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)

=>

=>

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)

=>

=>

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