Invariant Manifolds for Model Reduction

1
Invariant Manifolds for Model Reduction

• Alexander Gorban
• University of Leicester, UK,
• and Institute of Computational Modeling Russian
  Academy of Sciences

2
Plan

• Model reduction or solution?
• Fast-slow decomposition, invariant and
  approximate invariant manifolds
• Analytical, differential or continuous?
• Analytical invariant manifold and the Lyapunov
  auxiliary theorem
• Slowness and stability

3

• Four methods for construction of slow approximate
  invariant manifolds- Power series expansion-
  Newton method with incomplete linearization-
  Relaxation methods- Method of natural
  projector- Discretization of the invariance
  equation and invariant grids.
• Examples- The Chapman-Enskog method- The
  Bobylev instability and the Newton method-
  Natural projector and coarse-graining of the
  Liouville equation- Invariant grids for chemical
  kinetics equations
• Theorem about dissipativity preservation and
  thermodynamic projector

4
Two Universes of model reduction
Universe of models
Universe of ansatzs
a model
an ansatz
A reduced model that works
5
Idea of fast-slow decomposition
Bold dashed line slow invariant manifold bold
line approximate invariant manifold several
trajectories and correspondent directions of
fast motion are presented schematically.
6
Geometrical structures of model reduction
U
?(1-P)J(f)

• U is the phase space,
• J(f) is the vector field of the system under
  consideration df/dt J(f),
• O is an ansatz manifold,
• Tf is the tangent space to the manifold O at the
  point f,
• PJ(f) is the projection of the vector J(f) onto
  tangent space Tf,
• ? (1-P)J(f) is the defect of invariance,
• the affine subspace fkerP is the plain of fast
  motions, and ? belongs to kerP.

7
Invariant and Approximate Invariant Manifolds
Local condition of invariance (the invariance
equation)
Projector P depends on f PPf
?(f)0 on O ? O is (locally) invariant
manifold ?(f)lteJ(f) on O ? O is (locally)
approximate invariant manifold with
accuracy e These definitions are invariant with
respect to monotonic time change t ? t(t),
dt(t)/dtgt0.
8
Equivalence of linear and nonlinear systems
Analytical and differential isomorphismsexist if
and only if
Continuous isomorphism exists if and only if the
sets of signs coincide
A nonlinear system near a fixed point is
continuously equivalent to its linear part, if
all the eigenvalues of the operator of linear
part are non-zero (the Grobman Hartman theorem)
9
Equivalence of linear and nonlinear systems
A system near a fixed point is continuously
equivalent to its linear part, if all the
eigenvalues of the operator of linear part are
non-zero (the Grobman Hartman theorem). For
smooth equivalence it is not truth the following
two systems have no smooth isomorphism
But such examples are, in some sense, rare
10
The Lyapunov auxiliary theorem
The system df/dt J(f) is given near a fixed
point f0. We start from a decomposition f(y,z)
and study a problem does the analytic invariant
manifold of the form zF(y) (i.e. f(y,F(y)) )
with F(0)0, F(0)0 exists? The system has a form
For the linear approximation at point f0 we
assume the following structure.
It means that the linear space (y,0) is
invariant with respect to linearized equations.
Linearized dynamics of this subspace is defined
by matrix Ly .
11
The Lyapunov auxiliary theorem
12
The Lyapunov auxiliary theorem
The choice of projector ker P(0,z),
The invariance equation
The Lyapunov auxiliary theorem. For F(0)0, the
invariance equation (Inv) has the unique
analytical solution in the neighbourhood of 0
under the following spectral conditions.
13
The spectral conditions for the Lyapunov
auxiliary theorem
Let k1,km be the eigenvalues of
LyDyJy(y,z)(0,0), ?1,?p be the eigenvalues of
LzDzJz(y,z)(0,0).
14
(No Transcript)
15
Slowness as stability
We consider manifolds immersed into phase space
and parameterized by x fF(x). The equation of
motion for immersed manifolds is
Invariant manifolds are fixed points of this
equation, and we can try to define slow invariant
manifolds as stable fixed points.
16
Relaxation spectrum for manifolds
Let us consider the analytical manifolds
relaxation near an analytical invariant manifold
of the form zF(y) (i.e. f(y,F(y)) ). Assume
that F(0)0, F(0)0 and
The relaxation spectrum for manifolds relaxation
is Re(?i-kj), where k1,km are the eigenvalues
of Ly, ?1,?p are the eigenvalues of
Lz. The motion of manifolds is exponentially
stable, if
maxRe(?i-kj)lt0
17
(No Transcript)
18
Four methods for solution of the Invariance
equation
1. Power series expansion (a) in powers of a
small parameter (the Chapman Enskog expansion
for the Boltzmann equation, the Fenichel theory
of geometric singular perturbations) or (b) in
powers of phase variables (Lyapunov series) 2.
The Newton method (the Kolmogorov Arnold
theory) or the Newton method with incomplete
linearization 3. Relaxation methods (the Euler
method for the immersed manifolds motion, for
example) 4. Method of natural projector
(extension of the Hilbert method for the
Boltzmann equation) - Discretization of the
invariance equation and invariant grids.
19
Newton method with incomplete linearization
It converges to the slowest manifold under
assumptions of the Lyapunov auxiliary
theorem,orthogonal projectors and self-adjoint
linear parts.
20
Relaxation method
The equation of motion for immersed manifolds is
We are looking for a stable fixed point
The invariance equation is
The relaxation method
The choice of step h we can project the motion
on ?k and go ahead until the projection of
current ? on ?k becomes 0 (in the linear
approximation) it is the Newton method
projected on ?.
21
Ehrenfests coarse-graining
Coarse-grained density
Mechanical motion of ensemble, Sconst
Coarse-graining, S increases
22
Averaging in cells is a particular case of
entropy maximization
Averaging in cells
23
Ehrenfests chain general case
The chain
??? The macroscopic equation
24
Natural projector(projection of segments of
trajectories)
Tt evolution operator for initial
(microscopic) system Tt evolution operator for
projected (macroscopic) system The matching
condition
For any initial point from O
25
Natural projector and attractive invariant
manifolds
t
t
t
For large t, the natural projector gives the
approximation of projection of the genuine motion
from the attractive invariant manifold onto
initial ansatz manifold.
26
The classical example the Boltzmann equation
The Chapman Enskog expansion
27
Bobylev instability of Burnett equations
Acoustic dispersion curves for Burnett
approximation (dashed line), for the first Newton
iteration (solid line), and for regularization of
the Burnett approximation via partial summing of
the Chapman-Enskog expansion (punctuated dashed
line). Arrows indicate an increase of k2.
28
Negative viscosity for Burnett equations
Dependency of viscosity on compression for
Burnett approximation (dashed line), for the
first Newton iteration (solid line), and for
partial summing (punctuated dashed line).
29
Invariant grid for two-dimensional chemical
system A1 A2 ? A3 ? A2 A4
One-dimensional invariant grid (circles)
Projection onto the 3d-space of concentrations
c1, c4, c3. The trajectories of the system in the
phase space are shown by lines. The equilibrium
point is marked by square. The system quickly
reaches the grid and further moves along it.
30
Invariant grid for model Hydrogen burning
a) Projection onto the 3d-space of cH, cO, cOH
concentrations. b) Projection onto the
principal 3D-subspace.
31
Invariant grid as a screen for visualizing
dierent functions
Model Hydrogen burning
32
Visualizing functionsEntropy and entropy
production
Entropy production
Entropy
33
Natural projector(projection of segments of
trajectories)
Tt evolution operator for initial
(microscopic) system Tt evolution operator for
projected (macroscopic) system The matching
condition
For any initial point from O
34
Quasi-equilibrium approximation
35
Beyond QE first addition and entropy production
First order QE approximation
Second order dissipative kinetics
Entropy production
Entropic inner product
36
Example the Navier-Stokes eq.
from the Liouville eq.
Free flight kinetic eq.ff(r,v,t) is
one-particle distribution function
37
The compressible Euler eq. as QE approximation
38
The compressible Navier-Stokes eq. as the second
approximation
39
Stable post-Navier-Stokes hydrodynamics instead
of unstable Burnett eqs
40
The Fokker Planck equation,the source of
examples
41
Idea of fast-slow decomposition
Bold dashed line slow invariant manifold bold
line approximate invariant manifold several
trajectories and correspondent directions of
fast motion are presented schematically.
42
The Second Law and model reduction

• Entropy of a closed system increases dS/dt 0
• In a open system the entropy production is
  positive
• These properties should hold for every model all
  the procedures of model reduction should preserve
  the sign of entropy production.

43
The problem of thermodynamic projector

• For any given
• concave entropy function (functional) S
• ansatz manifold O which is not tangent to the
  levels of S
• Find a projector P that projects any vector
  field J with dS/dt 0 in a vector field on O
  with the same inequality.

44
Geometrical structures of model reduction
U
?(1-P)J(f)

• U is the phase space,
• J(f) is the vector field of the system under
  consideration df/dt J(f),
• O is an ansatz manifold,
• Tf is the tangent space to the manifold O at the
  point f,
• PJ(f) is the projection of the vector J(f) onto
  tangent space Tf,
• ? (1-P)J(f) is the defect of invariance,
• the affine subspace fkerP is the plain of fast
  motions, and ? belongs to kerP.

45
Example 1 QSS
R- concentrations of radicals, C- concentrations
of stable components
46
Geometry of QSS
47
Example 2 QE
48
QE
49
Geometry of QE
No fast components, but some fast directions
50
Geometrical structures of model reduction
U
?(1-P)J(f)

• U is the phase space,
• J(f) is the vector field of the system under
  consideration df/dt J(f),
• O is an ansatz manifold,
• Tf is the tangent space to the manifold O at the
  point f,
• PJ(f) is the projection of the vector J(f) onto
  tangent space Tf,
• ? (1-P)J(f) is the defect of invariance,
• the affine subspace fkerP is the plain of fast
  motions, and ? belongs to kerP.

51
Quasi-equilibrium approximation
fM S(f)?max, m(f)MdM/dtm(J(fM))S(M)S(fM).
The type of dynamics preservesdS(M)/dtdS(f)/
dt (for f fM)
52
Typical examples
53
Tamm-Mott-Smith approximation for shock waves
(1950s) f is a linear combination of two
Maxwellians (fTMSafhotbfcold) Variation of the
velocity distribution in the shock front at
M8,19 (Zharkovski at al., 1997)
54
The projection problem ?ta(x,t)? ?tb(x,t)?
The Boltzmann equation (BE) ?tf(x,v,t)(v,?xf(x,v,
t))Q(f,f) Coordinate functionals F1,2f(v).
Their time derivatives should persist (BE
?tF1,2TMS ?tF1,2) BE ?tF1,2f(x,v,t)?(?F1,2f
/?f)-(v,?xf(x,v,t))Q(f,f)dv TMS
?tF1,2fTMS ?t(a(x,t)) ?(? F1,2f/?f)fhot(v)dv
?t(b(x,t)) ?(?F1,2f/?f)fcold(v)dv. There
exists unique choice of F1,2f(v)without
violation of the Second Law F1n?fdv - the
concentration F2 s?f(lnf-1)dv - the entropy
density. Proposed by M. Lampis (1977).
Uniqueness was proved by A. Gorban I. Karlin
(1990).
55
Entropic inner product
56
Differential and gradient

• Differential (Gato) of a function S(f) at point f
  is a linear functional DfS that gives the best
  approximation of S near f S(fex)S(f)eDfS(x)o(
  e) for any vector x. Definition of differential
  does not depend on an inner product.
• Gradient is a vector that represents the
  differential by the inner product

Definition of gradient depends on the inner
product
57
Entropy gradient in entropic inner product
Vector gradS in entropic inner product gives the
Newtonian direction for S (an undergraduate
exercise). The gradient should belong to a
subspace of zero balances change for
58
The thermodynamic projector

• The thermodynamic projector P onto space Tf is

Where orthogonality and all gradients are defined
in the entropic inner product
is the orthogonal projector onto space Tf
59
Limit cases

• Ansatz manifold O is the union of
  quasi-equilibrium manifolds PP- (for example,
• where f is the equilibrium, fj are functions of
  one variable, lj are linear functions).
• Ansatz manifold O is one-dimensional and not
  quasi-equilibrium

It means just the entropic parametrization (as
for the Tamm Mott-Smith approximation).
60
Uniqueness theorem

• The thermodynamic projector is the unique
  operator which transforms the arbitrary vector
  field equipped with the given Lyapunov function S
  into a vector field with the same Lyapunov
  function (and also this happens on any manifold
  which is not tangent to the levels of S).

61
Free lunch

• The requirement is the thermodynamic projector
  preserves the sign of entropy production. As
  additional consequences from this requirement we
  have
• The thermodynamic projector preserves the value
  of entropy production (not only the sign of it)
• The thermodynamic projector transforms a system
  with Onsager reciprocity relations into a system
  with the Onsager reciprocity relations (it
  preserves the Onsager relations).

62
Three reasons to use the thermodynamic projector

• It guarantees the persistence of dissipation All
  the thermodynamic processes which should product
  the entropy conserve this property after
  projecting, moreover, not only the sign of
  dissipation conserves, but the value of entropy
  production and the reciprocity relations too

63

• Universality The coefficients (and, more
  generally speaking, the right hand part) of
  kinetic equations are known significantly worse
  then the thermodynamic functionals, so, the
  universality of the thermodynamic projector (it
  depends only on thermodynamic data) makes the
  thermodynamic properties of projected system as
  reliable, as for the initial system
• It is easy (much more easy than spectral
  projector, for example).

64
Thank you for your attention