Scenarios with eikonal
Describes the geometric properties of a wave with small wavelength moving through a medium
The eikonal equation is a PDE
Given a refractive index field, \(\eta\), what is a configuration, \(\Phi\), that satisfies the above equation?
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\(\Phi\) is the phase of the wavefront, and \(\nabla \Phi \) describes how rays will propagate through the volume
The eikonal equation can be solved in different ways
All of them solve for the phase (\(\Phi\)) given the underlying refractive field (\(\eta\))
"I want to find the refractive field, \(\eta\), that will admit the configuration, \(\Phi\), that I observed/desire"
With this formulation, we can conceptually treat \(\eta, \lambda,\) and \(\Phi\) as independent variables.
If we want to do gradient descent, our new optimization problem will be equivalent to our original problem if we insist that \(\lambda\) and \(\Phi\) are critical points of \(\mathcal{L}\).
where,
Solving the original forward problem yields a critical point with respect to \(\lambda\).
Assuming we have \(\Phi\), we can solve for \(\lambda\)
Using Divergence Theorem:
In order for both of these equations to be true, they need to be true for the whole domain of the integral
This is known as the adjoint equation, and \(\lambda\) is referred to as the adjoint state.
Forward System
Adjoint System
We now know \(\Phi^*\) and \(\lambda^*\) by solving for the stationary points.
We now know \(\Phi^*\) and \(\lambda^*\) by solving for the stationary points.
To solve the problem:
We can use these equations:
Using a known light source, or initial conditions, we can use the adjoint system to guess what the refractive index is and then refine the guess using gradient descent.
\(\lambda\) is effectively the derivative we are looking for.
What does it represent conceptually?
The observed residual only contributes to \(\lambda\) along the normal of our volume
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In other words, measurements that are parallel to the direction of rays will give us the most contribution to the derivative.
Expanding the above equation with product rule, yields:
One way to interpret this is that the adjoint state propagates parallel to the rays traveling through the medium
Solving the adjoint can then be described as propagating rays through the medium and then backtracing the residual value back through the volume.
If we instead use a discretized version of the eikonal equation, our Lagrangian becomes
We differentiate with respect to each \(\Phi_i\) and \(\lambda_i\) to get a linear system
\(\lambda\) here acts exactly in the same way as the continuous version