Multi-scale simulation and model order reduction for the radiative transfer equation
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Authors
Matsuda, Kimberly
Issue Date
2025-08
Type
Electronic thesis
Thesis
Thesis
Language
en_US
Keywords
Mathematics
Alternative Title
Abstract
The radiative transfer equation (RTE) models the propagation of radiation through a medium; it has applications in areas such as astrophysics, remote sensing, optical tomography, and radiative transfer. In applications such as uncertainty quantification and design optimization, the parametric RTE arises. Two computational challenges are considered here in solving the RTE. First, the RTE is multi-scale in nature due to the varying magnitude of the material properties throughout the spatial domain. Second, the unknown function in the RTE is defined in a high-dimensional phase space; simulating the RTE often requires one to solve a large algebraic system. This computational cost is further amplified when the parametric RTE is solved at many parameter values. In this thesis, we develop numerical schemes to solve the RTE and overcome these challenges. To address the multi-scale nature of the RTE, we consider a dimensionless form of the equation under a diffusive scaling. In the scattering dominated regime, the equation becomes stiff; this poses an additional challenge in designing appropriate time discretizations. In the first part of this thesis, we develop high-order asymptotic preserving (AP) schemes that are based on the micro-macro decomposition of the model and use implicit-explicit backward differentiation formula (IMEX-BDF) methods in time combined with discontinuous Galerkin (DG) methods in space and the discrete ordinates method in velocity/angle. AP schemes are advantageous for solving multi-scale equations as they correctly capture the behavior of the equations under different scales. In particular, our schemes follow from previous works that develop AP schemes with implict-explicit Runge-Kutta (IMEX-RK) methods in time combined with our spatial and velocity discretizations. Implicit-explicit (IMEX) methods are designed to solve ordinary differential equations (ODEs) containing stiff and non-stiff terms. IMEX-RK methods are a particular class of IMEX methods that use Runge-Kutta (RK) methods as the time integrator. In RK, the right-hand side of the ODE must be evaluated multiple times over one time step. In contrast, linear multistep methods such as backward differentiation formula (BDF) methods only need to evaluate the right-hand side once over one time step. This motivates us to develop AP schemes that use IMEX-BDF methods in time and systematically investigate the time discretization. We form three families of schemes that each use a particular choice of explicit and implicit terms for the time discretization and investigate their stability, accuracy, and AP property. The schemes are demonstrated to be stable and high-order accurate for the RTE under a broad range of scales. In particular, our proposed schemes compute high-resolution solutions faster than IMEX-RK schemes when the formal orders of accuracy are second or third order. In the second part of this thesis, we design reduced order models (ROMs) based on reduced basis methods (RBM) to efficiently solve the parametric steady-state RTE. Reduced order modeling seeks to build a surrogate model that computes accurate solutions to (non-)parametric partial differential equations (PDEs) at a much lower cost than more expensive full order models (FOMs). As a projection-based ROM, RBM involves the construction of a (low-dimensional) reduced basis space to represent the solution manifold over the parameter space of the PDE. A greedy procedure is used to iteratively build the reduced basis space. Because this procedure only requires the computation of the minimum number of FOM solutions, RBM has an advantage over other projection-based ROMs such as proper orthogonal decomposition (POD) that potentially require the computation of a large number of FOM solutions. Despite this, there has been limited work in developing RBM-based ROMs to solve the parametric RTE. Our work is the first to conduct a thorough formulation, investigation, and analysis of RBM-based ROMs to solve the parametric steady-state RTE. We develop four ROMs that are derived from the choice of one of two reduced basis projections combined with one of two possible error indicators. Implementation strategies are carefully designed to enhance the efficiency and robustness of our ROMs. The ROMs compute accurate solutions to the parametric RTE at a large collection of parameter values more efficiently than the FOM.
Description
August2025
School of Science
School of Science
Full Citation
Publisher
Rensselaer Polytechnic Institute, Troy, NY