Applied Math And Analysis Seminar
Monday, March 12, 2012, 4:30pm, 119 Physics
Hongkai Zhao (Dept of Mathematics, Univ. of California-Irvine)
Can iterative method converge in a finite number of steps?
Abstract:- When iterative methods are used to solve a discretized linear system for
partial differential equations, the key issue is how to make the convergence
fast. For different type of problems convergence mechanism can be quite
different. In this talk, I will present an efficient iterative method, the
fast sweeping method, for a class of nonlinear hyperbolic partial
differential equation, Hamilton-Jacobi equation, which is widely used in
optimal control, geometric optics, geophysics, classical mechanics, image
processing, etc. We show that the fast sweeping method can converge in a
finite number of iterations when monotone upwind scheme, Gauss-Seidel
iterations with causality enforcement and proper orderings are used. We
analyze its convergence, which is very different from that for iterative
method for elliptic problems. If time permit I will present a new
formulation to compute effective Hamiltonians for homogenization of a class
of Hamilton-Jacobi equations. Both error estimate and stability analysis
will be shown. [video]
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