Two-level domain decomposition methods for diffuse optical tomography

Diffuse optical tomography (DOT) in the near infrared involves the reconstruction of spatially varying optical properties of a turbid medium from boundary measurements based on a forward model of photon propagation. Due to the nonlinear nature of DOT, high quality image reconstruction is a computationally demanding problem which requires repeated use of forward and inverse solvers. Therefore, it is desirable to develop methods and algorithms that are computationally efficient. In this paper, we develop twolevel overlapping multiplicative Schwarz-type domain decomposition (DD) algorithms to address the computational complexity of the forward and inverse DOT problems. We use a frequency domain diffusion equation to model photon propagation and consider a nonlinear least-squares formulation with a general Tikhonov-type regularization for simultaneous reconstruction of absorption and scattering coefficients. In the forward solver, a two-grid method is used as a preconditioner to DD to enhance convergence. In the inverse solver, DD is initialized with a coarse grid solution to achieve local convergence. We show the strong local convexity of the nonlinear objective functional resulting from the inverse problem formulation and prove the local convergence of the DD algorithm for the inverse problem. We provide a computational cost analysis of the forward and inverse solvers and demonstrate their performance in numerical simulations.