Multigrid algorithms for optimization and inverse problems

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction can be formulated as the solution to a non-quadratic optimization problem. In this paper, we discuss the use of nonlinear multigrid methods as both tools for optimization and algorithms for the solution of difficult inverse problems. In particular, we review some existing methods for directly formulating optimization algorithm in a multigrid framework, and we introduce a new method for the solution of general inverse problems which we call multigrid inversion. These methods work by dynamically adjusting the cost functionals at different scales so that they are consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid optimization methods can efficiently compute the solution to a desired fine scale optimization problem. Importantly, the multigrid inversion algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions. An application of our method to optical diffusion tomography shows the potential for very large computational savings.