3:30 pm Thursday, February 2, 2006
Special Seminar: The Phase Flow Method and High Frequency Wave Propagation by
Lexing Ying (Caltech) in RLM 10.176
In many applications, we are faced with the problem of solving an ODE with multiple initial conditions. Standard ODE integrators compute the solution for each initial condition independently, which can be computationally expensive. The phase flow method (PFM) is a novel approach to construct phase maps for nonlinear autonomous ordinary differential equations on their compact invariant manifolds. It first constructs the phase map for a small time using a standard ODE integrator and then bootstraps the process with the help of a local interpolation scheme and the group property of the phase flow. This construction usually takes the time of tracing a couple of solutions and the resulting approximation to the phase map is accurate. Once the phase map is available, integrating the ODE for an initial condition on the invariant manifold only utilizes local interpolation, thus having constant complexity. We present the method and prove its properties in a general setting. As an example, the phase flow method is applied to the field of high frequency wave propagation. We concentrate on three problems: wavefront construction, multiple arrival time and amplitude computation. We also discuss the adaptive issues in the implementation. Numerical results are presented for both the 2D and 3D cases. (Joint work with Emmanuel Candes) (Dr. Ying is one of our aP candidates.) Submitted by
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