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Dimension Reduction by pre-image curve method Laniu S. B. Pope Feb. 24 th, 2005

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Part B: Dimension Reduction –Manifold Perspective Different methods impose different n u = n φ - n r conditions which determine the corresponding manifold φ m, which is used to approximate the attracting manifold Given a reduced composition r, according to the n u conditions to determine the corresponding full composition on the manifold φ m What is the attracting slow manifold? ---geometric significance ---invariant Could we define a manifold which has the same geometric significance and similar properties? Impose n u conditions=>

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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 U m 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 V c

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Part B: Manifold Given the reduced composition r, find a point which satisfy the above condition U c is from the sensitivity matrix A, which is the sensitivity of φ with respect to some point on the trajectory backward

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PartB: Simple Example I Slow attracting manifold QSSA manifold ILDM manifold Global Eigenvalue manifold

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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

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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 approaches to be invariant as approaches zero. The most compressive subspace approaches the QSSA species direction

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Part B: Simple Example II Slow attracting manifold QSSA manifold ILDM manifold Global Eigenvalue manifold

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Part B: Simple Example II (Contd) approaches the tangent plane of the slow manifold

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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 compressive subspace approaches the QSSA species direction

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Part B: Dimension Reduction by pre-image curve ---Manifold Perspective Ideas: Use pre-image curve to get a good U m, which is a good approximation to the tangent plane of the attracting slow manifold. H 2 /air system

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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

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