  Many inverse problems involve two or more sets of variables that represent
different physical quantities but are tightly coupled with each other. For
example, image super-resolution requires joint estimation of the image and
motion parameters from noisy measurements. Exploiting this structure is key for
efficiently solving these large-scale optimization problems, which are often
ill-conditioned.
  In this paper, we present a new method called Linearize And Project (LAP)
that offers a flexible framework for solving inverse problems with coupled
variables. LAP is most promising for cases when the subproblem corresponding to
one of the variables is considerably easier to solve than the other. LAP is
based on a Gauss-Newton method, and thus after linearizing the residual, it
eliminates one block of variables through projection. Due to the linearization,
this block can be chosen freely. Further, LAP supports direct, iterative, and
hybrid regularization as well as constraints. Therefore LAP is attractive,
e.g., for ill-posed imaging problems. These traits differentiate LAP from
common alternatives for this type of problem such as variable projection
(VarPro) and block coordinate descent (BCD). Our numerical experiments compare
the performance of LAP to BCD and VarPro using three coupled problems whose
forward operators are linear with respect to one block and nonlinear for the
other set of variables.
