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Numerical methods for SPDEs with TαS processes Mengdi Zheng, George Em Karniadakis Brown University/ Pizza Seminar March 21, 2014

2014 spring crunch seminar (SDE/levy/fractional/spectral method)

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Page 1: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Numerical methods for SPDEs with TαS processes

Mengdi Zheng, George EmKarniadakis

Brown University/ Pizza Seminar

March 21, 2014

Page 2: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Contents

� Background on TαS processes� Numerical simulations of TαS processes

� Compound Poisson approximation (CP)� Seires reprsentation

� Simulation of reaction diffusion equations with TαS white noises� MC/CP, MC/S, PCM/CP, PCM/S

� Simulation of overdamped Langevin equations with TαS whitenoises� Integral type PDE (by CP approximation)� TFPDEs

� Future work

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Page 3: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 1.1: Levy processes� Definition of a Levy 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 dependon t;

� RCLL: right continuous with left limits;� Stochastic continuity: ∀ε > 0, limh→0 P(|Xt+h − Xt | ≥ ε) = 0;� X0 = 0 P-a.s..

� Decomposition of a Levy process Xt = Gt + Jt + ct: a Gaussianprocess (Gt), a pure jump process (Jt), and a drift (ct).

� say some other facts here

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Page 4: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 1.2: Pure jump process Jt

� Definition of the jump: 4Jt = Jt − Jt− .� Definition of the Poisson random measure (an RV):

N(t,U) =∑

0≤s≤tI4Js∈U , U ∈ B(R0), U ⊂ R0. (1)

Figure : Represent a sample path of Jt in a space of jump times andsizes; N(t,U) is the number of dots in the box

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Page 5: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 1.2: Pure jump process Jt (continued)

� Definition of Levy measure ν(x):

ν(U) = E[N(1,U)], U ∈ B(R0), U ⊂ R0. (2)

� Definition of the compensated Poisson random measure:

N(dt, dz) = N(dt, dz)− ν(dz)dt = N(dt, dz)−E[N(dt, dz)]. (3)

� Pure jump process Jt in an integral form:

Jt =

∫ t

0

∫R0

zN(dτ, dz) =

∫ t

0

∫R0

z(N(dτ, dz) + ν(dz)dτ). (4)

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Page 6: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 1.3: Tempered α-stable processes (TαS) Lt

� A pure jump process

� Levy measure:

ν(x) =ce−λ|x |

|x |α+1, 0 < α < 2. (5)

� c : alters the intensity of jumps of all sizes.� α: determines the importance of smaller jumps.� λ: tempers the bigger jumps.

� Probability density function (PDF) for Lt is not known in a closedform unless α = 1

2 .

� Limit behavior: in a short time it looks like an α-stable process; ina long time it looks like a Brownian motion.

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Page 7: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 1.3: TαS processes Lt (continued)

� Parameters (c, α, λ) completely characterize a TαS process.

−0.05 0 0.050

5

10

15

20

25

30

35

40

x

i(x)

Levy measure of T_S processes

_=0.1,c=0.01,h=100_=0.1,c=0.01,h=1_=0.2,c=0.01,h=100

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

timeL t(t

)

sample paths of T_S processes w/ different h

h=0.01h=1h=10

Q=10000(truncation)_=1, c=1

Figure : Leve measures of TαS processes Lt (left); Samples paths of Lt

(right).

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Page 8: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2: Numerical simulation of TαS processes� Compound Poisson (CP) approximation

� R. Cont, P. Tankov, Financial Modelling with Jump Processes,Chapman & Hall/CRC Press, 2004.

� Series representation� J. Rosınski, Series representations of Levy processes from the

perspective of point processes in: Levy Processes - Theory andApplications, O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick(Eds.), Birkhauser, Boston, (2001), pp. 401–415.

� J. Rosınski, On series representations of innitely divisible randomvectors, Ann. Probab., 18 (1990), pp. 405–430.

� J. Rosınski, Series representations of infinitely divisible random vectorsand a generalized shot noise in Banach spaces, University of NorthCarolina Center for Stochastic Processes, Technical Report No. 195,(1987).

� J. Rosınski, Tempering stable processes, Stoch. Proc. Appl., (2007),pp. 117.

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Section 2.1: Simulation of TαS processes by CPapproximation

� Main idea: we simulate the large jumps as a CP process andreplace the small ones (size ≤ δ) with their expectation as a driftterm.

� The CP approximation X δt for this TαS subordinator Xt (with

ν(x) = ce−λx

xα+1 Ix>0, positive jumps only) is:

Xt ≈ X δt =

∑s≤t4Xs I4Xs≥δ+E[

∑s≤t4Xs I4Xs<δ] ≈

∞∑i=1

Jδi It≤Ti+bδt.

(6)

� Intensity of the CP process: U(δ) = c∫∞δ

eλxdxxα+1 (by num int).

� Jump size distribution: pδ(x) = 1U(δ)

ceλx

xα+1 Ix≥δ for Jδi (by rejection

sampling method).

� Drift: bδ = c∫ δ

0e−λxdx

xα (by num int).9 of 39

Page 10: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.1: Simulation of TαS processes by CPapproximation—-Algorithm of CP processes

Here we describe how to simulate the trajectories of a CP process

with intensity U(δ) and jump size distribution νδ(x)U(δ) , on a simulation

time domain [0,T ] at time t.

� Simulate an RV N from Poisson distribution with parameterU(δ)T , as the total number of jumps on the interval [0,T ].

� Simulate N independent RVs, Ti , uniformly distributed on theinterval [0,T ], as jump times.

� Simulate N jump sizes, Yi with distribution νδ(x)U(δ) .

Then the trajectory at time t is given by∑N

i=1 ITi≤tYi .

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Page 11: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.1: Simulation of TαS processes by CPapproximation—-Algorithm of rejection method

The distribution pδ(x) = 1U(δ)

ceλx

xα+1 Ix≥δ can be bounded by :

pδ(x) ≤ δ−αe−λδ

αU(δ)f δ(x), (7)

where f δ(x) = αδ−α

xα+1 Ix≥δ. The algorithm is: REPEATGenerate RVs W and V : independent and uniformly distributed on[0, 1]Set X = δW−1/α

Set T = f δ(X )δ−αe−λδ

pδ(X )αU(δ)

UNTIL VT ≤ 1RETURN X .

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Page 12: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.2: Simulation of TαS processes by seriesrepresentation

� Let {εj}, {ηj}, and {ξj} be sequences of i.i.d. RVs s.t.P(εj = ±1) = 1/2, ηj ∼ Exponential(λ), and ξj ∼Uniform(0, 1).Let {Γj} be arrival times in a Poisson process with rate one. Let{Uj} be i.i.d. uniform RVs on [0,T ].

� This representation converges almost surely as uniformly in t (by J.Rosınski):

Lt =+∞∑j=1

εj [(αΓj

2cT)−1/α ∧ ηjξ

1/αj ]I{Uj≤t}, 0 ≤ t ≤ T . (8)

� We will treat [(αΓj

2cT )−1/α ∧ ηjξ1/αj ] as one RV to reduce the # of

RVs.

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Section 2.2: Simulation of TαS processes by seriesrepresentation—-simplify the representation� We calculated the PDF of [(

αΓj

2cT )−1/α ∧ ηjξ1/αj ] RV

� When 0 < α < 1,

fAj∧Bj (x) = (α

xΓ(j)e−tt j |t= 2cT

αxα)

[αΓ(1− α)λα∫ +∞

x

(1− γinc(λz , 1− α))zα−1dz ]

+ [αΓ(1− α)λα(1− γinc(λx , 1− α)xα−1)]γinc(2cT

αxα, j)

(9)

� When 1 < α < 2,

fAj∧Bj (x) = (α

xΓ(j)e−tt j |t= 2cT

αxα)[

∫ +∞

x

fηjξ

1/αj

(z)dz ]+(fηjξ

1/αj

(x))γinc(2cT

αxα, j)

(10)

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Page 14: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.3.1: Inverse Gaussian (IG) process and K-Stest

� An IG subordinator has a Levy measure (α = 1/2) as:

νIG =ce−λx

x3/2Ix>0. (11)

� PDF:

pt(x) =ct

x3/2e2ct

√πλe−λx−πc

2t2/x , x > 0. (12)

� We perform the one-sample Kolmogorov-Smirnov statistic (K-Stest) between the empirical cumulative distribution function (CDF)and the exact reference CDF:

KS = supx|Fem(x)− Fex(x)|, x ∈ supp(F ). (13)

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Page 15: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.3.2: histograms of IG process by CP

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

CP =0.1reference PDF

KS = 0.152843

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

CP =0.02reference PDF

KS = 0.009250

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

CP =0.005reference PDF

KS = 0.003414

Figure : Empirical histograms of an IG subordinator (α = 1/2) simulated viathe CP approximation at t = 0.5: the IG subordinator has c = 1, λ = 3;each simulation contains s = 106 samples (we zoom in and plot x ∈ [0, 1.8]to examine the smaller jumps approximation); they are with different jumptruncation sizes as δ = 0.1 (left, dotted, CPU time 1450s), δ = 0.02(middle, dotted, CPU time 5710s), and δ = 0.005 (right, dotted, CPU time38531s); the reference PDFs are plotted in red solid lines; the one-sampleK-S test values are calculated for each plot; the RelTol of integration inU(δ) and bδ is 1e − 8.15 of 39

Page 16: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 2.3.3: histograms of IG process by Series rep

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

series rep Q=10reference PDF

KS = 0.360572

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

series rep Q=100reference PDF

KS = 0.078583

0 0.5 1 1.50

0.5

1

1.5

2

2.5

x

p t(x)

series rep Q=800reference PDF

KS = 0.040574

Figure : Empirical histograms of an IG subordinator (α = 1/2) simulated viathe series representation at t = 0.5: the IG subordinator has c = 1, λ = 3;each simulation is done on the time domain [0, 0.5] and contains s = 106

samples (we zoom in and plot x ∈ [0, 1.8] to examine the smaller jumpsapproximation); they are with different number of truncations in the seriesas Qs = 10 (left, dotted, CPU time 129s), Qs = 100 (middle, dotted, CPUtime 338s), and Qs = 1000 (right, dotted, CPU time 2574s); the referencePDFs are plotted in red solid lines; the one-sample K-S test values arecalculated for each plot.16 of 39

Page 17: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.1: Reaction diffusion equation with TαSwhite noises

� Problem: du(t, x ;ω) = (∂2u∂x2 + µu)dt + εdLt(ω), x ∈ [0, 2]

u(t, 0) = u(t, 1) periodic BCu(0, x) = u0(x) = sin(π2 x) initial condition

(14)where Lt(ω) is a one-dimensional TαS process .

� Integral form:

u(t, x) = eµt−π2

4tsin(

π

2x) + εeµt

∫ t

0e−µτdLτ , x ∈ [0, 2]. (15)

� Simulation done by: MC/CP, MC/S, PCM/CP, PCM/S.

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Page 18: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.1: Reaction diffusion equation with TαSwhite noises

� If θt(z), t ≥ 0, z ∈ R0 is F-adapted, we have the Ito isometry:

E[(

∫ T

0

∫R0

θt(z)N(dt, dz))2] = E[

∫ T

0

∫R0

θ2t (z)ν(dz)dt]. (16)

� The second moment is:

Eex [u2(t, x ;ω)] = e2µt−π2

2tsin2(

π

2x) +

cε2e2µt

µλ2−α (1− e−2µt)Γ(2−α).

(17)

� Define the error:

l2u2(t) =||Eex [u2(x , t;ω)]− Enum[u2(x , t;ω)]||L2([0,2])

||Eex [u2(x , t;ω)]||L2([0,2]). (18)

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Page 19: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.2.1: First-order Euler scheme in MC

� We say that∫ T

0 f (t)dLt is in an Ito sense so that the left end ofthe time interval is used:∫ T

0f (t)dLt = lim

n→∞

n∑i=1

f (ti−1)(Lti − Lti−1). (19)

� By the first order Euler’s method:

un+1 − un = (∂2u

∂x2+ µun)4t + ε(Ltn+1 − Ltn). (20)

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Page 20: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.2.2: Comparing MC/CP and MC/S

102 103 104 105

10−6

10−5

10−4

10−3

10−2

s

l2u2(T=1)

MC/S Qs=10

MC/CP b=0.1MC/CP b=0.01

C*s−1/2

102 103 104 10510−5

10−4

10−3

10−2

10−1

s

l2u2(T=1)

MC/S Qs=10

MC/PCM b=0.1MC/PCM b=0.01

C*s−1/2

Figure : l2u2(T ) of the solution for equation (??) versus the number ofsamples s obtained by MC with λ = 10 (left) and λ = 1 (right): T = 1,c = 0.1, α = 0.5, λ = 1, ε = 0.1, µ = 2 (left and right). Spatialdiscretization: Nx = 500 Fourier collocation points on [0, 2]; temporaldiscretization: first order Euler scheme with time steps 4t = 1e − 5. In theCP approximation: RelTol = 1e − 8 for integration in U(δ).20 of 39

Page 21: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.2.3: MC/CP V.s. MC/S

� Cost: the MC/CP costs much less CPU time than the MC/S, e.g.MC/CP w/ δ = 0.01 costs half than MC/S w/ Qs = 10 for thesame s.

� Accuracy: with only half of the CPU time, MC/CP is moreaccurate than MC/S, e.g. compare the δ = 0.01 and the Qs = 10lines.

� Convergence rate: on the left plot , the MC/CP w/ δ = 0.01 isone magnitude more accurate than δ = 0.1 for a smaller s; on theright plot, the MC/CP w/ δ = 0.01 has almost the same accuracywith δ = 0.1 for a smaller s, then δ = 0.01 starts to be moreaccurate than δ = 0.1 for larger s (Explain why.).

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Page 22: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.3.1: PCM/CP and PCM/S

� PCM: Suppose the solution is a function of a finite number ofindependent RVs ({Y 1,Y 2, ...,Y n)}) as v(Y 1,Y 2, ...,Y n), them-th moment of the solution is evaluated by

E[vm(Y 1,Y 2, ...,Y n)] =

d1∑i1=1

...

dn∑in=1

vm(y 1i1 , y

2i2 , ..., y

nin)w 1

i1 ...wnin .

(21)� # of sample points:

� In CP Xt ≈ X δt ≈

∑Qcp

i=1 Jδi It≤Ti + bδt: d2Qcp points.

� In series rep Lt =∑+∞

j=1 εj [(αΓj

2cT )−1/α ∧ ηjξ1/αj ]I{Uj≤t}: d3Qcp points.

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Page 23: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.3.1: PCM/CP and PCM/S—-less samplingpoints

� In CP d(Qcp + 1) instead of d2Qcp :

E[u2(t, x ;ω)] ≈ e2µt− 12π2tsin2(

π

2x) + ε2e2µtE[Jδ1 ]

Qcp∑i=1

E[e−2µTi ].

(22)� In series rep dQs instead of d3Qs :

E[u2(t, x ;ω)] ≈ e2µt− 12π2tsin2(

π

2x) + ε2e2µt 1

2µT(1− e−2µT )

Qs∑j=1

E[((αΓj

2cT)−1/α ∧ ηjξ

1/αj )2].

(23)� Indeed if E [F (X1, ...,Xd)] = G (E [f1(X1)], ...,E [fd(Xd)]).23 of 39

Page 24: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.3.2: PCM/CP and PCM/S—-Results

0 10 20 30 40 50 60 70 80

10−6

10−4

10−2

100

Qs or Qcp

l2u2

(T=1

)

PCM/CP b=0.1PCM/CP b=0.01PCM/CP b=0.001PCM/CP b=0.0001PCM/S

0 10 20 30 40 5010−10

10−8

10−6

10−4

10−2

100

Qs or Qcp

l2u2

(T=1

)

PCM/CP b=0.1PCM/CP b=0.01PCM/CP b=0.001PCM/S

Figure : l2u2(T ) versus Qcp (by PCM/CP) or Qs (by PCM/S) with λ = 1(left) and λ = 0.01 (right): T = 1, c = 0.1, α = 0.5, ε = 0.1, µ = 0.1,Nx = 500 Fourier collocation points on [0, 2] (left and right). In thePCM/CP: RelTol = 1e − 10 for integration in U(δ). In the PCM/S:

RelTol = 1e − 8 for integration of E[((αΓj

2cT )−1/α ∧ ηjξ1/αj )2].

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Page 25: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.3.3: PCM/CP and PCM/S—-Observationsin three stages

� For smaller values of Qs and Qcp: PCM/S is more accurate andconverges faster than PCM/CP. (Explain why.)

� For middle values of Qs and Qcp: the convergence rate of PCM/Sslows down but the convergence rate of PCM/CP goes up.(Explain why.)

� For larger values of Qs and Qcp: both PCM/CP and PCM/S stopconverging due to their own limitations.� Limitation of CP: when δ is smaller, the integration in

U(δ) = c∫∞δ

eλxdxxα+1

� Limitation of series rep: when j is larger, the density for

[(αΓj

2cT )−1/α ∧ ηjξ1/αj ] spreads out on a large domain

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Page 26: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 3.4: comparing MC and PCM

100 102 104 106 108 1010

10−4

10−3

10−2

10−1

s

l2u2(T=1)

MC/CP

PCM/CP, d=2, s=d2Qcp

PCM/CP, d=2, s=d(Qcp+1)

PCM/CP, d=3, s=d2Qcp

PCM/CP, d=3, s=d(Qcp+1)

100 105 1010 1015

10−5

10−4

10−3

10−2

10−1

s

l2u2(T=1)

MC/S, Qs=10

PCM/S, d=2, s=d3Qs

PCM/S, d=2, s=d*Qs

PCM/S, d=3, s=d3Qs

PCM/S, d=2, s=d*Qs

Figure : l2u2(T ) versus the number of samples s by: 1) MC/CP andPCM/CP w/ δ = 0.01 (left); 2) MC/S w/ Qs = 10 and PCM/S (right).T = 1 , c = 0.1, α = 0.5, ε = 0.1, µ = 2. Spatial discretization: Nx = 500Fourier collocation points on [0, 2]; temporal discretization: first order Eulerscheme in (??) with time steps 4t = 1e − 5 (left and right).RelTol = 1e − 8 in U(δ).26 of 39

Page 27: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.1: Generalized Fokker-Planck (FP)equations for overdamped Langevin equations

� It is known that for any overdamped Langevin equation:

dx(t) = f (x(t), t)dt + dηt(ω), x(0) = x0, (24)

� The PDF of the solution P(x , t) satisfies the following generalizedFP equation:

∂tP(x , t) = − ∂

∂x[f (x , t) P(x , t)] + F−1{Pk(t) lnSk}. (25)

� Sk = E[e−ikη1 ]� Pk(t) = E[e−ikx(t)]

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Page 28: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: FP eqns for overdamped Langevinequations

� We solve:

dx(t;ω) = −σx(t;ω)dt + dLt(ω), x(0) = x0. (26)

� Method 1: we approximate Lt by a CP process, the density satisfies

∂tPcp(x , t) = [σ − 2U(δ)]Pcp(x , t) + σx

∂Pcp(x , t)

∂x

+

∫ +∞

−∞dyPcp(x − y , t)

ce−λ|y |

|y |α+1.

(27)

We will solve this by RK2 in time and Fourier collocation in spaceon a large domain [−L, L].

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Page 29: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: FP eqns for overdamped Langevin eqns� Method 2: derive the tempered fractional PDEs (TFPDEs)

� When 0 < α < 1, Sk = exp[−D{(λ+ ik)α − λα}], where

D = cαΓ(1− α). Γ(t) =

∫ +∞0

x t−1e−xdx . The density Pts(x , t)satisfies:

∂tPts(x , t) =

∂x(σxPts(x , t))− D(α)∂α,λx Pts(x , t)

− D(α)∂α,λ−x Pts(x , t), 0 < α < 1,

(28)

with the initial condition Pts(x , 0) = δ(x − x0).

∂α,λx f (x) = eλxdα

dxα[e−λx f (x)]− λαf (x), 0 < α < 1, (29)

∂α,λ−x f (x) = e−λxdα

d(−x)α[eλx f (x)]− λαf (x), 0 < α < 1. (30)

We solve this by finite difference methods.

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Page 30: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: FP eqns for overdamped Langevin eqns� Method 2: derive the tempered fractional PDEs (TFPDEs)

� When 1 < α < 2, Sk = exp[D{(λ+ ik)α − λα − ikαλα−1}] [?, ?],where D(α) = c

α(α−1) Γ(2− α). The density Pts(x , t) satisfies:

∂tPts(x , t) =

∂x(σxPts(x , t)) + D(α)∂α,λx Pts(x , t)

+ D(α)∂α,λ−x Pts(x , t), 1 < α < 2,

(31)

with the initial condition Pts(x , 0) = δ(x − x0).

∂α,λx f (x) = eλxdα

dxα[e−λx f (x)]− λαf (x)− αλα−1f ′(x), 1 < α < 2,

(32)

∂α,λ−x f (x) = e−λxdα

d(−x)α[eλx f (x)]−λαf (x)+αλα−1f ′(x), 1 < α < 2.

(33)We solve this by finite difference methods.

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Page 31: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: FP eqns for overdamped Langevin eqns(finite difference)

� The Grunwald-Letnikov finite difference:

dxαf (x) = lim

h→0

+∞∑j=0

1

hαWj f (x − jh), 0 < α < 2, (34)

and

d(−x)αf (x) = lim

h→0

+∞∑j=0

1

hαWj f (x + jh), 0 < α < 2. (35)

� Note that Wk =

(αk

)(−1)k = Γ(k−α)

Γ(−α)Γ(k+1) can be derived

recursively via W0 = 1,W1 = −α,Wk+1 = k−αk+1 Wk .

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Page 32: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: TFPDEs for overdamped Langevin eqns(finite difference)

� When 0 < α < 1, fully implicit discretization scheme:

Pn+1i − Pn

i

dt= (σ + 2D(α)λα)Pn+1

i + σxiPn+1i+1 − Pn+1

i−1

2h

− D(α)

i∑j=0

Wje−λjhPn+1

i−j −D(α)

Nx−i∑j=0

Wje−λjhPn+1

i+j .

(36)� When 1 < α < 2, fully implicit discretization scheme:

Pn+1i − Pn

i

dt= (σ − 2D(α)λα)Pn+1

i + σxiPn+1i+1 − Pn+1

i−1

2h

+D(α)

i∑j=0

Wje−λjhPn+1

i−j +D(α)

Nx−i∑j=0

Wje−λjhPn+1

i+j .

(37)32 of 39

Page 33: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.2: dealing with initial condition

In both the CP approximation and the series representation, wenumerically approximate the initial condition by the deltasequenceseither with sinc functions

δDn =sin(nπ(x − x0))

π(x − x0), lim

n→+∞

∫ +∞

−∞δDn (x)f (x)dx = f (0), (38)

or with Gaussian functions

δGn = exp(−n(x − x0)2), limn→+∞

∫ +∞

−∞δGn (x)f (x)dx = f (0). (39)

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Page 34: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.3: evolution of density

0.5

1

1.5

2

10

12

0

1

2

3

4

5

t

x(t)

Pts(x,t)

Pts(x,t)

x(t)

t

0.20.4

0.60.8

0.5 0 0.5 1 1.5 2

0

2

4

6

tx(t)

Pts(x,t)

Pts(x,t)

x(t)

t

Figure : Zoomed in density Pts(t, x) plots for the solution of equation (??)at different times obtained from solving equation (??) for α = 0.5 (left) andequation (??) for α = 1.5 (right): σ = 0.4, x0 = 1, c = 1, λ = 10 (left);σ = 0.9, x0 = 1, c = 0.005, λ = 0.01 (right). We have Nx = 2000equidistant spatial points on [−12, 12] (left); Nx = 2000 points on [−12, 12](right). Time step is 4t = 1e − 4 (left and right). The initial conditions areapproximated by δD20 (left) and δG40 (right).

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Page 35: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.3: exact moments and definition of errors

� Solution in an integral form

� The second moment for the exact solution of equation (??) are:

E[x2(t)] = x20 e−2σt +

c

σ(1− e−2σt)

Γ(2− α)

λ2−α . (40)

� Let us define the errors of the first and the second moments to be

err1st(t) =|E[xnum(t)]− E[xex(t)]|

|E[xex(t)]|, err2nd =

|E[x2num(t)]− E[x2

ex(t)]||E[x2

ex(t)]|.

(41)

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Page 36: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.3: density with CP approximation (int eqn)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910 4

10 3

10 2

t

erro

rs

err

1st

err2nd

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910 4

10 3

10 2

10 1

100

t

erro

rs

err1st

err2nd

Figure : err1st and err2nd of the solution for equation (??) versus timeobtained by solving the density equation (??) with CP approximation for theTαS process Lt : c = 0.5, α = 0.95, λ = 10, σ = 0.01, x0 = 1 (left);c = 0.01, α = 1.6, λ = 0.1, σ = 0.02, x0 = 1(right). We used RK2 in timewith time steps 4t = 2e − 3, 1000 Fourier collocation points on [−12, 12] inspace, δ = 0.012, RelTol = 1e − 8 for U(δ), and initial condition as δD20 (leftand right).36 of 39

Page 37: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.3: TFPDEs V.s. PCM via moments

0 0.2 0.4 0.6 0.8 110 4

10 3

10 2

10 1

100

t

err 2n

d

fractional density equation

PCM/CP

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510 3

10 2

10 1

100

t

err 2n

d

fractional density equation

PCM/CP

Figure : err2nd versus time by: 1) TFPDEs; 2) PCM/CP. 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/CP: δ = 1e − 5, Qcp = 50, d = 2, RelTol = 1e − 8for U(δ) (left); δ = 1e − 3, Qcp = 30, d = 2, RelTol = 1e − 8 for U(δ)(right). For density approach: 4t = 2.5e − 5, 2000 points on [−12, 12],δ = 0.012, IC is δD40 (left); 4t = 1e − 5, 2000 points on [−20, 20], δ = 0.02,i.c. given by δG40 (right).37 of 39

Page 38: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Section 4.3: TFPDEs V.s. MC via histograms

4 2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x(T = 0.5)

dens

ity P

(x,t)

histogram by MC/CPdensity by fractional PDEs

4 2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x(T=1)

dens

ity P

(x,t)

histogram by MC/CPdensity by fractional PDEs

Figure : Zoomed in plots of Pts(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 (leftand right). In MC/CP: s = 105, 316 bins, δ = 0.01, RelTol = 1e − 8 forU(δ), 4t = 1e − 3 (left and right). In the TFPDEs: 4t = 1e − 5, andNx = 2000 points on [−12, 12] in space (left and right).

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Page 39: 2014 spring crunch seminar (SDE/levy/fractional/spectral method)

Future work

� Derive the systems of TFPDEs corresponding to the solution ofSPDEs with TαS processes

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