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, (ii) multigroup-in-energy, and (iii) discrete ordinate
collocation leads to huge 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 and solving the
tensorized generalized eigenvalue problem in a fixed-point scheme with tensor
network optimization techniques. We validate our approach by applying it to two
realistic examples of 3D neutron transport problems, currently solved by the
PARallel TIme-dependent SN (PARTISN) solver. We demonstrate that our TT/QTT
method, executed on a standard desktop computer, leads to a yottabyte
compression of the memory storage, and more than 7500 times speedup with a
discrepancy of less than 1e-5 when compared to the PARTISN solution.Comment: 38 pages, 9 figure