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.
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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).
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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