  Iterative hybrid projection methods have proven to be very effective for
solving large linear inverse problems due to their inherent regularizing
properties as well as the added flexibility to select regularization parameters
adaptively. In this work, we develop Golub-Kahan-based hybrid projection
methods that can exploit compression and recycling techniques in order to solve
a broad class of inverse problems where memory requirements or high
computational cost may otherwise be prohibitive. For problems that have many
unknown parameters and require many iterations, hybrid projection methods with
recycling can be used to compress and recycle the solution basis vectors to
reduce the number of solution basis vectors that must be stored, while
obtaining a solution accuracy that is comparable to that of standard methods.
If reorthogonalization is required, this may also reduce computational cost
substantially. In other scenarios, such as streaming data problems or inverse
problems with multiple datasets, hybrid projection methods with recycling can
be used to efficiently integrate previously computed information for faster and
better reconstruction. Additional benefits of the proposed methods are that
various subspace selection and compression techniques can be incorporated,
standard techniques for automatic regularization parameter selection can be
used, and the methods can be applied multiple times in an iterative fashion.
Theoretical results show that, under reasonable conditions, regularized
solutions for our proposed recycling hybrid method remain close to regularized
solutions for standard hybrid methods and reveal important connections among
the resulting projection matrices. Numerical examples from image processing
show the potential benefits of combining recycling with hybrid projection
methods.
