Nonlinear parametric inverse problems appear in many applications in science and engineering. We
focus on one such application, namely di_use optical tomography (DOT) in medical image reconstruction. In
DOT, we aim to recover an unknown image of interest, such as the absorption coe_cient in tissue to locate
tumors in the body. We use a mathematical model (the forward model) to predict the measurements given a
certain parametrization of the tissue. The main computational bottleneck in solving such inverse problems is
the need for repeated evaluations of this large-scale forward model, which corresponds to solving large linear
systems for each source and frequency at each optimization step. Moreover, to e_ciently compute the derivative
information, we need to solve linear systems with the adjoint for each detector and frequency at each optimization
step. As rapid advances in technology allow for large numbers of sources and detectors, these problems become
computationally prohibitively expensive. In this talk, I focus on two methods to reduce this cost.
Using simultaneous random sources and detectors has been proposed to reduce this cost. However, convergence
tends to be slow for DOT. First, we propose a combination of simultaneous random sources and detectors and
optimized (for the problem) sources and detectors in order to improve convergence drastically.
Using reduced order models (ROM) is an alternative approach to drastically reduce the size of the linear systems
to be solved in each optimization step while still solving the inverse problem accurately. However, the construction
of the ROM basis still incurs a substantial cost. The second improvement we propose is using randomization to
drastically reduce the number of large linear solves for constructing the global ROM basis.
We show the e_ciency of the above approaches with 2-dimensional and 3-dimensional examples from DOT,
however, we believe our methods have the potential to be useful for other applications as well. This is joint work
with Eric de Sturler (VT), Serkan Gugercin (VT) and Misha E. Kilmer (Tufts University).