Thesis defense improved

Numerical methods for stochastic systems

subject to generalized Levy noiseby Mengdi Zheng!

Thesis committee: George Em Karniadakis (Ph.D., advisor)!Hui Wang (Ph.D., reader, APMA, Brown)!

Xiaoliang Wan (Ph.D., reader, Mathematics, LSU)

Motivation from 2 aspects

Mathematical!Reasons

Reasons from applications

Mathematical!Finance!

!Levy flights!in Chaotic!

flows

Goal of my thesisWe consider:!

!SPDEs driven by !

1. discrete RVs !2. jump processes!

!(jump systems/

memory systems)

Deterministic method:!!

Fokker-Planck (FP)!equation

Probabilistic method:!!

Monte Carlo (MC)!general Polynomial chaos (gPC)!

probability collocation method (PCM)!multi-element PCM (MEPCM)

Uncertainty !Quantification

We are the first ones who solved such systems through both deterministic & probabilistic methods.

Contents

Outline ♚ Adaptive multi-element polynomial chaos with discrete

measure: Algorithms and applications to SPDEs

♚ Adaptive Wick-Malliavin (WM) approximation to nonlinear

SPDEs with discrete RVs

♚ Numerical methods for SPDEs with 1D tempered -stable (T S) processes

♚ Numerical methods for SPDEs with additive multi-dimensional Levy jump processes

♚ Future work

αα

Probability collocation method (PCM) in UQXt (ω ) ≈ Xt (ξ1,ξ2,...,ξn ) ω ∈Ω Ε[um (x,t;ω )] ≈ Ε[um (x,t;ξ1,ξ2,...,ξn )]

ξ1

ξ2ξ3

... ξn

O Ω

PCM

ξ1

ξ2Ω

O

MEPCMB1 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}

Gauss quadrature and weights

generate orthogonal to{P j

k (ξj )}µ j (ξ j )

moment statistics

5 ways

tensor product

or sparse

grid

Ε[um (x,t;ω )]

017.5

3552.5

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 = σ iξi ,

i=1

n

∑ x ∈!

u(x,0) = a2sech2 ( 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 −1k + n− | i |

⎛⎝⎜

⎞⎠⎟(Ui1 ⊗ ...⊗Uin )

k+1≤|i|≤k+n∑

Sparse grids in Smolyak algorithm: level k, dimension n

sparse!grid

tensor!product!

grid

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

100

polynomial order i

CP

U ti

me

to e

valu

ate

orth

(i)

NowakStieltjesFischerModified ChebyshevLanczos

C*i2n=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

100

polynomial order i

orth

(i)

NowakStieltjesFischerModified ChebyshevLanczos

N=100, p=1/2

polynomial order i

Bino(100, 1/2)

orthogonality ?

cost ?

Bino(N , p)Binomial distribution

| f (ξ )µ(ξ )− QmBi f

i=1

Nes

∑Γ∫ |≤Chm+1 || EΓ ||m+1,∞,Γ | f |m+1,∞,Γ

{Bi}i=1Nes : elementsNes : number of elements

: number of elementsΓµ : discrete measure

QmBi 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.

Multi-element Gauss integration over discrete measures

2 3 4 5 6 7 8

10−3

10−2

d

error

l2u1aPCl2u2aPC

2 3 5 10 15 20 3010−7

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

Adaptive MEPCM for stochastic KdV equation with 1RV

ut + 6uux + uxxx = σ iξi ,i=1

n

∑ x ∈!


2(x − x0 ))

σ i2local 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

erro

rs

2 el, even grid2 el, uneven grid4 el, even grid4 el, uneven grid5 el, even grid5 el, uneven grid

(MEPCM)!adaptive!

vs.!non-

adaptive!meshes

error of Ε[u2 ]

Improved !

ut + 6uux + uxxx = σ iξi ,i=1

n

∑ x ∈!


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 4,84510−10

10−9

10−8

10−7

10−6

10−5

10−4

r(k)

erro

rs

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

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 efficiency in computing moment

statistics by sparse grid (on discrete measure)



♚ Adaptive Wick-Malliavin (WM) approximation to

nonlinear SPDEs with discrete RVs



♚ Future work

αα

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 +σ c1(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 ∂2u! k∂x2

+σδ1k ,

general Polynomial Chaos (gPC) propagator

k = 0,1,...,P.

ξ ∼ Pois(λ)

ck : Charlier polynomiale−λλ k

k!cm (k;λ)cn (k;λ) =< cmcn >= n!λ

nδmnk∈!∑

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

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

!✰ approximate the product of uv as:

Γ(x) = e−λλ k

k!δ (x − k)

k∈!∑

cm (x;λ)◊cn (x;λ) = cm+n (x;λ),

Dpci (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+n2

∑Kmnp = a(m + n − 2p,m,n)

uv =Dpu◊ pD

pvp!

≈p=0

∞

∑ Dpu◊ pDpv

p!p=0

Q

∑

WM approximation simplifies the gPC propagator !ut + uux =υuxx +σ c1(ξ;λ), x ∈[−π ,π ]

∂u! k∂t

+ u! m∂u! n∂t

< cmcnck >=m,n=0

P

∑ υ ∂2u! k∂x2

+σδ1k , k = 0,1,...,P.gPC propagator:

∂u! k∂t

+ u! ii=0

P

∑ ∂u! k+2 p−i∂x

Ki,k+2 p−i,Q =p=0

Q

∑ υ ∂2u! 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

Spectral convergence when Q ≥ P −1

ut + uux =υuxx +σ c1(ξ;λ) u(x,0) = 1− sin(x)

ξ ∼ Pois(λ) x ∈[−π ,π ] Periodic B.C.

concept of P-Q refinement

ut + uux =υuxx +σ c1(ξ;λ) u(x,0) = 1− sin(x)

ξ ∼ Pois(λ) x ∈[−π ,π ] Periodic B.C.

WM for stochastic Burgers equation w/ multiple RVs

ut + uux =υuxx +σj=1

3

∑ c1(ξ j;λ)cos(0.1 jt)

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

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

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.

R. Mikulevicius and B.L. Rozovskii, On distribution free Skorokhod-Malliavin calculus, submitted.







♚ Future work

αα

sample path of a Poisson process

Jt

t

Introduction of Levy processes

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 Pricing, Springer Finance, 2013.

S.I. Denisov, W. Horsthemke, P. Ha ̈nggi, Generalized Fokker-Planck equation: Derivation and exact solutions, Eur. Phys. J. B, 68

(2009), pp. 567–575.

Generalized Fokker-Planck Equation for Overdamped Langevin EquationOverdamped 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

Generalized FP Equation for Overdamped Langevin Equation w/ 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)

1

23

Histogram from MC vs. Density from TFPDEs

jump intensity jump size distribution

Moment Statistics from PCM/CP vs. TFPDE

1. TFPDE costs less than PCM 2. PCM depends on the series representation 3. TFPDE depends on the initial condition 4. Convergence in TFPDE by refinement

λ = 10 λ = 1







♚ Future work

αα

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

How to describe the dependence structure among components!of a multi-dimensional Levy jump process ?

LePage’s representation of Levy measure:1Series representation:

Levy measure:

J. Rosinski, Series representations of Levy processes from the perspective of point processes in: Levy Processes - Theory and Applications, O. E. Barndor-Nielsen, T. Mikosch and S. I. Resnick (Eds.), Birkh ̈auser, Boston, (2001), pp. 401–415.

How to describe the dependence structure among components!of a multi-dimensional Levy jump process ?

Dependence structure by Levy copula:

J. Kallsen, P. Tankov, Characterization of dependence of multidimensional Levy processes using Levy copulas, Journal of Multivariate Analysis, 97 (2006), pp. 1551–1572.

Levy copula + Marginal Levy measure = Levy measure

Series rep:

τ = 1

τ = 100

Construction:

2

Analysis of variance (ANOVA) + FP = marginal distributionFP equation

ANOVA decomposition

ANOVA terms are related to marginal distributions

1D-ANOVA-FP for marginal distributions

2D-ANOVA-FP for marginal distributionsLePage’s representation

TFPDEs

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[u1D−ANOVA−FP]

E[u2D−ANOVA−FP]

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 13.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2x 10−4

T

L 2 nor

m o

f diff

eren

ce in

E[u

]

||E[u1D−ANOVA−FP−E[uPCM]||L2([0,1])/||E[uPCM]||L2([0,1])

||E[u2D−ANOVA−FP−E[uPCM]||L2([0,1])/||E[uPCM]||L2([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%

Moments: 1D-ANOVA-FP is not accurate for in 10D

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

x

E[u

2 (x,T

=1)]

E[u2PCM]

E[u21D−ANOVA−FP]

E[u22D−ANOVA−FP]

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

T

L 2 nor

m o

f diff

eren

ce in

E[u

2 ]

||E[u21D−ANOVA−FP−E[u2

PCM]||L2([0,1])/||E[u2PCM]||L2([0,1])

||E[u22D−ANOVA−FP−E[u2

PCM]||L2([0,1])/||E[u2PCM]||L2([0,1])

1D-ANOVA-FP 2D-ANOVA-FP PCM

1D-ANOVA-FP

2D-ANOVA-FP

Ε[u2 ]

NSR ≈18.24%

Moments: PCM vs. FP (TFPDE)

Initial condition of FP equation introduce error

0.2 0.4 0.6 0.8 110−10

10−8

10−6

10−4

10−2

l2u2

(t)

t

PCM/S Q=5, q=2PCM/S Q=10, q=2TFPDE

NSR 5 4.8%

Moments: PCM vs. MC

LePage’s representation (2D)

Ε[u2 ]

Ε[u2 ]LePage’s representation (2D)

100 102 104 10610−4

10−3

10−2

10−1

s

l2u2

(t=1)

PCM/S q=1PCM/S q=2MC/S Q=40

PCM costs less than MC

Q — # of truncation in series representation q — # of quadrature points on each dimension

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

Levy!copula

t = 1

t = 1

t = 1.5

t = 1.5

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

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







♚ Future work

αα

Future work For methodology:!✰ 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 !For applications:!✰ Application to the Energy Balance Model in climate modeling: P.

Imkeller, Energy balance models - viewed from stochastic dynamics, Stochastic climate models, Basel: Birkhuser. Prog. Probab., 49 (2001), pp. 213– 240.

!✰ Application to Mathematical Finance such as the stock price model

associated with multi-dimensional Levy jump processes: R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Press, 2004.

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 National Science

Foundation, "Overcoming the Bottlenecks in Polynomial chaos: Algorithms and Applications to Systems Biology and Fluid Mechanics" (Grant #526859)

✰ Thanks for the support from the Air Force Office of Scientific Research: Multidisciplinary Research Program of the University Research Initiative, "Multi-scale Fusion of Information for Uncertainty Quantification and Management in Large-Scale Simulations" (Grant #521024)

Thanks for attending !

Thesis defense improved

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