Using the levenberg-marquardt method for the solution of inverse transport problems with scattering

The Levenberg-Marquardt optimization method is applied to inverse transport problems in a multilayered one-dimensional spherical source/shield system. This method computes a set of unknown parameters using gradients of an error function with respect to each of the unknown parameters. Two problems are considered separately: 1) the unknown parameters are the locations (radii) of material interfaces and 2) the unknown parameters are weight fractions of the constituents of one of the shield materials. The Marquardt method has previously been applied to similar inverse problems in systems which included only unscattered decay gamma-ray lines. Here it is implemented in a system that includes a neutron source surrounded by a material that produces neutron-induced gamma rays. Scattering of both neutrons and photons is accounted for in a 250-energy group structure (130 neutron and 120 photon groups) that includes downscatter, self-scatter, and neutron upscatter. Gradients of the calculated leakages with respect to unknown parameters are calculated using an adjoint-based method. These gradients compare very favorably with gradients calculated using a central-difference approximation. Numerical test cases using two different methods of measurement simulation are considered. Exact radii and material constituents are recovered when measurements are generated using the same spatial and angular discretization as the inverse method. Less accurate results are obtained when more realistic measurements are used.