a treatment of multi-scale hybrid systems involving
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A treatment of multi-scale hybrid systems involving
deterministic and stochastic approaches, coarse
graining and hierarchical closures
Alexandros Sopasakis
Department of Mathematics and Statistics
University of North Carolina at Charlotte
Work partially supported by NSF-DMS-0606807 and DOE-FG02-05ER25702
Alexandros Sopasakis, UNCC
Outline
• Motivation – climate modeling, catalysis, traffic flow
• The prototype hybrid systemintroduction and examples
• Part I. Deterministic closures: stochastic averaging, local mean field models
• Part II. Stochastic closures: coarse-graining in hybrid systems
• The role of rare events and phase transitions
• Intermittency and metastability phenomena.
• Conclusions
Alexandros Sopasakis, UNCC
Motivation
Pure Stochastic
• Micromagnetics (P. Plechac)
• Traffic Flow (N. Dundon, T. Alperovich)
• Epidemiology (R. Jordan)
Coupled Systems
• Catalytic Reactors (M. Katsoulakis, D. Vlachos)
• Climate Modeling (M. Katsoulakis, A. Majda)
• General Biological Systems
• …
phase Gas
Surface
Alexandros Sopasakis, UNCC
Some challenges and questions:
• Disparity in scales and models;
DNS require ensemble averages for large systems
• Model reduction, however no clear scale separation;
need hierarchical coarse-graining
• Deterministic vs stochastic closures;
when is stochasticity important?
• Error control, stability of the hybrid algrithm;
efficient allocation of computational resources
adaptivity, model and mesh refinement
Alexandros Sopasakis, UNCC
A mathematical prototype hybrid model
• We introduce the microscopic spin flip
stochastic Ising process
• Coupled to a PDE/ODE that serves as
a caricature of an overlying gas phase
dynamics.
ott }{
),(1
XgXdt
d
c
)(1
)(
ELfEfdt
d
I
Alexandros Sopasakis, UNCC
• We assume a lattice L and denote by
(x) the value of the spin at location x
• A spin configuration is an element of
the configuration space
and we write
• The stochastic process is a
continuous time jump Markov process
on with generator L
L }1,0{
}:)({ L xx
0}{ tt
),( RL
L
x
x ffxcLf )]()()[,()(
0 11 10 )(x
The Stochastic Component: )(1
)(
ELfEfdt
d
I
Alexandros Sopasakis, UNCC
The Arrhenius spin-flip rate c(x,s) at lattice site x and spin configuration is given by
with interaction potential
and local interaction via
Parameters/Constants:
• where is the characteristic time of the stochastic process
• L denotes the interaction radius
• V is usual taken to be some uniform constant
1)(en wh
0)(n whe),(
)(
xc
xecxc
a
xU
d
I
da cc
1 I
L
zxz
XhzzxJxU )()(),()(
The Stochastic Component:Arrhenius
Spin-Flip Dynamics
12
||
12
1 ),(
L
yxV
LyxJ
Alexandros Sopasakis, UNCC
Equilibrium states of the stochastic model are described by the Gibbs measure at the
prescribed temperature T
where is the energy Hamiltonian for
and with
)(1
)( )(
, dPe
Zd N
H
N
L
x
N xddP ))(()( 2
1)1)(( ,
2
1)0)(( xx
kT
1
The Stochastic Component:
L
x
xxUH )()(2
1 )(
)(1
)(
ELfEfdt
d
I
Alexandros Sopasakis, UNCC
The probability of a spin-flip at x during time [t, t+Dt] is
The dynamics as described here leave the Gibbs measure invariant since they satisfy
detailed balance,
where
)(),( 2tOtxc DD
))(exp(),(),( Hxcxc x
x D
)()()( HHH x
x D
The Stochastic Component: )(1
)(
ELfEfdt
d
I
Alexandros Sopasakis, UNCC
Some examples for the ODE
CGL:
Bistable:
Saddle:
Linear:
where depends linearly on
*2 ˆ||))((),( XXXXiaXg
3)(),( XXaXg
2)(),( XaXg
cXbaXg )(),(
The Deterministic Component ),( Xgdt
Xdc
)(a
Alexandros Sopasakis, UNCC
The stochastic system and the ODE are coupled via, respectively:
• The external field,
• The area fraction (or total coverage) defined as the spatial average
of the stochastic process ,
)(Xhh
L
x
xN
)(1
Hybrid System Coupling
Alexandros Sopasakis, UNCC
Part I. Deterministic Closure
The main requirement is the ergodicity property of the stochastic process
dtXgT
T
tT 0
),(1
lim
)()(1
)( )(
,
N
x
H
N PexZ
hut
L
Due to special structure of g (depends linearly on )
we have
)(Xg
),()(,
XgEXgN
),(
,
NEXg
)))((,( , XhuXg N
)(, hu NWhere can be found from the known Gibbs equilibrium measure
Alexandros Sopasakis, UNCC
• On an arbitrary bounded time interval [0, T] with fixed N and c we have:
0any for 0)|)()(|sup(P lim00
txtX
TtI
xX
TXgXdt
d
ELffdt
d
t
c
I
0
],0[for t ),,(1
)(1
)(
xx
xhuxgxdt
d
xhu
tNt
c
t
tN
0
,
,
T][0,for t )))((,(1
))((Given
)(tXX
)(txx
Suppose is the solution of the
hybrid system:
And is the solution of the
(reduced) averaged system:
Part I. Deterministic Closure
Alexandros Sopasakis, UNCC
Stability and Potential Wells
Alexandros Sopasakis, UNCC
Solutions of the coupled system
Direct Numerical Simulation vs Deterministic Closure
Alexandros Sopasakis, UNCC
A Rare Event
Direct Numerical Simulation vs Deterministic Closure
Deterministic closures can fail in extended time simulations.
Alexandros Sopasakis, UNCC
Another example: Coupled system with Hopf ODEDirect Numerical Simulation vs Deterministic Closure
Deterministic
closures
can fail if 1
Fast Stochastic Case Equivalent Characteristic Times
Slow Stochastic Case
Alexandros Sopasakis, UNCC
Part II. Stochastic Closures
We define the coarse random process, for k=1, … m
and with and N=mq
where (naturally )
Fine Lattice L
Coarce Lattice Lc
kDx
xk )()(
}:)({ ckk L
]1,0[1
L Zm
c LLc
qk ,...1,0)(
Total number of micro cells: N
1 2 3 4 … mCoarse
cells
1 2 3 4 NMicro
cells:
Alexandros Sopasakis, UNCC
The coarse grained generator for the Markovian process is defined to be,
For any test function where the coarse level adsorption/desorption rates are,
With a corresponding coarse potential
)]()()[,(
)]()()[,()(
ffkc
ffkcfL
kd
k
kac
c
L
))((
0
0
0)(),(
)]([),(
kUU
d
a
ekdkc
kqdkc
L
kll c
XhkJllkJkU )()1)()(0,0()(),()(
);( , RHLf qn
Part II. Stochastic Closures
Alexandros Sopasakis, UNCC
Compare Stochastic Transient and Equilibrium Dynamics.
)(1
)( cMicroscopi
ELfEfdt
d
I
)(1
)( Grained Coarse vs
fELfEdt
dc
I
Alexandros Sopasakis, UNCC
Coarse Grained Stochastic Closure
Our Coarse Grained system therefore becomes,
and gives a stochastic closure at this level.
• This stochastic closure is expected to be valid for all time since no
linearization arguments or use of expected values are involved.
• This stochastic closure is an approximation to the original hybrid system
)(1
)(
),(1
fELfEdt
d
XgXdt
d
c
I
c
Alexandros Sopasakis, UNCC
Direct Numerical Simulation vs Stochastic Coarse Grained Closure (q=20).
Fast Stochastic Case
Equivalent Characteristic Times
Slow Stochastic Case
Alexandros Sopasakis, UNCC
Hopf Bifurcation
Direct Numerical Simulation vs Stochastic Coarse Grained Closure (q=20)
Alexandros Sopasakis, UNCC
Rare events
and
Phase Transitions
Alexandros Sopasakis, UNCC
Revisit the rare event: Direct Numerical Simulation vs Stochastic Coarse Grained Closure
We can capture the rare event
Alexandros Sopasakis, UNCC
In the case of phase transitions the Coarse Grained closure
still agrees with the DNS solution.
q DNS 2 4 5 10 20 25 50 100
Rel Error % 0 .01 .22 .38 .82 3.42 4.91 17.69 77.73
CPU(sec) 309647 132143 85449 58412 38344 16215 7574 4577 345
to error estimate
Alexandros Sopasakis, UNCC
Externally Driven Phase TransitionsDirect Numerical Simulation vs Coarse Grained Closure (for q = 10)
Alexandros Sopasakis, UNCC
Externally Driven Phase TransitionsDirect Numerical Simulation vs Coarse Grained Closure (q=50, 100)
Alexandros Sopasakis, UNCC
Theorem: Suppose the process defined by the coarse generator is the coarse
approximation of the microscopic process then for any q < L and N where mq=N
The information loss as q/L -> 0 is
Theorem: Let be a test function on the coarse level s.t there exist a test function
with property Given the initial configuration we define the coarse
configuration Assume the microscopic process with the initial condition
and the approximating coarse process with the initial condition then
the weak error satisfies for final time T,
L
qOTQTQR
N
c
TTT )|(1
],0[,],0[, 00
],0[}{ Ttt
],0[}{ Ttt cL
)( cL
)( L ).()( T0
00 T },}({ 0 Ltt
0 },}({ 0 ctt L 00 T
2
|)]([)]([|
L
qCETE TTT
Error Estimates
Alexandros Sopasakis, UNCC
Intermittency and
Metastability Phenomena
Alexandros Sopasakis, UNCC
Phase transitions II Strong particle/particle interactions (FhN equation)
Step 1: Mean field approximations (ODEs):
Bistable, excitable, oscillatory regimes (strong interactions)
),()(
ELfEfdt
d
cXbaXdt
d
))((01)(txhuJ
ueuEfdt
d
cxbauxdt
d
Alexandros Sopasakis, UNCC
Alexandros Sopasakis, UNCCto end
Alexandros Sopasakis, UNCC
Alexandros Sopasakis, UNCC
Alexandros Sopasakis, UNCC
Conclusions
Presented a coupled Prototype Hybrid System consisting of:
capable of describing the behavior of several different physical systems
Studied two types of closures for this system:a) Deterministic (averaging principle, mean field)b) Stochastic (coarse grained Monte Carlo)
Examined solutions under extreme phenomena exhibiting • rare events,
• phase transitions and
• intermittency/metastability effects.
Deterministic type closures (averaging principle / mean field) are valid for finite time intervals and for the case of fast stochastic dynamics relative to the ODE
The (stochastic) Coarse Grained Monte Carlo closure seems to be always valid under certain conditions:
ODE gBifurcatin General typeCGL
Model Noise Stochastic
)(
Lq
)( T
Alexandros Sopasakis, UNCC
•Last but not least the CGMC method offers much more than just
agreement in average quantities.
There is spatial agreement as well
Alexandros Sopasakis, UNCC
Microscopic System vs Coarse Grained Closure
Spatial Lattice Simulations (non-averaged)
through a phase transition!
Alexandros Sopasakis, UNCC
Recent Relevant Publications
Stochastic Models
• Sopasakis, Physica A, (2003).
• Sopasakis and Katsoulakis, SIAM J. Applied Math. (2005).
• Alperovich and Sopasakis, Physica Review E, submitted, (2007).
• Dundon and Sopasakis, Transportation and Traffic Theory, to appear, (2007).
Hybrid Deterministic/Stochastic Systems
• Katsoulakis, Majda, Sopasakis, Comm. Math. Sci. (2004).
• Katsoulakis, Majda, Sopasakis, Comm. Math. Sci. (2005).
• Katsoulakis, Majda, Sopasakis, Nonlinearity, (2006).
• Katsoulakis, Majda, Sopasakis, AMS Contemporary Math. to appear, (2007).
Coarse-Grained Models
•Katsoulakis, Plechac, Sopasakis, IMA preprtint 2019, (2005)
•Katsoulakis, Plechac, Sopasakis, SIAM J. Numerical Analysis, (2006).
Partially Supported by:
• NSF-DMS-0606807
• DE-FG02-05ER25702
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