Recycling Subspace Information for Diffuse Optical Tomography

We discuss the efficient solution of a large sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging. In particular, we analyze a number of strategies for recycling Krylov subspace information for the most efficient solution. We reconstruct three-dimensional absorption and scattering information by matching computed solutions from a parameterized model to measured data. For this nonlinear least squares problem we use the Gauss-Newton method with a line search. This algorithm requires the solution of a large sequence of linear systems. Each choice of parameters in the nonlinear least squares algorithm results in a different matrix describing the optical properties of the medium. These matrices change slowly from one step to the next, but may change significantly over many steps. For each matrix we must solve a set of linear systems involving both multiple shifts and multiple right-hand sides. We discuss strategies that minimize the overall solution time. In particular, we show how we can tune the linear solver for both the nonlinear optimization algorithm and the underlying application. Although we focus on a particular application and optimization algorithm, we feel that our approach is applicable generally to problems where many linear systems must be solved. We describe extensions to the GCRO algorithm to deal efficiently with symmetric problems and to combine subspace recycling with solving for multiple shifts using a single Krylov subspace. We provide results for two sets of numerical experiments to demonstrate the effectiveness of the resulting method