Tensor networks for solving the time-independent Boltzmann neutron transport equation

Truong, Duc Phan; Ortega, Mario Ivan; Boureima, Ismael Djibrilla; Manzini, Gianmarco; Rasmussen, Kim Ørskov; Alexandrov, Boian S.

doi:10.1016/j.jcp.2024.112943

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Journal Article · Tue Mar 19 00:00:00 EDT 2024 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2024.112943· OSTI ID:2328630

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Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by $(i)$ diamond differencing in space, $(ii)$ multigroup-in-energy, and $(iii)$ discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10^-5.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2328630

Alternate ID(s):: OSTI ID: 2999962

Report Number(s):: LA-UR--23-29906; 10.1016/j.jcp.2024.112943

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 507; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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