Conjugate gradient based acceleration for inverse problems
For researchers solving inverse problems with non-differentiable functionals, this work provides practical acceleration techniques, though it is incremental.
The paper reviews and applies iteratively reweighted least squares and convolution smoothing to enable conjugate gradient methods for non-differentiable inverse problems, demonstrating advantages in geotomography and multi-scale reconstruction.
The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite precision arithmetic) the exact solution after a finite number of iterations. It is thus well suited for many types of inverse problems. On the other hand, the method requires the computation of the gradient. Here difficulty can arise, since the functional of interest to the given inverse problem may not be differentiable. In this paper, we review two approaches to deal with this situation: iteratively reweighted least squares and convolution smoothing. We apply the methods to a more generalized, two parameter penalty functional. We show advantages of the proposed algorithms using examples from a geotomographical application and for synthetically constructed multi-scale reconstruction and regularization parameter estimation.