Solving Eikonal equation using Matlab
Nicholas Kinar
I would like to find an efficient way to solve an Eikonal-type equation
using Matlab. I've looked at the Matlab Fast Marching Toolbox
(http://www.mathworks.com/matlabcentral/fileexchange/6110), but I cannot
immediately see how to apply it to my problem. I believe that the Fast
Marching Toolbox will find the shortest distance between two points.
What I would like to do is to solve the following Eikonal-type PDE with
Matlab, perhaps using a finite-difference algorithm:
(dT/dv)^2 + (1/(lamba^2))*(dT/dt)^2 = exp(-2 * alpha(t,v))
In this equation, (dT/dv)^2 is the second partial derivative with
respect to v, lambda is a real-valued constant, (dT/dt)^2 is the second
derivative with respect to t, exp(.) is the exponential function, and
alpha is a function of t and v. The time is given by t, and the
velocity is given by v. I've calculated alpha(t,v) on a regular grid,
which is stored in a Matlab 2D matrix.
I would like to find an optimal trajectory by solving this equation
using Matlab. What is the best way to do this?
Nicholas Kinar
Hmm, I think that I've managed to sort things out. Essentially what I
did was use the Fast Marching Toolbox:
(1) Select a "surface velocity" at the top of the plot. This is the
starting point.
(2) Use the Fast Marching Toolbox to compute minimum distances.
(3) Select the minimum value at the bottom of exp(-alpha(t,v)). You may
have to play around with alternate RHS of this equation, such as exp(-2
* alpha(t,v)) or exp(-alpha(t,v)).
(3) Use the Fast Marching Toolbox to determine a trajectory between the
points at the top and the bottom of the semblance plot.
Nicholas