laniu s. b. pope feb. 24 th , 2005
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Dimension Reduction by pre-image curve method. Laniu S. B. Pope Feb. 24 th , 2005. Part B: Dimension Reduction –Manifold Perspective. Impose n u conditions. =>. - PowerPoint PPT PresentationTRANSCRIPT
Dimension Reduction by pre-image curve method
LaniuS. B. Pope
Feb. 24th, 2005
Part B: Dimension Reduction –Manifold Perspective
Different methods impose different nu= nφ-nr conditions which determine the corresponding manifold φm , which is used to approximate the attracting manifoldGiven a reduced composition r, according to the nu conditions to determine the corresponding full composition on the manifold φm
What is the attracting slow manifold? ---geometric significance ---invariant
Could we define a manifold which has the same geometric significance and similar properties?
Impose nu conditions =>
The sensitivity matrix is defined as
Part B: Geometric significance of sensitivity matrices
The initial ball is squashed to a low dimensional object, and this low dimensional object aligns with the attracting manifold
The principal subspace Um should be a good approximation to the tangent space of the attracting manifold at the mapping point
The “maximally compressive” subspace of the initial ball is that spanned by Vc
dd AR
Part B: Manifold
Given the reduced composition r, find a point which satisfy the above condition
Uc is from the sensitivity matrix A, which is the sensitivity of φ with respect to some point on the trajectory backward
PartB: Simple Example I
Slow attracting manifold
QSSA manifold
ILDM manifold
Global Eigenvalue manifold
PartB: Simple Example I (Contd)
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Tangent plane of the manifold
The manifold is approaching to be invariant
approaches the tangent plane of the slow manifold
Part B: Simple Example I (Contd)Comments:
For this linear system, ILDM predicts the exact slow manifold. The ILDM fast subspace seems weird
The new manifold approaches the slow manifold and approachesto be invariant as approaches zero.
The most compressive subspace approaches the QSSA species direction
Part B: Simple Example II
Slow attracting manifold
QSSA manifold
ILDM manifold
Global Eigenvalue manifold
Part B: Simple Example II (Contd)
approaches the tangent plane of the slow manifold
Part B: Simple Example II (Contd)
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Tangent plane of the manifold approaches the tangent plane of the slow manifold; The manifold is approaching to be invariant; the most compressivesubspace approaches the QSSA species direction
Part B: Dimension Reduction by pre-image curve ---Manifold Perspective
Ideas: Use pre-image curve to get a good Um, which is a good approximation to the tangent plane of the attracting slow manifold.
H2/air system
Conclusion and Future work
Identify The geometric significance of the sensitivity matrix
Identify the principal subspace and the compressive subspace
Identify the tangent plane of the pre-image manifold
Species reconstruction by attracting-manifold pre-image curve method is implemented
The manifold perspective of dimension reduction by pre-image curve method is discussed
Thanks to Professor Guckenheimer