Tensorized Discontinuous Isogeometric Analysis Method for the 2-D Time-Independent Linearized Boltzmann Transport Equation

TL;DR

TDIGA introduces a TT-based Tensorized Discontinuous Isogeometric Analysis framework for the 2-D time-independent LBTE discretized with discrete ordinates in angle and multigroup energy, leveraging NURBS-based geometry for exact CAD representations. The method assembles angular, interior spatial, and boundary operators in TT format and solves fixed-source and k-eigenvalue problems with unpreconditioned restarted GMRES, complemented by a mixed TT/CSR approach to manage boundary-rank growth. Across homogeneous and multi-patch geometries, interior operators exhibit three- to four-order-of-magnitude compression relative to CSR, while highly curved boundaries induce higher TT ranks and slower time-to-solution unless mixed representations are used. Compared with Monte Carlo and analytic benchmarks, TDIGA delivers high-fidelity transport on expensive high-order IGA meshes, though practical use will require tensorized preconditioners and potentially domain-decomposition strategies, especially for 3-D cases. The work highlights strong interior-operator compression, identifies boundary-coupling limitations, and outlines clear paths toward preconditioning, adaptive refinement, and 3-D extensions to fully exploit TT formats in nuclear transport applications.

Abstract

We present the novel Tensorized Discontinuous Isogeometric Analysis (TDIGA) method applied to the discontinuous Galerkin (DG) time-independent 2-D linearized Boltzmann transport equation (LBTE) with higher-order scattering, discretized with discrete ordinates in angle, multigroup in energy, and isogeometric analysis (IGA) in space. We formulate operator assembly in the tensor train (TT) format, producing seven-dimensional operators for both fixed-source and $k$-eigenvalue neutron transport problems solved using the restarted Generalized Minimum Residual Method (GMRES) and power iteration with an uncompressed solution vector. Our results on single-patch homogeneous and multi-patch heterogeneous problems, including a cruciform-shaped fuel array inspired by advanced reactor fuel designs, demonstrate the TT format's ability to compress interior operators from petabytes to megabytes, whereas the Compressed Sparse Row (CSR) matrix format requires gigabytes of storage. However, highly coupled boundary operators present a significant challenge for TT. Despite the storage savings, TT formatted operators increase time-to-solution relative to CSR as an uncompressed solution vector forces operator-vector product scaling of $O(dr^2N^d\log(N))$ for TT while CSR scales at $O(\text{nnz})$. We mitigate this discrepancy by using mixed formats with interior operators in TT, while high-rank boundary operators remain in CSR format. We compare all results to Monte Carlo (MC) and analytic reference solutions. While CSR remains $<10\times$ faster than this mixed format, the TDIGA method enables high-fidelity transport for expensive high-order IGA meshes.

Figures (38)

• Figure 1: NURBS patches for a pinell CAD model.
• Figure 1: Example tensor network diagrams for a vector $\mathcal{X}\in\mathbb{R}^{N_1}$, matrix $\mathcal{A}\in\mathbb{R}^{N_1\times N_2}$, and a three-dimensional tensor $\mathcal{T}\in\mathbb{R}^{N_2\times N_3\times N_4}$. We also show the diagram for a matrix vector product of $\mathcal{A}$ and $\mathcal{X}$ over $i_1$ and for a contraction of $\mathcal{A}$ and $\mathcal{T}$ over $i_2$.
• Figure 2: Mapping of a knot span in a NURBS circle for the (a) physical, (b) parametric, and (c) parent element spaces.
• Figure 3: Visualization of a tensor train (TT) (a) vector and (b) operator (matrix). For a tensor $\mathcal{X}\in\mathbb{R}^{N_1\times N_2\times\dots\times N_d}$ we can decompose it into $d$ three-dimensional tensors connected by tensor products over the ranks $r_j\in\{r_1, \dots,r_{d - 1}\}$. The operator tensor $\mathcal{A}\in\mathbb{R}^{N_1\times N_1'\times N_2\times N_2'\times\dots\times N_d\times N_d'}$ follows similarly with an additional dimension on each core, resulting in a linear network of $d$ four-dimensional tensors. The labeled circles represent tensors, the lines connecting the tensors indicate a tensor contraction over the contracted dimensions, and the dangling lines labeled with an index are the free indices that are open to contract with other tensor networks (TNs).
• Figure 4: A diagram of the tensor Kronecker product of a tensor train (TT) $\mathcal{X}\in\mathbf{R}^{N_1\times N_2\times \cdots \times N_d}$ and a vector $\mathbf{c}\in\mathbb{R}^{N_{d + 1}}$, \ref{['eq:kronecker']}. The result, $\mathcal{Y}\in\mathbb{R}^{N_1\times N_2\times \cdots \times N_d\times N_{d + 1}}$, is a TT with the cores of $\mathcal{X}$ coupled to $\mathbf{c}$ by a tensor product of rank $r_{d} = 1$ between $\mathcal{G}_d^{\mathcal{X}}$ and $\mathbf{c}$.

Theorems & Definitions (7)

• Definition 1: Tensor Kronecker Product
• Definition 2: Core Transpose
• Definition 3: Core Diagonalization
• Definition 4: Permutation
• Definition 5: TT-vector of Ones
• Definition 6: Identity Matrix in TT Format
• Definition 7: Indicator Vector