  PDE-constrained optimization problems are often treated using the reduced
formulation where the PDE constraints are eliminated. This approach is known to
be more computationally feasible than other alternatives at large scales.
However, the elimination of the constraints forces the optimization process to
fulfill the constraints at all times. In some problems this may lead to a
highly non-linear objective, which is hard to solve. An example to such a
problem, which we focus on in this work, is Full Waveform Inversion (FWI),
which appears in seismic exploration of oil and gas reservoirs, and medical
imaging. In an attempt to relieve the non-linearity of FWI, several approaches
suggested to expand the optimization search space and relax the PDE
constraints. This comes, however, with severe memory and computational costs,
which we aim to reduce. In this work we adopt the expanded search space
approach, and suggest a new formulation of FWI using extended source functions.
To make the source-extended problem more feasible in memory and computations,
we couple the source extensions in the form of a low-rank matrix. This way, we
have a large-but-manageable additional parameter space, which has a rather low
memory footprint, and is much more suitable for solving large scale instances
of the problem than the full rank additional space. In addition, we show how
our source-extended approach is applied together with the popular simultaneous
sources technique -- a stochastic optimization technique that significantly
reduces the computations needed for FWI inversions. We demonstrate our
approaches for solving FWI problems using 2D and 3D models with high frequency
data only.
