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Systematic model reduction techniques for chemically reactive
systems Eliodoro Chiavazzo, Pietro Asinari
Multi-scale Modeling Lab, Department of Energetics, Politecnico di Torino
PHD COURSE: COMPLEX SYSTEMS IN ENGINEERING SCIENCES, Organizer: Prof. Nicola Bellomo, Deparment of Mathematics, Politecnico di Torino 21°-22° November 2011.
Brief outline • Detailed chemical kinetics and governing equations of reactive
flows;
• Combustion in homogeneous reactors;
• Model reduction and low dimensional manifolds;
• Invariant manifolds and construction techniques;
• A few words on model reconstruction for biological systems
• More references
Motivations • Energy conversion systems are today dominated and will for several
decades depend very significantly on combustion processes (still around 80%).
• Particularly in transportation systems (internal combustion engines and gas turbines), efficient and “near-zero” pollutant combustion processes can only be designed if complex reaction kinetics and their interaction with thermofluidics can be investigated and understood in depth.
The simplest fuel: H2
OHOH 2222
1
9 species, 21 reactions
+ Large disparity of time scales
Elementary reactions
OHHHO 2
• Detailed mechanisms aim at describing a chemical reaction at a more
fundamental level as it results from a large number of reversible collision
processes (elementary reaction steps) involving a usually remarkably large
number of chemical components (species):
O
2H
O
OH
5
Detailed chemistry modeling • The detailed description of combustion chemistry of practical fuels
(typically blends of higher hydrocarbons) can include hundreds of species participating in thousands of reactions.
)22(2 nnHC
Reactive flows: Governing equations
0
i
i
uxt
Conservation of mass,
NkYVVuxt
Ykkik
c
ii
i
k ,...,1,,
t
pTVvhY
t
h
k
kkk
Conservation of species
Conservation of energy
i
j
j
i
ji
ji
j
i
x
u
x
u
xx
puu
xt
u
Conservation of momentum
RTp Eq. of state
N. Fields at each grid node: Nf=3+D+N
Bell et al., PNAS 2005
N. of equations to solve at each time step:
f
D
x NN
Ex. dof107.1100256 93
A few examples: Chaotic laminar flames
Yu et al., MCS, Cagliari (Italy) 2011
Evolution of corrugated flame front:
Altantzis et al., Proc. Comb. Inst., 2011
Cellular structure of flames:
A few examples: Oscillating laminar flames
Pizza et al., Comb. Flame., 2008
Kerkemeier, PhD thesis ETH Zurich, 2011
A few examples: Turbulent flames Auto-ignition in turbulent non-premixed flows (hydrogen injected in co-flowing air, run on circa 130.000 CPUs):
Time scales in combustion
11
12
Starting point: Homogeneous reactors
NkWdt
dYkk
k ,...,1,1
N
k
kkk
p
WTHcdt
dT
1
1
Autonomous Dynamical system are to be solved. Two main drawbacks: 1) The number of degrees of
freedom is typically huge!
2) The dynamics is characterized by a wide range of time scales. This significantly hinders the numerical solution of them (due to stiffness)
• Those problems have been pointed out from the very beginning, and several methods aiming at simplifying the solution of the above governing equations have been investigated;
• Among others, the very popular techniques are surely: 1) the Quasi-Steady-State-Assumption – QSSA – (for fast species) and 2) the Partial Equilibrium Approximation (for fast reactions);
• Although those methods present a simple implementation, they are often out of reach for non-experts and not automated!!! (Chemist’s intuition needed).
13
Thermodynamic Lyapunov function
H,P,ϕ: fixed
2H 2O
OH2
H
Physical system: Homogeneous reactive mixture
Autonomous Dynamical system: Kinetic equations for a batch reactor
NkWdt
dYkk
k ,...,1,1
N
k
kkk
p
WTHcdt
dT
1
1
The Lyapunov function candidate L with respect to the above kinetic system of equations is suggested by the second law of thermodynamics. L can be constructed on the basis of the specific mixture-averaged entropy (mass based):
0~
0~
lnln1~
1 L
L
P
PRXRTSX
WSL
N
k ref
kkk
eq
14
Elemental constraints • In a given batch reactor, the equilibrium point is unique. Notice that, the
kinetic system of equations typically has several zeros, however only one is
physically acceptable (positive concentrations):
• Due to conservation of the number of moles of “d” different atoms involved
in the reaction, elemental linear constraints have to be fulfilled:
• Hence, another Lyapunov function candidate G can be written as follows:
0Y,,,YYY eq
1 fYYfdt
dN
dCYW
dN
k
k
k
k,...,1,
1
0Y~
Y~ eq
,
eq
11
LddLYW
dClLL PHk
N
k k
kd
15
Phase-Space
• In homogeneous reactive mixtures, the state evolves in N-dimensional
phase space (N is the number of chemical species), however it is only a
sub-set of
• The reason for that are the elemental constraints (equality constraints) and
the positivity of the concentrations (inequality constraints) due to the fact
that negative concentrations are physically not acceptable:
N
0
0
,...,1,0
1
1
N
N
k
k
k
k
Y
Y
dCYW
d
OHY2
OY
2HY
Phase-Space: Convex N-dimensional Polytope (hyper-polyhedron)
Steady-State
16
Simplification of a description 1/2
OHY2
OY
2HY
Steady-State
Initial conditions
NkWdt
dYkk
k ,...,1,1
N
k
kkk
p
WTHcdt
dT
1
1
Detailed description of the chemical phenomenon:
Simplified description of the same phenomenon (valid after a “long time”):
0Y ,
eq PHdL
2H 2O
OH2
HH,P,ϕ: fixed
17
Simplification of a description 2/2
NkWdt
dYkk
k ,...,1,1
N
k
kkk
p
WTHcdt
dT
1
1
Original problem: ODE’s system
Under which condition can we trust the above simplification (thermodynamic description)? t
• Thermodynamic data is reliable after a very long time!!!
• Finding the equilibrium state can be regarded as a simple model
reduction technique!!!
• However, the validity condition may be a rather restricting one!!!
0Y ,
eq PHdL
Reduced problem: Non-linear algebraic system
Dr Eliodoro Chiavazzo 18
Can we generalize that idea? • A useful generalization of the latter model reduction technique is
accomplished assuming the existence of hierarchical structures in the
phase-space due to disparate time scales in the dynamics, as exemplified
in the following cartoon: Arbitrary
initial conditions
Two-dimensional surface (2D manifold)
One-dimensional curve (1D manifold)
Steady State (0D manifold)
Solution trajectory (say 3D)
t
0tLOW DIMENSIONAL MANIFOLDS:
19
Low dimensional manifolds • Constructing low dimensional manifolds (of various dimensions) allows to
generalize the idea of model reduction, whereas the higher the manifold
dimension the more time-scales are included in the reduced model.
• The entire solution trajectory is the detailed model (dashed line): full model
= maximal computational effort = best accuracy
0D manifold, valid for: t
00 tt
1tt
2tt
...210 ttt
1D manifold, valid for: tt2
2D manifold, valid for: But necessary for:
tt121 ttt
Full dimension (3D), valid for: But necessary for:
0tt
10 ttt
20
Reduced degrees of freedom • In general, a state can evolve in the full phase-space (N-dimensions = N
degrees of freedom (DoF) to describe a given phenomenon),
• However, the occurrence of low dimensional manifolds (after a given time)
forces the states to evolve in low dimensional subspaces, which can be
therefore described by a fewer number “M” of degrees of freedom:
Hopefully M<<N.
• Fewer degrees of freedom to take into account means both a less
computational effort and a better understanding of a complex phenomenon,
without renouncing (after a given time) to a good accuracy.
Elapsed time 0 1t2t
Full model with N=3 DoF
Dynamics with M=2 DoF, 2D manidold
Dynamics with M=1 DoF, 1D manidold
M=0 DoF, Steady state
• Model reduction techniques typically operate a cut with respect to the
time scales to be represented.
21
General strategy • The problem of simplification of large ODE’s system (system of kinetic
equation) turns out to be the computation of low dimensional manifolds in
the phase-space:
N DoF N-1 DoF 1 DoF
“M=0” easy: Equilibrium point (Thermodynamic description)
Construction of a low dimensional manifold with a given dimension “M”.
Very fast time scales
Fast time scales
Intermediate Time scales
22
Fast/Slow motions have different directions in the phase-space:
1x
2x
Initial condition
Steady state
Separation of motions
23
How do we compute low dimensional manifolds for model reduction in
combustion?
Beyond Equilibrium (0D): 1-2-3…ND?
N DoF N-1 DoF 1 DoF
Fast time scales
Intermediate Time scales
0 Elapsed time
Quasi-Equilibrium manifold (QEM)
Dr Eliodoro Chiavazzo 24
1Y
2Y
Based on the implication due to: Motion separation + Lyapunov function
0t
1t
2t
3t
0t
1t
2t
3t
...
...
3210
3210
tLtLtLtL
tLtLtLtL
The conseguence is: The minimum of L constrained to the fast direction gives a point “around” the low dimensional manifold
0 1 2 3 3 2 1 0
Fast direction
L
Dr Eliodoro Chiavazzo 25
Iso-lines of the Lyapunov function L (convex function)
1Y
2Y
The geometry of a QEM When searching for a QEM, we guess the fast directions and assume that have fixed inclination in the full phase-space:
The locus of the constrained minima is called QEM
Fast directions are assumed to be known and fixed in the entire phase-space
2N
Dr Eliodoro Chiavazzo 26
Constrained equilibrium point of L or QEM point
Does it remind you anything?
1Y
2Y
2211 YY
CE
STATESTEADY
• The blue dashed line represents an
ideal fast direction;
• The blue line can be interpreted as
the locus where the following linear
combination has the constant value ξ:
• The quantity ξ is a “slow variable”
because it does not change during
fast movements;
• In general, more constraints can be
imposed.
• We need to guess one inclination of
the fast direction (1D manifold) and
fix the slow variable ξ :
2211
1
YYYN
k
kk
2N
21,
27
How do we find a QE-point?
MjY
dCYW
dts
L
N
k
j
k
j
k
N
k
k
k
k
PH
,...,1,
,...,1,..
minimum~
1
1
,
Mathematically speaking, a QE-point is the solution of the problem:
N
k ref
kkk
eq
P
PRXRTSX
WSL
1
lnln1~
This ensures that the number of atoms are conserved!!!
This imposes that the slow variables ξj are conserved during the fast motions
The locus of all the QE-points is called QE-manifold (considering ξj as variable parameters). The number “M” of additional constraints dictates the dimension of the QE-manifold
How do we solve the above problem?
k
N
k
j
k
jM
j
jk
N
k k
kd
YlYW
dClLL
1111
~
Global minimization of the following Lagrange function:
28
RCCE parameterization • A M-dimensional QE-manifold can be computed as soon as the following M
parameterization vectors are specified:
• In combustion problems, it is rather popular the RCCE (Rate Controlled
Constrained Equilibrium) parameterization (by Keck & Gillespie, 1971),
where the above vectors are linked to known slow variables.
• For example, the total number of moles is a slow variable due to the fact
that recombination/dissociation reactions (that make the total number of
moles to change) are quite slow:
M
N
M
N
,,
,,
1
11
1
N
N
N
k k
k
WWW
Y 1,,
1,,
1
11
1
1
1
29
Spectral Quasi Equilibrium • RCCE parameterization requires that slow quantities are know in advance
(there are some recipes).
• A more systematic parameterization is given by the Spectral Quasi
Equilibrium SQE (by Chiavazzo, Gorban, Karlin, 2007):
0Y,,,YYY eq
1 fYYfdt
dN
N
NN
N
Yf
Yf
Yf
Yf
J
1
1
1
1
System of kinetic equations:
Jacobian matrix J of the above system:
• According to the SQE, the parameterization vectors are given by the left
eigenvectors of the Jacobian matrix evaluated at the equilibrium point:
eqYJ
30
Slow Invariant Manifolds – SIM and construction techniques (for homogeneous reactors)
31
• Manifolds suitable for model reduction collect all the trajectories after a short
relaxation (fast motion);
• After reaching such manifolds, fast dynamics is exhausted and only the slow
dynamics is left;
• Hence, those manifolds can be termed as low-dimensional manifolds of the
slow motions;
• Trajectories, once attracted to the manifold, do not leave any more the manifold
• The latter properties is typically referred to as “Invariance Condition”:
Invariance condition
Fast relaxation toward the manifold
Slow dynamics toward the steady state (always on the manifold)
Slow Invariant Manifolds (Best for Model Reduction)
32
Invariant manifolds • A given manifold is termed invariant with respect to a dynamical system if,
starting from any of its points (as initial condition at time t=0), the solution
trajectory proceeds along the manifold without leaving it any longer at a future
time t>0:
0Y,,,YYY eq
1 fYYfdt
dN
Initial Condition
Steady state
2D invariant manifold with respect to the dynamical system
How do we check if a given manifold is invariant?
33
Modern model reduction methods • Various modern model reduction methods are all about techniques aiming at
accurately compute slow invariant manifolds - SIM;
• Constructing SIM is everything but a simple task;
• Usually, only approximations are computed;
• Quasi-Equilibrium Manifolds are rough approximations of the SIM, since they
are typically not invariant:
Yf
Non invariant manifold:
Tangent space
T~
Tf~
Y
Yf
Invariant manifold:
T~
Tf~
Y
34
Using the invariance condition
MODEL REDUCTION OF N-dimensional systems
SEARCHING FOR SLOW-INVARIANT MANIFOLDS
HOW CAN WE FIND SIMs? • SIM are some very special INVARIANT MANIFOLDS;
• Hence, an option is to write the invariance condition in a form of an equation e
try to solve it: INVARIANCE EQUATION;
• However, this is not a trivial task because INVARIANCE IS ONLY
NECESSARY CONDITION FOR SIM!!!
• Notice, for example, that all trajectories are 1D invariant manifolds
• SIM are only a small subset of invariant manifolds.
• Model reduction is understood as the construction of SIM in the phase-space:
Invariant manifolds SIM
Davis-Skodje toy-model
2
1
2
112
2
11
1
1
y
yyy
dt
dy
ydt
dy
• Simple system with two degrees of freedom (y1,y2). A reduced description with
one degree of freedom is required:
• The Davis-Skodje example is characterized by a Slow Invariant Manifold (SIM)
whose analycal form is explicitly known, thus can be used for testing SIM
construction methods.
1
35
Singularly perturbed systems
• The Davis-Skodje example is a classical example of singularly perturbed systems
(derivatives multiplied with small number) and can be rewritten as follows:
11
2
1
1
1
12
2
11
11 y
y
y
yy
dt
dy
ydt
dy
• At a glance, we expect that if ε is small, then the variable y2 will be
characterized by a fast dynamics compared to y1. In other words, the
dynamics of y2 is expected to be slaved by the dynamics of y1 on a larger time
scale.
36
Slaving and invariance condition
• The idea of slaving can be elucidated by saying that fast variables have not
independent dynamics in time. Mathematically speaking, this can be expressed
as follows:
tyyty 122
• Hence, time derivatives of fast variables can be computed applying the chain
rule (and recording the first equation of the D-S system):
1
21
1
1
22
dy
dyy
dt
dy
dy
dy
dt
dy
• Upon substitution in the second equation of the D-S system:
21
1
1
12
1
21
11 y
y
y
yy
dy
dyy
INVARIANCE EQUATION
37
Invariance condition 1/2
21
1
1
12
1
21
11 y
y
y
yy
dy
dyy
• The invariance equation (IE) states that: For any dependence of the slow
variable y1, function y2(y1) has to be such that the above equation is satisfied;
• Importantly, in the invariance equation there are no more time-derivatives;
• A certain function y2(y1) has to be found to describe the SIM;
• The latter function has to satisfy the IE at least up to some orders of accuracy;
• The IE itself does not provide the unknown function y2(y1), due to the fact that
there are simply too many solutions. Additional hypothesis are to be done
before trying to solve it: IE only necessary condition!!!
38
Invariance condition 2/2
f
1y
2y
12: yy does not satisfy the invariance condition
21, yyy
yfdt
dy
12: yydoes satisfy the invariance condition
f
1y
2y
39
Chapman-Enskog solution to the IC
21
1
1
12
1
21
11 y
y
y
yy
dy
dyy
,...,2,1,0...,... )(
2
)2(
2
2)1(
2
)0(
22 nyyyyy nn
• Chapman-Enskog expansion is a perturbation method that attempts a solution
of the invariance condition, by exploiting the smallness of ε and assuming the
following shape of the unknown solution function y2(y1):
• Where:
tyyy nn
1
)(
2
)(
2
21
1
1
1)2(
2
2)1(
2
)0(
2
)2(
2
2)1(
2
)0(
2
1
111
......y
y
y
yyyyyyy
dy
dy
• Upon substitution into the invariance condition:
• IC:
40
Dynamics at different scales • Grouping together terms of the same order in ε:
1
1)0(
2
0
10:
y
yy
1
1)0(
21 y
yy
2
1
1)1(
2
1
)0(
21
1
1:
y
yy
dy
dyy 0)1(
2 y
)(
2
1
)1(
21: n
nn y
dy
dyy
ny n 0)(
2
• The full summation can be computed and it gives the exact SIM equation (very special case, typically we stop after a few terms):
1
1
0
)(
221 y
yyy
n
nn
Exact expression for the SIM 41
Trajectories solutions in the phase-space
• Solution trajectories (blue circles) in the phase-space (y1,y2) of the D-S
example with ε=1/50 – by Euler scheme with constant dt=5e-3:
• Asymptotic solution y2(y1) (black curve): 1
12
1 y
yy
2
1
2
112
2
11
1
1
y
yyy
dt
dy
ydt
dy
50
42
CE method for combustion
• In combustion, CE method is typically adopted for analysis purposes of fairly
simple systems, mainly due to the fact that small parameters are not explicitly
expressed in the RHS of dynamical systems;
• Usually, dynamical systems are regarded as a black box, hence there is a need
of automated procedure for constructing numerical approximations of the SIMs.
2H 2O
OH2
HH,P,ϕ: fixed
NkWdt
dYkk
k ,...,1,1
N
k
kkk
p
WTHcdt
dT
1
1
Chemical system Black box (from commercial codes)
f Automated
reduction
Desired technological chain for realistic systems:
Method of Invariant Manifold
44
A.N. Gorban I.V. Karlin
• This approach is about an alternative method for
solving the invariance equation. Introduced (for
chemical kinetics) in 2004 by Gorban and Karlin
How do we write the invariance equation in a general form?
T~ fP
~
f • Define a Projector operator (matrix)
onto the tangent space, such that:
fPfPPPP
TfP
~~~~~
~~
2
• General form of the invariance equation:
0~
fPfTangent space
Leicester University (UK)
ETH-Zurich
45
General form of the IE • In general, invariance can be imposed by introducing a projector operator onto
the tangent space and writing down the following condition:
T~
f
iY
fP~
fPf ~
0~
cfcPcf
Example: It does not satisfy the invariance equation (local condition):
jY
c
NYYc ,,1
0~
fPf
The idea here is to solve the (non-linear) Invariance equation in order to find SIM
How do we solve non-linear equations?
Local correction
Initial guess – Typically non-invariant manifold (E.G. QE-Manifolds)
Slow invariant manifold – to be found
Intermediate iteration
Steady state
c
0c1c
46
Iterative solution of the IE • In the same spirit of the Newton-Raphson method for solving non-linear
equations, The Method of Invariant Manifold (MIM) operates a linearization of
the Invariance Equation (non linear because of both the Projector and the
vector field) around an initial guess and try to solve it.
0~~ 00000 ccJcfcPccJcffPf
Incomplete linearization of the IE:
Plus a solvability condition:
0~ 0 ccP
Plus the conservation of atoms
dCYW
dN
k
k
k
k,...,1,0
1
47
The MIM algorithm 1/2
0
0~
0~
cD
cP
ccJcfcPccJcf nnnnn
N
dNd
N
N
nn
W
d
W
d
W
d
W
d
Dccc
1
1
1
1
11
1 ,
• Linear system to solve with respect to cn+1
Iterations terminate when the
invariance defect is small:
cfcfcPcffc ~
48
The MIM algorithm 2/2
0
0~
0~
(*)
cD
cP
ccJcfcPccJcf nnnnn
• Algebraic system of equations to solve at each node of the manifold:
• It proves convenient to introduce the following vectors:
ib
Basis of the subspace: DP ker~
ker
• The last two equations of (*) are automatically fulfilled if we impose:
i
h
i ibc
1
• The sistem (*) can be recast as follows by scalar product with bk, to be solved
with respect to the coefficients δi:
hwith Being the dimension of DP ker~
ker
hkbJPbbJP kki
h
i i ,...,1,~
1~
11
49
Construction of the projector: An example (1D)
u
u
uu
ˆ
Unity vector:
T~
f
fP~
uuffP ˆˆ~
Euclidean projector:
yyxy
yxxx
uuuu
uuuuP
ˆˆˆˆ
ˆˆˆˆ~ yx uuu ˆ,ˆˆ Tensorial notation (N=2):
Tu~
50
How do we construct an Euclidean 2D projector?
T~
1u
2u
f
0
~
~
21
2
1
uu
Tu
Tu
2D tangent space (hyper-plane)
Gram-Schmidt ortho-normalization:
11
21
112121
122
11
0
uu
uu
uuuuuu
uuu
uu
2
22
1
11
ˆ,ˆu
uu
u
uu
2211ˆˆˆˆ
~uufuuffP
Relaxation of the Film Equation (FE)
51
A.N. Gorban I.V. Karlin U. Maas
• The Film equation is dynamical equation
written for manifolds, aiming at refining an
initial non-invariant manifold. Used by Maas
& Collaborators for Combustion problems,
and further developed by Gorban and Karlin.
12: YY
fPfdt
d ~
1Y
2Y
Initial guess – Typically non-invariant manifold (E.G. QE-Manifolds)
Hint: Write by yourself an explicit first-order scheme for solving the FE (e.g. for the Davis-Skodje model)!
SIM
Leicester University (UK)
ETH-Zurich University of Karlsruhe (Germany)
Example: H2 + Air
D1 D2
Chiavazzo, PHD thesis, ETH-Zurich 2009
Reduced system to solve (not closed):
MY
ik YfMPt
NqM Matrix, q<<N
SIM do provide closures for reduced systems.
NiYft
Yi
k ,...,1,
Detailed system:
Spectral decomposition of J 1/2
53
U. Maas S.B. Pope
• The Intrinsic Low dimensional Manifold (ILDM) method is a very popular method in combustion (introduced in 1992 by Maas and Pope) for computing accurate approximations of SIM, though typically NOT INVARIANT!!!
Basic idea: The slow invariant manifolds (SIM) represent the locus of points “c” in the phase-space where we observe very strong contractions along some directions (fast directions) compared to other transversal directions (slow directions along the SIM):
c
c
fcc
fc
Perturbation along the fast direction:
ffff
ff
cccJcfccf
ccJcfccf
The fast direction is almost an eigen-direction of the Jacobian matrix “J” evaluated at “c”.
University of Karlsruhe (Germany)
Cornell University (NY - US)
54
c
scc
sc
c
Perturbation along the slow direction:
ssss
ss
cccJcfccf
ccJcfccf
0 sf
The slow direction (SIM) is “almost” an eigen-direction of the Jacobian matrix “J” evaluated at “c”. Due to the disparate time scales, there exists a separation between “fast” and “slow eigenvalues”:
• The above idea can be generalized by noticing that it can be established a hierarchy of eigen-vectors on the basis of the corresponding eigen-values of J:
NMNMN vvvv
,,,,, 11
011 NMNMN
Fast subspace Slow subspace
Spectral decomposition of J 2/2
55
ILDM
c
• The ILDM approach divides the eigenvectors of the Jacobian matrix “J” in two
sub-groups: The first one identifies the slow sub-space, whereas the other
identifies the fast sub-space.
• By decomposing the vector field “f” into components of the fast- and slow-
subspace, the equation of the ILD-Manifold is found imposing that the fast
component is null:
f
slowf
fastf
slowfast fff
1v
2v
21 vvfff slowfastslowfast
0fast
ILDM condition:
56
Non-Cartesian coordinate systems 1v
2v
1v
2v
f
How do we find the components in a Non-Cartesian coordinate system?
2
2
1
1 vvf
ijji vv
Let’s introduce the dual spaces as follows:
2211 vfvvfvf
Kronecker delta
57
The ILDM equation
c
f
slowf
fastf
slowfast fff
1v
2v
21 vvf slowfast
00 1 fvfast
ILDM equation:
2211 vfvvfvf
ijji vv
Slow-Fast decomposition of the vector field “f”:
In general, let’s compute the matrix “Q” and its dual:
NMNMN vvvvQ
,,,,, 11
Fast Slow
slow
fast
N
MNQ
Q
v
v
v
Q
1
1
IQQ 1
0 fQ fast
ILDM equation:
Hint: Find the analytical ILDM of the Davis-Skodje model!
CSP
58
H. Lam
Computational Singular Perturbation (CSP) method
D. Goussis Princeton University (NJ - USA)
?
• Similarly to ILDM, the CSP method looks for a decomposition into fast and slow
modes of the right-hand side of the kinetic equation system “f”:
0Y,,,YYY eq
1 fYYfdt
dN
MNf vvA
1
N x (N-M) Matrix of the fast vectors
NMNs vvA
1 N x M Matrix of the slow vectors
TMN
f vvB
1
(N-M) x N Dual of Af: IAB f
f
TNMN
s vvB
1
M x N Dual of Af: IAB s
s
fBAfBAf s
s
f
f
The four matrices are updated according to the CSP iterative algorithm, and the SIM equation is:
0fB f
Lebiedz’s Method
59
D. Lebiedz University of Freiburg (Germany)
• This method is called Minimal Entropy Production
Trajectories (MEPT);
• It looks for approximations of SIM, and it typically
does not deliver the exact SIM;
• It has been introduced in 2004-2006;
• It is based on a variational problem: Minimization
of the Entropy production under constraints
0Y,,,YYY eq
1 fYYfdt
dN
MjY
dCYW
d
fdt
d
dt
N
k
j
k
j
k
N
k
k
k
k
t
t
f
,...,1,
,...,1,0
YY
Ymin
1
1
0
An approximate SIM is given by all solution trajectories that minimize an objective function (time integral of a function ϕ).
Two options have been adopted:
dt
Sd.1
fJdt
d
d
fd
dt
fdY
Y
Y.2
Entropy production
Trajectory Curvature
Atom conservation
Parameterization
Kevrekidis’s method
60
I. Kevrekidis Princeton University (NJ - USA)
Reverse integration method:
Initial condition
SIM
• According to this approach, we let a system relax from an arbitrary initial condition;
• After some time, the trajectory is expected to land on the SIM (non simple to check it);
• From this point on (yellow), we can integrate the system backward and extend the manifold;
• Simple and fascinating idea, though with a quite tricky implementation when the dimension of the manifold is high;
• Still under investigations.
Forward integration
Backward integration Steady state
ICE-PIC Method
61
S. B. Pope Cornell University (NY - USA)
Z. Ren
?
• This method is called: Invariant Constrained
Equilibrium edge PreImage Curve method (ICE-
PIC);
• This method is based on the notion of Quasi-
Equilibrium Manifold (Constrained Equilibrium
Manifold) with the RCCE parameterization and an
algorithm to improve it;
• It has been introduced in 2006 by Prof. Pope and
collaborators
QE-Manifold
Steady-State
Convex Polytope (Phase-Space) boundary
Back to the edge of the domain
Forward integration
CE-point
ICE-PIC point
• First, the Quasi-Equilibrium problem is solved and the CE-point found;
• Second, an appropriate boundary point “cbound” is found by the Preimage Curve Method;
• Finally, forward integration allows to find a much more accurate state than the CE point. boundc
Relaxation Redistribution Method Yet another (and numerically stable) way of solving the Film equation of dynamics, and the problem of minimal reduced description (choice of the manifold dimension q):
Chiavazzo & Karlin, PRE 2011
Global construction in the Phase-Space: Local construction in the Phase-Space:
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.1618v2.pdf
http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.0730v2.pdf
More info at:
fPfdt
d ~
Minimal description: Example
63
Chiavazzo & Karlin, PRE 2011
Homogeneous reactor: Detailed vs reduced with variable dimension:
Closure of reduced equations (flows)
FSIM for homogeneous reactor
Transport term
Chemistry term
iiY
Primitive variable
Collection of SIM for homogeneous reactor
Full dynamics (transport + chemistry)
Reduced dynamics (projection onto tangent)
Full dynamics
Reduced description of reactive flows
NqiYm k
k
i
ki ,...,1, Reduced observables
NkYYYFt
Yikiik
k ,...,1,, 2
Species equations
Energy eq. + Momentum Eq. + Eq. of state
qkFt
kkk ,...,1,
Reduced species
equations
Energy eq. + Momentum Eq. + Eq. of state
REDUCTI ON
Typical strategy:
NOT CLOSED!
Closure provided by the collection of SIMs (typically stored in the form of a LUT)
Example: Detailed vs Reduced
Fiorina et al, Comb. Theo. Model. 2003
(A) Detailed solution;
(B) Reduced solution.
Methane: 29 species, 300 reactions
Example: Detailed vs Reduced
Gicquel et al, Comb. Theo. Model. 1999
13 species 2 species 13 species 2 species
Biological systems • Biological systems are often characterized by several interacting agents (biological
network) and processes with disparate time scales;
• Sometimes, model reduction techniques may be reversed in order to construct models for describing time evolution of e.g. biological systems (Model reconstruction);
• The simplest instance of bio-network (Micaelis-Menten catalytic reaction):
Network:
Mechanism:
Kinetic model:
An example
Schematics: Sugar from CO2 Modeling: Agents interacting in a network
An approach to dissipative bio-systems
Basic assumption for dissipative bio-systems:
We assume that dynamics is described by a hyerarchy of relaxation processes towards low dimensional manifolds, and each process can be approximated by BGK (Bhatnagar–Gross–Krook) – like expressions:
Fasano, Chiavazzo, Asinari, Unpublished 2011
Gibbs free energy (ideal systems):
Dynamical system: Approximation of SIMs (QEM approximation):
Parameter tuning
Chose parameters: Mi , That ensure minimal deviation from experimental data
min
Mitogen-activated protein kinase (MAPK)
Optimized system:
Fasano, Master thesis, Politecnico Torino 2011
http://www.ebi.ac.uk/biomodels-main/
MAPK: Network inference
ii y
dt
dx
MORE INFO AT…
http://e-collection.library.ethz.ch/view/eth:41898?q=chiavazzo
Useful links:
http://angkinetics.narod.ru/pdf/CiCP2007vol2_n5_p964.pdf
http://www.global-sci.com/freedownload/v8_701.pdf
Gorban&Karlin 2005, Springer
http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.0730v2.pdf
http://www.springerlink.com/content/x737842j7438mu30/
http://iopscience.iop.org/1478-3975/8/5/055011
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