  We develop hybrid projection methods for computing solutions to large-scale
inverse problems, where the solution represents a sum of different stochastic
components. Such scenarios arise in many imaging applications (e.g., anomaly
detection in atmospheric emissions tomography) where the reconstructed solution
can be represented as a combination of two or more components and each
component contains different smoothness or stochastic properties. In a
deterministic inversion or inverse modeling framework, these assumptions
correspond to different regularization terms for each solution in the sum.
Although various prior assumptions can be included in our framework, we focus
on the scenario where the solution is a sum of a sparse solution and a smooth
solution. For computing solution estimates, we develop hybrid projection
methods for solution decomposition that are based on a combined flexible and
generalized Golub-Kahan processes. This approach integrates techniques from the
generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The
benefits of the proposed methods are that the decomposition of the solution can
be done iteratively, and the regularization terms and regularization parameters
are adaptively chosen at each iteration. Numerical results from photoacoustic
tomography and atmospheric inverse modeling demonstrate the potential for these
methods to be used for anomaly detection.
