Characterization of the proper generalized decomposition method for fixed-source diffusion problems

Abstract

Reduced-order models can greatly decrease the computational time required to solve a large numerical problem. Proper Generalized Decomposition (PGD) is a relatively new solution method that builds the reduced-order basis on the fly and thus does not require an expensive learning phase. By generating a reduced-order basis in the form of a finite sum of separable functions, a multi-dimensional partial differential equation can be transformed into a set of coupled one-dimensional partial differential equations, which greatly reduces the overall degrees of freedom. To date, some preliminary work has been done to implement PGD as a neutronics solver. The PGD method has been implemented to solve one-group, two-dimensional neutron diffusion eigenvalue and time-dependent problems. The work done so far has demonstrated the viability of PGD neutronics solvers, but more research is needed to warrant the deployment of PGD for practical analysis applications. The purpose of this work is to characterize the performance of the PGD solution method over a broad range of problem specifications and demonstrate the application of several proposed methodological improvements. One-group and two-group derivations of the separated PGD equations are provided along with detailed descriptions of the solution algorithms. Several practical details are discussed, including a new algorithm for selecting the inner solver tolerances in order to avoid over-solving. PGD is found to be a robust solver, with performance only moderately influenced by the scattering ratio or whether the system is absorption dominated or diffusion dominated. Heterogeneous systems with arbitrary cross sections can be solved with PGD, although performance is affected by the separability of the cross sections. By direct comparison with a standard finite element solver, substantial time savings are observed in agreement with the theoretical linear scaling behavior with respect to mesh refinement for both one-group and two-group problems.