Data-Driven, Projection-Based Model Order Reduction with Proper Orthogonal Decomposition for Parametric Radiation Transport

Abstract

This work explores the application of data-driven, projection-based, reduced-order models (ROMs) to parametric steady-state, neutral particle, no-fission, source-driven radiation transport. In the offline phase, ROM basis vectors are obtained using Proper Orthogonal Decomposition (POD) with the method of snapshots. In the online phase, the ROM is obtained by projecting the parametric full-order model (FOM) onto a reduced subspace to obtain a low rank system to solve for the expansion coefficients. The projection step of the ROM involves the computation of the action of the parametric transport matrices on each basis vector. There are several ways to perform this computation, each having implications on the intrusiveness and speedup of the ROM. Two such methods are studied in this work.

The first method, a sweep-based ROM, is based on the sweep-preconditioned form of transport. The action of the transport operator on each basis vector is computed by performing a full-order transport sweep. This is minimally invasive to the full-order code. Because the transport operators are parametric, the full-order sweeps must be recomputed for each new parameter realization in the online phase, slowing the ROM considerably. While the sweep-based ROM yields accurate transport solutions relative to the FOM, the speedup is modest for the reasons listed above.

The second method, called an affine decomposition-based ROM, is based on the un-preconditioned form of the transport. When written in this form, the transport operators are affine with respect to their parameters, which may be factored out. This allows for the precomputation of the parameter-independent components of the projection terms in the offline phase. The parametric reduced operators are rapidly built in the online phase using these precomputed terms. Because all full-order computations are performed in the offline phase, the affine-based ROM has potential for great speedup in the online phase. However, decomposing and projecting the transport operators is very code-intrusive to the FOM solver. This ROM was found to yield accurate solutions relative to the FOM, as well as large speedups.

For the subset of transport problems that are streaming-dominated with localized sources, solutions often vary over many orders of magnitude. This variation can be so large that significant data corruption occurs in POD when computing the singular value decomposition (SVD) of the snapshot matrix. A domain decomposition-like approach called multiresolution POD (mrPOD) is proposed to mitigate this data corruption issue. The method works by splitting the spatial domain into regions where the snapshots vary less than over the whole domain. POD is performed separately on each region to yield local basis vectors for the ROM. Because a region’s data has less variance, a more accurate SVD is possible. mrPOD ROMs are found to yield more accurate transport solutions for deep-penetration problems than POD ROMs.