This document summarizes research on numerical methods for solving stochastic partial differential equations (SPDEs) driven by Lévy jump processes. It discusses both probabilistic methods like Monte Carlo simulation and polynomial chaos methods, as well as deterministic methods based on generalized Fokker-Planck equations. Specific examples presented include the overdamped Langevin equation driven by a tempered α-stable Lévy process, and heat equations with jumps modeled by multi-dimensional Lévy processes using either Lévy copulas or Lévy measure representations. Comparisons are made between probabilistic and deterministic methods in terms of accuracy and computational efficiency for moment statistics.

1. Numerical Methods for SPDEs driven by L´evy Jump
Processes: Probabilistic and Deterministic Approaches
Mengdi Zheng, George Em
Karniadakis (Brown University)
2015 SIAM Conference on
Computational Science and Engineering
March 17, 2015

2. Contents
Motivation
Introduction
L´evy process
Dependence structure of multi-dim pure jump process
Generalized Fokker-Planck (FP) equation
Overdamped Langevin equation driven by 1D TαS process
by MC and PCM (probabilistic methods)
by FP equation (deterministic method, tempered fractional PDE)
Diffusion equation driven by multi-dimensional jump processes
SPDE w/ 2D jump process in LePage’s rep
SPDE w/ 2D jump process by L´evy copula
SPDE w/ 10D jump process in LePage’s rep (ANOVA decomposition)
Future work
2 of 25

3. Section 1: motivation
Figure : We aim to develop gPC method (probabilistic) and generalized FP
equation (deterministic) approach for UQ of SPDEs driven by non-Gaussian
L´evy processes.
3 of 25

4. Section 2.1: L´evy processes
Definition of a L´evy process Xt (a continuous random walk):
Independent increments: for t0 < t1 < ... < tn, random variables
(RVs) Xt0
, Xt1
− Xt0
,..., Xtn−1
− Xtn−1
are independent;
Stationary increments: the distribution of Xt+h − Xt does not depend
on t;
RCLL: right continuous with left limits;
Stochastic continuity: ∀ > 0, limh→0 P(|Xt+h − Xt| ≥ ) = 0;
X0 = 0 P-a.s..
Decomposition of a L´evy process Xt = Gt + Jt + vt: a Gaussian
process (Gt), a pure jump process (Jt), and a drift (vt).
Definition of the jump: Jt = Jt − Jt− .
Definition of the Poisson random measure (an RV):
N(t, U) =
0≤s≤t
I Js ∈U, U ∈ B(Rd
0 ), ¯U ⊂ Rd
0 . (1)
4 of 25

5. Section 2.2: Pure jump process Jt
L´evy measure ν: ν(U) = E[N(1, U)], U ∈ B(Rd
0 ), ¯U ⊂ Rd
0 .
3 ways to describe dependence structure between components of a
multi-dimensional L´evy process:
Figure : We will discuss the 1st (LePage) and the 3rd (L´evy copula)
methods here.
5 of 25

6. Section 2.2: LePage’s multi-d jump processes
Example 1: d-dim tempered α-stable processes (TαS) in spherical
coordinates (”size” and ”direction” of jumps):
L´evy measure (dependence structure):
νrθ(dr, dθ) = σ(dr, θ)p(dθ) = ce−λr
dr
r1+α p(dθ) = ce−λr
dr
r1+α
2πd/2
dθ
Γ(d/2) ,
r ∈ [0, +∞], θ ∈ Sd
.
Series representation by Rosinksi (simulation)1
:
L(t) =
+∞
j=1 j [(
αΓj
2cT )−1/α
∧ ηj ξ
1/α
j ] (θj1, θj2, ..., θjd )I{Uj ≤t},
for t ∈ [0, T].
P( j = 0, 1) = 1/2, ηj ∼ Exp(λ), Uj ∼ U(0, T), ξj ∼U(0, 1).
{Γj } are the arrival times in a Poisson process with unit rate.
(θj1, θj2, ..., θjd ) is uniformly distributed on the sphereSd−1
.
1
J. Ros´ınski, On series representations of infinitely divisible random vectors,
Ann. Probab., 18 (1990), pp. 405–430.6 of 25

7. Section 2.2: dependence structure by L´evy copula
Example 2: 2-dim jump process (L1, L2) w/ TαS components2
(L++
1 , L++
2 ), (L+−
1 , L+−
2 ), (L−+
1 , L−+
2 ), and (L−−
1 , L−−
2 )
Figure : Construction of L´evy measure for (L++
1 , L++
2 ) as an example
2
J. Kallsen, P. Tankov, Characterization of dependence of
multidimensional L´evy processes using L´evy copulas, Journal of Multivariate
Analysis, 97 (2006), pp. 1551–1572.7 of 25

8. Section 2.2: dependence structure by (L´evy copula)
Example 2 (continued):
Simulation of (L1, L2) ((L++
1 , L++
2 ) as an example) by series
representation
L++
1 (t) =
+∞
j=1 1j (
αΓj
2(c/2)T )−1/α
∧ ηj ξ
1/α
j I[0,t](Vj ),
L++
2 (t) =
+∞
j=1 2j U
++(−1)
2 F−1
(Wi U++
1 (
αΓj
2(c/2)T )−1/α
∧ ηj ξ
1/α
j ) I[0,t](Vj )
F−1
(v2|v1) = v1 v
− τ
1+τ
2 − 1
−1/τ
.
{Vi } ∼Uniform(0, 1) and {Wi } ∼Uniform(0, 1). {Γi } is the i-th
arrival time for a Poisson process with unit rate. {Vi }, {Wi } and {Γi }
are independent.
8 of 25

9. Section 2.3: generalized Fokker-Planck (FP) equations
For an SODE system du = C(u, t) + dL(t), where C(u, t) is a
linear operator on u.
Let us assume that the L´evy measure of the pure jump process
L(t) has the symmetry ν(x) = ν(−x).
The generalized FP equation for the joint PDF satisfies3:
∂P(u, t)
∂t
= − ·(C(u, t)P(u, t))+
Rd −{0}
ν(dz) P(u+z, t)−P(u, t) .
(2)
3
X. Sun, J. Duan, Fokker-Planck equations for nonlinear dynamical systems
driven by non-Gaussian L´evy processes. J. Math. Phys., 53 (2012), 072701.9 of 25

10. Section 3: overdamped Langevin eqn driven by 1D
TαS process
We solve:
dx(t; ω) = −σx(t; ω)dt + dLt(ω), x(0) = x0.
L´evy measure of Lt is: ν(x) = ce−λ|x|
|x|α+1 , 0 < α < 2
FP equation as a tempered fractional PDE (TFPDE)
When 0 < α < 1, D(α) = c
α Γ(1 − α)
∂
∂t P(x, t) = ∂
∂x σxP(x, t) −D(α) −∞Dα,λ
x P(x, t)+x Dα,λ
+∞P(x, t)
When 1 < α < 2, D(α) = c
α(α−1) Γ(2 − α)
∂
∂t P(x, t) = ∂
∂x σxP(x, t) +D(α) −∞Dα,λ
x P(x, t)+x Dα,λ
+∞P(x, t)
−∞Dα,λ
x and x Dα,λ
+∞ are left and right Riemann-Liouville tempered
fractional derivatives4
.
4
M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional
Calculus, De Gruyter Studies in Mathematics Vol. 43, 2012.10 of 25

11. Section 3: PCM V.s. TFPDE in moment statistics
0 0.2 0.4 0.6 0.8 1
10
4
10
3
10
2
10
1
10
0
t
err2nd
fractional density equation
PCM/CP
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
10
3
10
2
10
1
10
0
t
err2nd
fractional density equation
PCM/CP
Figure : err2nd versus time by: 1) TFPDEs; 2) PCM. Problem: α = 0.5,
c = 2, λ = 10, σ = 0.1, x0 = 1 (left); α = 1.5, c = 0.01, λ = 0.01, σ = 0.1,
x0 = 1 (right). For PCM: Q = 50 (left); Q = 30 (right). For density
approach: t = 2.5e − 5, 2000 points on [−12, 12], IC is δD
40 (left);
t = 1e − 5, 2000 points on [−20, 20], i.c. given by δG
40 (right).
11 of 25

12. Section 3: MC V.s. TFPDE in density
4 2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x(T = 0.5)
densityP(x,t)
histogram by MC/CP
density by fractional PDEs
4 2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x(T=1)
densityP(x,t)
histogram by MC/CP
density by fractional PDEs
Figure : Zoomed in plots of P(x, T) by TFPDEs and MC 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: s = 105
, 316 bins, t = 1e − 3 (left and right). In the
TFPDEs: t = 1e − 5, and Nx = 2000 points on [−12, 12] in space (left
and right).
12 of 25

13. Section 4: heat equation w/ multi-dim jump process
We solve :



du(t, x; ω) = µ∂2u
∂x2 dt + d
i=1 fi (x)dLi (t; ω), x ∈ [0, 1]
u(t, 0) = u(t, 1) = 0 boundary condition
u(0, x) = u0(x) initial condition,
(3)
L(t; ω), {Li (t; ω), i = 1, ..., d} are mutually dependent.
fk(x) =
√
2sin(πkx), x ∈ [0, 1], k = 1, 2, 3, ... is a set of
orthonormal basis functions on [0, 1].
By u(x, t; ω) = +∞
i=1 ui (t; ω)fi (x) and Galerkin projection onto
{fi (x)}, we obtain an SODE system, where Dmm = −(πm)2:



du1(t) = µD11u1(t)dt + dL1,
du2(t) = µD22u2(t)dt + dL2,
...
dud (t) = µDdd ud (t)dt + dLd ,
(4)
13 of 25

14. Section 4.1: SPDEs driven by multi-d jump processes
Figure : An illustration of probabilistic and deterministic methods to solve
the moment statistics of SPDEs driven by multi-dim L´evy processes.
14 of 25

15. Section 4.2: FP eqn when Lt (2D) is in LePage’s rep
When the L´evy measure of Lt is given by
νrθ(dr, dθ) = ce−λr dr
r1+α
2πd/2dθ
Γ(d/2) , for r ∈ [0, +∞], θ ∈ Sd
The generalized FP equation for the joint PDF P(u, t) of solutions
in the SODE system is:
∂P(u,t)
∂t = − d
i=1 µDii (P + ui
∂P
∂ui
)
− c
α Γ(1 − α) Sd−1
Γ(d/2)dσ(θ)
2πd/2 r Dα,λ
+∞P(u + rθ, t) , where θ is a
unit vector on the unit sphere Sd−1.
x Dα,λ
+∞ is the right Riemann-Liouville Tempered Fractional (TF)
derivative.
Later, for d = 10, we will use ANOVA decomposition to obtain
equations for marginal distributions from this FP equation.
15 of 25

16. Section 4.2: simulation if Lt (2D) is in LePage’s rep
Figure : FP vs. MC/S: joint PDF P(u1, u2, t) of SODEs system from FP
Equation (3D contour) and by MC/S (2D contour), horizontal and vertical
slices at the peak of density. t = 1 , c = 1, α = 0.5, λ = 5, µ = 0.01,
NSR = 16.0% at t = 1.
16 of 25

17. Section 4.2: simulation when Lt (2D) is in LePage’s
rep
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%
0.2 0.4 0.6 0.8 1
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
l2u2(t)
t
PCM/S Q=10, q=2
PCM/S Q=20, q=2
TFPDE
NSR 6.4%
Figure : FP vs. PCM: L2 error norm in moments obtained by PCM and FP
equation. α = 0.5, λ = 5, µ = 0.001 (left and right). c = 0.1 (left); c = 1
(right). In FP: initial condition is given by δG
2000, RK2 scheme.
17 of 25

18. Section 4.3: FP eqn if Lt (2D) is from L´evy copula
The L´evy measure of Lt is given by L´evy copula on each corners
(++, +−, −+, −−)
dependence structure is described by the Clayton family of copulas
with correlation length τ on each corner
The generalized FP eqn is :
∂P(u,t)
∂t = − · (C(u, t)P(u, t))
+
+∞
0 dz1
+∞
0 dz2ν++(z1, z2)[P(u + z, t) − P(u, t)]
+
+∞
0 dz1
0
−∞ dz2ν+−(z1, z2)[P(u + z, t) − P(u, t)]
+
0
−∞ dz1
+∞
0 dz2ν−+(z1, z2)[P(u + z, t) − P(u, t)]
+
0
−∞ dz1
0
−∞ dz2ν−−(z1, z2)[P(u + z, t) − P(u, t)]
18 of 25

19. Section 4.3: FP eqn if Lt (2D) is from L´evy copula
Figure : FP vs. MC: P(u1, u2, t) of SODE system from FP eqn (3D
contour) and by MC/S (2D contour). t = 1 , c = 1, α = 0.5, λ = 5,
µ = 0.005, τ = 1, NSR = 30.1% at t = 1.
19 of 25

20. Section 4.3: if Lt (2D) is from L´evy copula
0.2 0.4 0.6 0.8 1
10
−5
10
−4
10
−3
10
−2
t
l2u2(t)
TFPDE
PCM/S Q=1, q=2
PCM/2 Q=2, q=2
NSR 6.4%
0.2 0.4 0.6 0.8 1
10
−3
10
−2
10
−1
10
0
t
l2u2(t)
TFPDE
PCM/S Q=2, q=2
PCM/S Q=1, q=2
NSR 30.1%
Figure : FP vs. PCM: L2 error of the solution for heat equation α = 0.5,
λ = 5, τ = 1 (left and right). c = 0.05, µ = 0.001 (left). c = 1, µ = 0.005
(right). In FP: I.C. is given by δG
1000, RK2 scheme.
20 of 25

21. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA
The unanchored analysis of variance (ANOVA) decomposition is 5:
P(u, t) ≈ P0(t) + 1≤j1≤d Pj1 (uj1 , t) + 1≤j1<j2≤d Pj1,j2 (uj1 , uj2 , t)
+... + 1≤j1<j2...<jκ≤d Pj1,j2,...,jκ (uj1 , uj2 , ..., uκ, t)
κ is the effective dimension
P0(t) = Rd P(u, t)du
Pi (ui , t) = Rd−1 du1...dui−1dui+1...dud P(u, t) − P0(t) =
pi (ui , t) − P0(t)
Pij (xi , xj , t) = Rd−1 du1...dui−1dui+1...duj−1duj+1...dud P(u, t)
−Pi (ui , t) − Pj (uj , t) − P0(t) =
pij (x1, x2, t) − pi (x1, t) − pj (x2, t) + P0(t)
5
M. Bieri, C. Schwab, Sparse high order FEM for elliptic sPDEs, Tech.
Report 22, ETH, Switzerland, (2008).21 of 25

22. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA
When the L´evy measure of Lt is given by
νrθ(dr, dθ) = ce−λr dr
r1+α
2πd/2dθ
Γ(d/2) , for r ∈ [0, +∞], θ ∈ Sd (for
0 < α < 1)
∂pi (ui ,t)
∂t = − d
k=1 µDkk pi (xi , t) − µDii xi
∂pi (xi ,t)
∂xi
−cΓ(1−α)
α
Γ( d
2
)
2π
d
2
2π
d−1
2
Γ( d−1
2
)
π
0 dφsin(d−2)(φ) r Dα,λ
+∞pi (ui +rcos(φ), t)
∂pij (ui ,uj ,t)
∂t =
− d
k=1 µDkk pij −µDii ui
∂pij
∂ui
−µDjj uj
∂pij
∂uj
−cΓ(1−α)
α
Γ( d
2
)
2π
d
2
2π
d−2
2
Γ(d−2
2
)
π
0 dφ1
π
0 dφ2sin8(φ1)sin7(φ2) r Dα,λ
+∞pij (ui + rcosφ1, uj +
rsinφ1cosφ2, t)
22 of 25

23. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA
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])
Figure : 1D-ANOVA-FP V.s. 2D-ANOVA-FP in 10D: the mean (left) for the
solution of heat eqn at T = 1. The L2 norms of difference in E[u](right).
c = 1, α = 0.5, λ = 10, µ = 10−4
. I.C. of ANOVA-FP: MC/S data at
t0 = 0.5, s = 1 × 104
. NSR ≈ 18.24% at T = 1.
23 of 25

24. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA
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[u2
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
L2
normofdifferenceinE[u
2
]
||E[u2
1D−ANOVA−FP
−E[u2
PCM
]||
L
2
([0,1])
/||E[u2
PCM
]||
L
2
([0,1])
||E[u2
2D−ANOVA−FP
−E[u2
PCM
]||
L
2
([0,1])
/||E[u2
PCM
]||
L
2
([0,1])
Figure : 1D-ANOVA-FP V.s. 2D-ANOVA-FP in 10D: the 2nd moment (left)
for heat eqn.The L2 norms of difference in E[u2
] (right).
c = 1, α = 0.5, λ = 10, µ = 10−4
. I.C. of ANOVA-FP: MC/S data at
t0 = 0.5, s = 1 × 104
.NSR ≈ 18.24% at T = 1.
24 of 25

25. Future work
multiplicative noise (now we have additive noise)
nonlinear SPDE (now we have linear SPDE)
higher dimensions (we computed up to < 20 dimensions)
thanks!
25 of 25