Lecture

Projection of Flows: Dual Basis

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Description

This lecture discusses the projection of flow evolution on stable and unstable manifolds, exploring the use of Gram-Schmidt and the concept of dual basis. It covers the definition and application of dual basis, finding dual vectors, and the adjoint operator of the Navier-Stokes equations. The implementation of computing dual basis, handling travelling waves, and quantifying errors in the projection process are also explained. An example application demonstrates the projection of an optimal seed on the unstable manifold using the dual basis.

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

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In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

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