We consider the problem of acoustic scattering as described by the free-space, time-harmonic scalar wave equation given by
We describe an algorithm for efficiently solving the inverse scattering
problem for the low-frequency time-harmonic scalar wave equation with
multi-point illumination.
The following approach is based on the FaIMS algorithm described by
Chaillat and Biros in [1].
First, we compress the number of incident fields (computed by solving the
variable coefficient Helmholtz equation with point sources) using a
randomized QR factorization to compute a low rank approximation.
By compressing the incident fields, we greatly reduce the problem size
and by using a randomized QR factorization we can compute an
approximation to within a specified tolerance, ensuring that there is not
significant information loss.
After compressing the incident fields, we can compute the medium
perturbation by solving
, where the scattered
field
is measured as the difference between the incident field and
the measured total field.
We transform the Helmholtz equation above into a linear integral equation to obtain
We have extended the results of [1] in a few different ways.
In the original paper the background medium was assumed to have a
constant wavenumber except for the perturbation, i.e. was zero,
but here we allow for variable background medium.
We expand the types of scatterers allowed; the original paper allowed
only point scatterers while we allow for multiple continuous scattering
objects.
We also make use use of the iterated Born approximation in our solver.
Finally, we achieve superior scalability to the approached referenced in the FaIMS paper.
We rapidly evaluate the integral for the forward scatterer using a fast
multipole method for volume potentials in parallel.
Furthermore, we use PETSc for in our implementation for its fast
iterative solvers.