In this dissertation, we attempt to overcome the "curse of dimensionality" inherent to radiation diffusion kinetics problems by employing a novel reduced order modeling technique known as proper generalized decomposition (PGD). After verifying a proposed PGD algorithm and associated solvers through various tests, we explore its performance for computing reduced-order models (ROMs) for the TWIGL benchmark as well as a diffusion variant (-DV) of the C5G7-TD0 benchmark by comparing to a reference high-fidelity solution.
For problems that exhibit sufficient spatial regularity, we show that our PGD algorithm computes accurate ROMs in less time than a reference high-fidelity calculation. By considering a variation of the TWIGL benchmark that maintains the same delayed supercritical behavior but has a homogeneous core configuration, we compute PGD ROMs with a total power history within 2.2% of a reference calculation with a speedup of nearly 13x. Additionally, for an infinite lattice variation of the C5G7-TD0-DV problem, we compute PGD ROMs with a total power history within 5% of a reference calculation; though using significantly fewer degrees of freedom, due to the problem size and the choice of solvers, the PGD ROM is as computationally expensive to calculate as the reference solution.
When introducing the stronger spatial heterogeneities of the reference benchmarks, the accuracy and timing of the proposed PGD algorithm diminishes. For TWIGL, we observe a maximum relative difference in total power of -36.048% and PGD ROMs that have significantly fewer degrees of freedom, but are as computationally expensive to calculate as the reference solution. For the fully heterogeneous C5G7-TD0-DV benchmark, the PGD solution diverges.