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

LAP: a Linearize and Project Method for Solving Inverse Problems with Coupled Variables