random ordinary differential equations (rode) · pdf filetechnische universit¨at munc¨...

Technische Universitat Munchen

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Random Ordinary Differential Equations (RODE)

F. Menhorn, T. Neckel, A. Parra, C. Riesinger, F. RuppTechnische Universitat Munchen

March 14, 2016

Tobias Neckel: Random Ordinary Differential Equations (RODE)

HIGH TEA @ SCIENCE, March 14, 2016 1


Motivation

• goal: efficient tools for numerical simulation of random effects

• RODE as an alternative setting for SODE⇒ massive simulations of pathes

• approach:1. HPC aspects of RODE simulations: parallelisation2. math aspects: relation of RODE to classical UQ




Motivation

• goal: efficient tools for numerical simulation of random effects• RODE as an alternative setting for SODE⇒ massive simulations of pathes

• approach:1. HPC aspects of RODE simulations: parallelisation2. math aspects: relation of RODE to classical UQ




Outline

Motivation

Random Differential Equations (RODEs)Definition of RODEsProperties of RODEsApplication Example: Kanai-Tajimi Earthquake Model

Solving RODEs on GPU clusters




Definition of RODEs

Assumptions• X : I × Ω→ Rm: stochastic process (continuous sample paths)• f : Rd × I × Ω→ Rd continuous

DefinitionRODE on Rd

dXt

dt= f (Xt (·), t , ω) , Xt (·) ∈ Rd (1)

is a non-autonomous ordinary differential equation

x =dxdt

= Fω(x , t) , x := Xt (ω) ∈ Rd (2)

for almost all pathes ω ∈ Ω.[Bun72]

⇒ path-wise solutions!




RODE vs. SODE

SODEdXt = a(Xt , t)︸︷︷︸

drift

dt + b(Xt , t)︸︷︷︸diffusion

dWt .

typically: considerable mathematical effort (Ito’s calculus etc.)

Doss-Sussmann/Imkeller-Schmalfuss correspondence (DSIS)

Under sufficient regularity conditions [IS01]:

SODEs and RODEs are conjugate!

⇒ rewrite SODE as RODE by changing the driving stochastic process:

typically: white noise stationary Ornstein-Uhlenbeck (OU) process




Application Example: Kanai-Tajimi Earthquake Model

4-storey wireframe building

ug ⇐⇒

u + Cu + Ku = F (t) ,

Kanai-Tajimi excitation & storeydisplacements

0 5 10 15 20−20

0

20ug

t

0 5 10 15 20−0.2

0

0.2

u1

0 5 10 15 20−0.2

0

0.2

u2

0 5 10 15 20−0.2

0

0.2

u3

0 5 10 15 20−0.5

0

0.5

u4




Kanai-Tajimi Earthquake Model as RODE

SODEug = xg + ξt = −2ζgωg xg − ω2

gxg

xg + 2ζgωg xg + ω2gxg = −ξt , xg(0) = xg(0) = 0

ξt : white noise

ζg , ωg : model parameters (e.g. ζg = 0.64, ωg = 15.56[rad/s])

SODE in vector form

d

(xy

)=

(−y

−2ζgωgy + ω2gx

)dt +

(01

)dWt

RODE (DSIS correspondence)

(z1

z2

)=

(−(z2 + Ot )

−2ζgωg(z2 + Ot ) + ω2gz1 + Ot

)




Kanai-Tajimi Earthquake Model as RODE

SODEug = xg + ξt = −2ζgωg xg − ω2

gxg

xg + 2ζgωg xg + ω2gxg = −ξt , xg(0) = xg(0) = 0

ξt : white noise

ζg , ωg : model parameters (e.g. ζg = 0.64, ωg = 15.56[rad/s])

SODE in vector form

d

(xy

)=

(−y

−2ζgωgy + ω2gx

)dt +

(01

)dWt

RODE (DSIS correspondence)

(z1

z2

)=

(−(z2 + Ot )

−2ζgωg(z2 + Ot ) + ω2gz1 + Ot

)Tobias Neckel: Random Ordinary Differential Equations (RODE)



Kanai-Tajimi Earthquake Model: OU Process

Definition of Ornstein-Uhlenbeck process:

dOt = θ(µ−Ot )dt + σdWt

Numerical scheme (exact integration [Gil96])

Ot+h = µx Ot + σx n1

with• h: timestep size• µx = e−h·θ

• σ2x := σ2

2θ (1− e−h·2θ)

• n1: sample value of N (0, 1)




Numerical Schemes for RODEs

Specific methods to achieve order of convergence [GK01, KJ07, NR13]:

• Averaged Euler• Averaged Heun• K -RODE Taylor schemes

h

︸︷︷︸

︷︸︸︷δtn tn+1

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time

x−po

sitio

n

Averaging a sample path of an OU Process (step size h = 1)

image courtesy: A. Parra




Numerical Schemes for RODEs II

For an RODE in the following form

dXdt

= G(t) + g(t)H(X ),

use averaged values for g(t) on δ:

g(1)h,δ(t) =

1N

N−1∑j=0

g(t + jδ),

=⇒ averaged Euler scheme (timestep h for xn → xn+1)same averaging procedure for G(t):

xn+1 =1N

N−1∑j=0

xn + hG(tn + jδ) + hg(tn + jδ)H(xn)

= xn +1N

N−1∑j=0

hG(tn + jδ) +H(xn)

N

N−1∑j=0

hg(tn + jδ) .

=⇒ global discretisation error of order 1




Numerical Schemes for RODEs II

For an RODE in the following form

dXdt

= G(t) + g(t)H(X ),

use averaged values for g(t) on δ:

g(1)h,δ(t) =

1N

N−1∑j=0

g(t + jδ),

=⇒ averaged Euler scheme (timestep h for xn → xn+1)same averaging procedure for G(t):

xn+1 =1N

N−1∑j=0

xn + hG(tn + jδ) + hg(tn + jδ)H(xn)

= xn +1N

N−1∑j=0

hG(tn + jδ) +H(xn)

N

N−1∑j=0

hg(tn + jδ) .

=⇒ global discretisation error of order 1Tobias Neckel: Random Ordinary Differential Equations (RODE)



Solving RODEs on GPU clusters

and now let’s switch to the dark side: Computer Science ;-)

http://bgr.com/2015/11/19/darth-vader-daily-life-photos/



http://bgr.com/2015/11/19/darth-vader-daily-life-photos/


Literature I

H. Bunke.

Gewohnliche Differentialgleichungen mit zufalligen Parametern.Akademie-Verlag, Berlin, 1972.

D. T. Gillespie.

Exact numerical simulation of the ornstein-uhlenbeck process and its integral.Physical Review E, 54(2):2084–2091, 1996.

L. Grune and P. E. Kloeden.

Pathwise approximation of random ordinary differential equations.BIT Numerical Mathematics, 41(4):711–721, 2001.

P. Imkeller and B. Schmalfuss.

The conjugacy of stochastic and random differential equations and the existenceof global attractors.Journal of Dynamics and Differential Equations, 13(2):215–249, 2001.




Literature II

P. Kloeden and A. Jentzen.

Pathwise convergent higher order numerical schemes for random ordinarydifferential equations.Proc. R. Soc. A, 463:2929–2944, 2007.

T. Neckel and F. Rupp.

Random Differential Equations in Scientific Computing.Versita, De Gruyter publishing group, Warsaw, 2013.open access.



random ordinary differential equations (rode) · pdf filetechnische universit¨at munc¨...

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