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Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry

Markus Bachmayr, Riccardo Bardin, Matthias Schlottbom

TL;DR

This paper develops a low-rank tensor product framework for the RTE in plane-parallel geometry by exploiting the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products.

Abstract

The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applications, and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products. This equation is solved by a preconditioned and rank-controlled Richardson iteration in Hilbert spaces. Using exponential sums approximations we construct a preconditioner that is compatible with the low-rank tensor product framework. The use of suitable preconditioning techniques yields a transformation of the operator equation in Hilbert space into a sequence space with Euclidean inner product, enabling rigorous error and rank control in the Euclidean metric.

Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry

TL;DR

This paper develops a low-rank tensor product framework for the RTE in plane-parallel geometry by exploiting the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products.

Abstract

The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applications, and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature of the phase space to recover an operator equation where the operator is given by a short sum of Kronecker products. This equation is solved by a preconditioned and rank-controlled Richardson iteration in Hilbert spaces. Using exponential sums approximations we construct a preconditioner that is compatible with the low-rank tensor product framework. The use of suitable preconditioning techniques yields a transformation of the operator equation in Hilbert space into a sequence space with Euclidean inner product, enabling rigorous error and rank control in the Euclidean metric.
Paper Structure (20 sections, 12 theorems, 82 equations, 5 figures, 11 tables, 2 algorithms)

This paper contains 20 sections, 12 theorems, 82 equations, 5 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Weak formulation and Galerkin approximation of the even-parity equations
  3. Function spaces
  4. Weak formulation
  5. Galerkin approximation
  6. Structure of system matrix
  7. Matrix representation for the Riesz map
  8. Preconditioning via exponential sums approximations
  9. Transformed linear system and iterative method
  10. Rank controlled iteration
  11. Numerical realization
  12. PN-FEM method
  13. SN-FEM method
  14. Computational complexity and storage requirements
  15. Inexact evaluation of residuals
  16. ...and 5 more sections

Key Result

Lemma 2.1

\newlabellem:trace0 There exists a bounded linear operator $\tau: W^2(\Omega) \to L^2(\partial\Omega_-;\mu)$, which satisfies for any $v\in W^2(\Omega)$ and $C_{\mathrm{tr}} \colonequals {2}/\sqrt{1-\exp(-2Z)}$. Moreover, $\tau v$ can be identified with $\tau_0 v=v(0,\cdot)$ and $\tau_Z v=v(Z,\cdot)$ if $v\in C^0(\overline{\Omega})$.

Figures (5)

  • Figure 1: Behaviour of iterates ranks (left) and thresholding parmeter (right) for the $P_N$-FEM (top) and $S_N$-FEM (bottom) methods applied to test case \ref{['eq:test_case_1']} with Richardson tolerance $\varepsilon=10^{-7}$.
  • Figure 2: Ranks of the intermediate objects $\mathbf W_{\eta_k}$ produced in lines $5$ and $12$ of \ref{['alg:inexact_residuals']} for the $S_N$-FEM method, with $J$ elements in space and $N$ subdivisions in angle, applied to test case \ref{['eq:test_case_1']}.
  • Figure 3: Behaviour of iterates ranks for the $P_N$-FEM and $S_N$-FEM methods applied to test case \ref{['eq:test_case_2']} with algebraic (top row) and exponential (bottom row) decay of the singular values.
  • Figure 4: Behavior of iterates ranks (top left), thresholding parameter (top right), Frobenius norm of the residuals (bottom left) and singular values of both $\mathbf W^{\varepsilon}$ and $\mathbf U^\varepsilon$ (bottom right), for the $P_N$-FEM method applied to test case \ref{['eq:test_case_3']}.
  • Figure 5: Behavior of iterates ranks (top left), thresholding parameter (top right), Frobenius norm of the residuals (bottom left) and singular values of both $\mathbf W^{\varepsilon}$ and $\mathbf U^\varepsilon$ (bottom right), for the $S_N$-FEM method applied to test case \ref{['eq:test_case_3']}.

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.3
  • Proof 1
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7: Change of basis
  • Remark 3.1
  • Lemma 3.2
  • Proof 2
  • ...and 15 more