FOR RANDOM DIFFERENTIAL EQUATIONS* G(x, t)= E[H(x, c(t/e, to ...
Random Ordinary Differential Equations (RODE) · PDF fileTechnische Universit¨at Munc¨...
Transcript of Random Ordinary Differential Equations (RODE) · PDF fileTechnische Universit¨at Munc¨...
Technische Universitat Munchen
HIGH TEA @ SCIENCE
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
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 2
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 2
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 2
Technische Universitat Munchen
Outline
Motivation
Random Differential Equations (RODEs)Definition of RODEsProperties of RODEsApplication Example: Kanai-Tajimi Earthquake Model
Solving RODEs on GPU clusters
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 3
Technische Universitat Munchen
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!
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 4
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 5
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 5
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 6
Technische Universitat Munchen
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)
HIGH TEA @ SCIENCE, March 14, 2016 7
Technische Universitat Munchen
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)
HIGH TEA @ SCIENCE, March 14, 2016 7
Technische Universitat Munchen
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)
HIGH TEA @ SCIENCE, March 14, 2016 7
Technische Universitat Munchen
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)
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 8
Technische Universitat Munchen
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)
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 8
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 9
Technische Universitat Munchen
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
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 10
Technische Universitat Munchen
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)
HIGH TEA @ SCIENCE, March 14, 2016 10
Technische Universitat Munchen
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/
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 11
Technische Universitat Munchen
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.
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 12
Technische Universitat Munchen
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.
Tobias Neckel: Random Ordinary Differential Equations (RODE)
HIGH TEA @ SCIENCE, March 14, 2016 13