Solving forward and inverse Helmholtz equations via controllability methods

Waves are useful for probing an unknown medium by illuminating it with a source.
To infer the characteristics of the medium from (boundary) measurements,
for instance, one typically formulates inverse scattering problems
in frequency domain as a PDE-constrained optimization problem.
Finding the medium, where the simulated wave field
matches the measured (real) wave field, the inverse problem
requires the repeated solutions of forward (Helmholtz) problems.
Typically, standard numerical methods, e.g. direct solvers or iterative methods,
are used to solve the forward problem.
However, large-scaled (or high-frequent) scattering problems
are known being competitive in computation and storage for standard methods.
Moreover, since the optimization problem is severely ill-posed
and has a large number of
local minima, the inverse problem requires additional regularization
akin to minimizing the total variation.
Finding a suitable regularization for the inverse problem is critical
to tackle the ill-posedness and to reduce the computational cost and storage requirement.
In my thesis, we first apply standard methods to forward problems.
Then, we consider the controllability method (CM)
for solving the forward problem: it
instead reformulates the problem in the time domain
and seeks the time-harmonic solution of the corresponding wave equation.
By iteratively reducing the mismatch between the solution at
initial time and after one period with the conjugate gradient (CG) method,
the CMCG method greatly speeds up the convergence to the time-harmonic
asymptotic limit. Moreover, each conjugate gradient iteration
solely relies on standard numerical algorithms,
which are inherently parallel and robust against higher frequencies.
Based on the original CM, introduced in 1994 by Bristeau et al.,
for sound-soft scattering problems, we extend the CMCG method to
general boundary-value problems governed by the Helmholtz equation.
Numerical results not only show the usefulness, robustness, and efficiency
of the CMCG method for solving the forward problem,
but also demonstrate remarkably accurate solutions.
Second, we formulate the PDE-constrained optimization
problem governed by the inverse scattering problem
to reconstruct the unknown medium.
Instead of a grid-based discrete representation combined with
standard Tikhonov-type regularization, the unknown medium is
projected to a small finite-dimensional subspace,
which is iteratively adapted using dynamic thresholding.
The adaptive (spectral) space is governed by solving
several Poisson-type eigenvalue problems.
To tackle the ill-posedness that the Newton-type optimization
method converges to a false local minimum,
we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy.
Numerical examples illustrate the usefulness of the ASI approach,
which not only efficiently and remarkably reduces the dimension of the
solution space, but also yields an accurate and robust method.