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Summer 2010 Seminars

Thursday, July 15, 2010
Inverse Problems Seminar
Speaker: Daniel Renzi, Weill Cornell Medical College in Qatar
Title: High Order Methods for the Eikonal Equation       
Location: Amos Eaton Rm 436
Time: 12:00 PM

Abstract: Fast Eikonal solvers generally consist of two parts. The first part is a local solver, which when applied at every grid point results in a non-linear system of equations. The second part is an iteration strategy that will lead to fast convergence of the non-linear system. Ideally, a local solver for the Eikonal Equation will: 1) depend only on grid points with smaller solution (causality); and 2) use a fixed grid. The first condition is necessary for fast global convergence of the non-linear system. If a causal local solver is used the non-linear system completely decouples if the grid points are ordered in terms of increasing solution. Of course this ordering is not known in advance, but there are a number of iteration strategies that take advantage of this property. Generally convergence can be obtained by executing the local solver about m times at each grid point, where m is the dimension of the problem. The second condition is necessary for the local solver itself to be efficient. While first and second order methods that satisfy these conditions have been around for some time (mid 90’s), finding local solvers of order greater than two that are both causal and Eulurian has proven to be very difficult. Previously methods have been developed that weaken conditions (1) or (2). Weakening (1) dramatically increases the number of global iterations, while weakening (2) substantially increases the cost of the local solver due to local iterations and high dimensional interpolation. We note that high order Eikonal solvers often take about two orders of magnitude longer than the fastest low order Eikonal solvers. In this talk we develop high-order local solvers that are both causal and Eulerian. Much like the low order Eikonal solvers, combining our high order solver with an appropriate iteration strategy leads to fast convergence with only a few executions of the local solver per grid point. We demonstrate the efficiency of this technique with a number of 2d examples.

 

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