=0
m+n
∑ Kmnpcm+n−2 p (x;λ)
p=0
m+n
2
∑
Kmnp = a(m + n − 2p,m,n)
◊p
cm◊pcn =
p!m!n!
(m + p)!(n + p)!
Km+p,n+p,pcm◊cn
cmcn =
Dp
cm◊pDp
cn
p!p=0
m+n
2
∑
uv =
Dp
u◊pDp
v
p!
≈
p=0
∞
∑
Dp
u◊pDp
v
p!p=0
Q
∑

14. WM approximation simplifies the gPC propagator !
ut + uux = υuxx +σc1(ξ;λ), x ∈[−π,π]
∂u!k
∂t
+ u!m
∂u!n
∂t
< cmcnck >=
m,n=0
P
∑ υ
∂2
u!k
∂x2
+σδ1k , k = 0,1,...,P.gPC propagator:
∂u!k
∂t
+ u!i
i=0
P
∑
∂u!k+2 p−i
∂x
Ki,k+2 p−i,Q =
p=0
Q
∑ υ
∂2
u!k
∂x2
+σδ1k , k = 0,1,...,P.WM propagator:
How much less? Let us count the dots !
k=0 k=1 k=2 k=3 k=4
P=4, Q=1/2

15. Spectral convergence when Q ≥ P −1
ut + uux = υuxx +σc1(ξ;λ) u(x,0) = 1− sin(x)
ξ ∼ Pois(λ) x ∈[−π,π] Periodic B.C.

16. concept of P-Q refinement
ut + uux = υuxx +σc1(ξ;λ) u(x,0) = 1− sin(x)
ξ ∼ Pois(λ) x ∈[−π,π] Periodic B.C.

17. WM for stochastic Burgers equation w/ multiple RVs
ut + uux = υuxx +σ
j=1
3
∑ c1(ξj;λ)cos(0.1jt)
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
Q1
=1,Q2
=Q3
=0
Q1
=Q2
=1,Q3
=0
Q
1
=Q
2
=Q
3
=1
u(x,0) = 1− sin(x) ξ1,2,3 ∼ Pois(λ)
x ∈[−π,π] Periodic B.C.
How about 3 discrete RVs ? How about the cost in d-dim ?
C(P,Q)d
the # of terms u!i
∂u! j
∂x
(P +1)3d
the # of terms u!m
∂u!n
∂t
Let us find the ratio
C(P,Q)d
(P +1)3d
P=3
Q=2
P=4
Q=3
d=2 61.0% 65.3%
d=3 47.7% 52.8%
d=4 0.000436% 0.0023%
C(P,Q)d
(P +1)3d
??
Cost: WM vs. gPC

18. Summary of contributions (2)
References
✰ Extend the numerical work on WM approximation for
SPDEs driven by Gaussian RVs to discrete RVs with
arbitrary distribution w/ finite moments
✰ Discover spectral convergence when for
stochastic Burgers equations
✰ Error control with P-Q refinements
✰ Computational complexity comparison of gPC and
WM in d dimensions
Q ≥ P −1
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.

19. M. Zheng, G.E. Karniadakis, ‘Numerical Methods for SPDEs with Tempered Stable Processes’,SIAM J. Sci. Comput., accepted.
N. Hilber, O. Reichmann, Ch. Schwab, Ch. Winter, Computational Methods for Quantitative Finance: Finite Element Methods for
Derivative Pric- ing, Springer Finance, 2013.
S.I. Denisov, W. Horsthemke, P. Hänggi, Generalized Fokker-Planck equation: Derivation and exact solutions, Eur. Phys. J. B, 68
(2009), pp. 567–575.
Generalized Fokker-Planck Equation for Overdamped Langevin Equation
Overdamped Langevin equation (1D, SODE, in the Ito’s sense)
Density satisfies tempered fractional PDEs (by Ito’s formula)
1D tempered stable (TS) pure jump process has this Levy measure

20. Generalized FP Equation for Overdamped Langevin Equation driven by TS white noise
Left Riemann-Liouville tempered fractional derivatives (as an example)
Fully implicit scheme in time, Grunwald-Letnikov for fractional derivatives
MC for Overdamped Langevin Equation driven by TS white noise
TFPDE
PCM for Overdamped Langevin Equation driven by TS white noise
Compound Poisson (CP)
approximation
MC!
(probabilistic)
PCM!
(probabilistic)
TFPDE!
(deterministic)

21. Histogram from MC vs. Density from TFPDEs
Zoomed in plots of P(x,T) by TFPDEs and MC/CP at T = 0.5 (left) and T = 1 (right): α = 0.5, c = 1, λ = 1, x0
= 1 and σ = 0.01 (left and right). In MC/CP: s = 105, δ = 0.01, △t = 1e −3 (left and right). In the TFPDEs:
△t = 1e −5, and Nx = 2000 points on [−12, 12] in space (left and right).
jump intensity jump size distribution

22. Moment Statistics from PCM/CP vs. TFPDE
TFPDE vs. PCM/CP: error of the 2nd moment of the solution versus time with λ=10 (left) and λ=1 (right).
α=0.5,c=2,σ=0.1,x0 =1 (left and right). For the TFPDE: finite difference scheme with △t = 2.5 × 10−5 , Nx
equidistant points on [−12, 12], initial condition given by δD (left and right).
TFPDE costs much less computational time
but more accurate than PCM/CP

23. M. Zheng, G.E. Karniadakis, Numerical methods for SPDEs with additive multi-dimensional Levy jump processes, in
preparation.

24. How to describe the dependence structure among components!
of a multi-dimensional Levy jump process ?
J. Kallsen, P. Tankov, Characterization of dependence of multidimensional Levy processes using Levy copulas, Journal of
Multivariate Analysis, 97 (2006), pp. 1551–1572.
LePage’s representation of Levy measure:
Series representation:
τ = 1
τ = 100
Levy copula!
+!
Marginal Levy!
measure!
=!
Levy measure
1
2

25. Analysis of variance (ANOVA) + FP = marginal distribution
FP equation
ANOVA decomposition
ANOVA terms are related to marginal distributions
1D-ANOVA-FP for marginal distributions
2D-ANOVA-FP for marginal distributions
LePage’s
representation
TFPDEs

26. 0 0.2 0.4 0.6 0.8 1
−2
0
2
4
6
8
10
12
x
E[u(x,T=1)]
E[uPCM
]
E[u
1D−ANOVA−FP
]
E[u
2D−ANOVA−FP
]
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
x 10
−4
T
L2
normofdifferenceinE[u]
||E[u
1D−ANOVA−FP
−E[u
PCM
]||
L
2
([0,1])
/||E[u
PCM
]||
L
2
([0,1])
||E[u
2D−ANOVA−FP
−E[u
PCM
]||
L
2
([0,1])
/||E[u
PCM
]||
L
2
([0,1])
Moments: 1D-ANOVA-FP is accurate for E[u] in 10D
1D-ANOVA-FP
2D-ANOVA-FP
PCM
1D-ANOVA-FP
2D-ANOVA-FP
noise-to-signal!
ratio NSR ≈18.24%

27. Moments: 1D-ANOVA-FP is not accurate for in 10D
0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
x
E[u
2
(x,T=1)]
E[u
2
PCM
]
E[u2
1D−ANOVA−FP
]
E[u
2
2D−ANOVA−FP
]
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
T
L
2
normofdifferenceinE[u2
]
||E[u2
1D−ANOVA−FP
−E[u2
PCM
]||
L
2
([0,1])
/||E[u2
PCM
]||
L
2
([0,1])
||E[u
2
2D−ANOVA−FP
−E[u
2
PCM
]||
L
2
([0,1])
/||E[u
2
PCM
]||
L
2
([0,1])
1D-ANOVA-FP
2D-ANOVA-FP
PCM
1D-ANOVA-FP
2D-ANOVA-FP
Ε[u2
]
NSR ≈18.24%

28. Moments: PCM vs. FP (TFPDE)
Initial condition of FP
equation introduce error
0.2 0.4 0.6 0.8 1
10
−10
10
−8
10
−6
10
−4
10
−2
l2u2(t)
t
PCM/S Q=5, q=2
PCM/S Q=10, q=2
TFPDE
NSR 4.8%
Moments: PCM vs. MC
LePage’s representation (2D)
Ε[u2
]
Ε[u2
]
LePage’s representation (2D)
10
0
10
2
10
4
10
6
10
−4
10
−3
10
−2
10
−1
s
l2u2(t=1)
PCM/S q=1
PCM/S q=2
MC/S Q=40
PCM costs less than MC
Q — # of truncation in series representation
q — # of quadrature points on each
dimension

29. Density: MC vs. FP equation (2D Levy)
LePage’s !
representation!
2D — MC
3D — FP/TFPDE
Levy!
copula

30. General picture of solving SPDEs w/ multi-dim jump processes

31. Summary of contributions (3, 4)
✰ Established a framework for UQ of SPDEs w/ multi-
dimensional Levy jump processes by probabilistic
(MC, PCM) and deterministic (FP) methods
✰ Combined the ANOVA & FP to simulate moments of
solution at lower orders
✰ Improved the traditional MC method’s efficiency and
accuracy
✰ Link the area of fractional PDEs & UQ for SPDEs w/
Levy jump processes

32. Future work
✰ Simulate SPDEs driven by higher-dimensional Levy
jump processes with ANOVA-FP
✰ Consider other jump processes than TS processes
✰ Consider nonlinear SPDEs w/ multiplicative multi-
dimensional Levy jump processes
✰ Application to the Energy Balance Model in climate
modeling
✰ Application to Mathematical Finance

33. Acknowledgements
✰ Thanks Prof. George Em Karniadakis for advice and
support
✰ Thanks Prof. Xiaoliang Wan and Prof. Hui Wang to be
on my committee
✰ Thanks Prof. Xiaoliang Wan and Prof. Boris Rozovskii
for their innovative ideas and collaboration
✰ Thanks for the support from the NSF/DMS (grant
DMS-0915077) and the Airforce MURI (grant
FA9550-09-1-0613)

34. Thanks
for
attending !