Optimal control adjoint method for inversion in wave propagation
Μέθοδος βελτιστοποίησης με τη χρήση του συζυγούς τελεστή για τη λύση αντίστροφων προβλημάτων κυματικής διάδοσης
Καρασμάρη, Ευτυχία Β
Karasmani, Eftychia V
We suppose that the wave propagation inside a medium is modelled by a well
posed mathematical problem. Speci¯cally, the adopted model consists of:
1. the properties of the medium (e.g., density, wave speed, attenuation, etc.)
2. the properties and the location of the source
3. the boundary conditions
4. the propagated ¯eld.
We choose to write the above problem in terms of a system of partial di®erential
equations with initial and boundary conditions.
The task of specifying the propagated ¯eld when the properties of the medium,
the properties and the location of the source as well as the boundary conditions
are assumed to be known, is called the Direct Problem. We suppose that we
can obtain a numerical solution for this problem, using an appropriate model.
On the other hand, an Inverse Problem arise when the propagated ¯eld
is assumed to be known by experimental measurements and the objective is to
recover the properties of the medium, or ¯nd the location and the properties of
the source, or even determine the boundary conditions. A solution for the Inverse
problem can be derived by minimizing the mis¯t between the measured ¯eld and
the ¯eld predicted by the model using as control parameters the unknown model
In this work, the wave propagation in a waveguide is modelled via the parabolic
approximation and a non-local boundary condition in the form of a Neumann to
Dirichlet map is used. An Optimal Control Method using the Adjoint Operator
of the problem is exhibited for recovering the properties of the medium.