Random Ordinates Method (ROM)

• Random Ordinates Method (ROM) is an angular discretization technique that mitigates ray effects in radiative transport using randomized sampling and ensemble averaging.
• It partitions the velocity space into cells and randomly selects one direction per cell to perform DOM-like solves, thereby reducing non-physical streaks and oscillations.
• ROM maintains comparable computational cost to DOM while naturally supporting parallel execution, making it ideal for distributed computing environments.

Searching arXiv for papers on the Random Ordinates Method and related ray-effect mitigation in radiative transport. The Random Ordinates Method (ROM) is an angular discretization technique for the Radiative Transport Equation (RTE) developed to mitigate the ray effect, a longstanding artifact of the classical Discrete Ordinates Method (DOM). ROM partitions the velocity space into cells, randomly selects one direction from each cell, solves a DOM-like discrete transport system for each random realization, and estimates macroscopic quantities by averaging over multiple independent samples. In the formulation reported for radiative transport, ROM is presented as having computational cost comparable to that of DOM, requiring minimal changes to existing DOM-based code, and being naturally parallel because each random sampling is independent (Li et al., 2024).

1. Problem setting and motivation

DOM is the most widely used velocity discretization method for simulating the RTE, but it is prone to the ray effect when the velocity discretization is not sufficiently refined (Li et al., 2024). In the reported benchmarks, the ray effect is associated with low regularity in the velocity variable of the solution. Its visible manifestations include artificial streaks, non-physical oscillations, and loss of rotational invariance in computed macroscopic quantities such as particle density, especially when the number of ordinates is small or the source and boundary data are non-smooth (Li et al., 2024).

Standard mitigation in DOM consists of increasing the number of ordinates, but this is computationally expensive and does not fundamentally alter the mechanism that produces directional artifacts (Li et al., 2024). ROM addresses the same difficulty by changing the angular discretization itself: instead of confining transport to a fixed set of deterministic directions, it introduces randomized ordinates whose artifacts vary from run to run and can be reduced by ensemble averaging (Li et al., 6 Oct 2025).

A central point in the reported analysis is that the gain from ROM is most pronounced when regularity in the velocity variable is low. In that regime, deterministic quadrature can be accuracy-limited, whereas randomization and averaging improve both the visual behavior of the solution and the convergence of macroscopic observables (Li et al., 2024).

2. Construction of the method

The ROM construction begins by partitioning the velocity space $S$ into $n$ cells $S_1,S_2,\ldots,S_n$. One direction is then selected randomly from each cell, typically with uniform probability inside the cell, producing a randomized quadrature set for one sample (Li et al., 2024). The associated angular quadrature has the form

$$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$

with weights

$\omega_\ell = |S_\ell|/|S|.$

This preserves the DOM workflow at the level of a single solve while replacing the fixed ordinate set by a random one (Li et al., 2024).

For one realization, the discrete transport system is written as

$$u_\ell \cdot \nabla \psi_\ell(z) + \sigma_T(z)\psi_\ell(z) = \sigma_S(z) \sum_{\ell'=1}^n \omega_{\ell'} P_{\ell',\ell}\psi_{\ell'}(z) + q_\ell(z),$$

with $P_{\ell',\ell} = P(u_{\ell'},u_\ell)$ (Li et al., 2024). In the 2025 convergence paper, the same idea is expressed as a DOM-like solve indexed by a random sample $\xi$, followed by averaging across independent realizations (Li et al., 6 Oct 2025).

Macroscopic quantities are computed after each realization in the usual manner. For example, the average density is

$$\phi^\xi(\mathbf{z}) = \sum_{\ell} \omega_\ell \psi_\ell(\mathbf{z}).$$

The ROM estimator is then obtained from multiple independent runs, each with a different randomized ordinate set, by averaging the resulting macroscopic fields (Li et al., 2024). Optional symmetry in the selection of ordinates is reported as a possible performance improvement, but the basic method does not require tuning to a specific problem setup (Li et al., 2024).

3. Error, bias, and convergence theory

The theoretical analysis distinguishes between the error of a single randomized solve and the bias of the ensemble mean. In the notation used in the convergence study, the single-run error is

$\mathcal{E}:=E\|\phi^\xi-\phi\|,$

and the bias is

$n$0

where $n$1 is the reference density (Li et al., 6 Oct 2025).

For 1D slab geometry, the reported estimates are

$n$2

and

$n$3

with $n$4 the number of cells or ordinates, $n$5 a constant, and $n$6 depending on the problem parameters (Li et al., 6 Oct 2025). The 2024 paper states the same one-sample error and bias rates and reports formal higher-dimensional behavior: the bias is $n$7 and the root-mean-square error is $n$8 in $n$9 dimensions (Li et al., 2024).

The interpretation given in the 2025 analysis is that, for rough or low-regularity solutions, DOM may exhibit only low-order convergence, while a single ROM realization achieves a higher order and the ensemble average improves the order further, reaching order $S_1,S_2,\ldots,S_n$0 in the slab setting reported there (Li et al., 6 Oct 2025). This is tied to statistical cancellation: different random realizations generate different ray patterns, so averaging suppresses systematic angular artifacts that remain visible in any single deterministic DOM solve (Li et al., 6 Oct 2025).

The mathematical framework reported for these estimates uses operator expansions via Neumann series, an error decomposition based on differences between random and mean operators, and a Rosenthal-type inequality for operator-valued random terms. A stated bound is

$S_1,S_2,\ldots,S_n$1

which controls sample-to-sample fluctuation in the operator discretization (Li et al., 6 Oct 2025). The same study also emphasizes that bias is systematic because, in general,

$S_1,S_2,\ldots,S_n$2

so increasing the number of samples reduces variance but does not remove bias unless the angular mesh is refined (Li et al., 6 Oct 2025).

4. Computational structure and implementation

ROM is presented as a drop-in modification of the quadrature stage of DOM rather than a replacement of the broader transport-solver infrastructure. Each random sample corresponds to a standard DOM run with a different set of ordinates and weights; the spatial discretization, iterative solvers, and the rest of the existing DOM framework remain unchanged (Li et al., 2024). The reported implementation change is therefore concentrated at ordinate initialization and the storage and averaging of outputs across samples (Li et al., 2024).

A stated advantage is that each sample is independent, so the method is embarrassingly parallel (Li et al., 2024). This matters both operationally and analytically. Operationally, many small solves can be distributed across processors or cores. Analytically, the method exploits multiple independent realizations to average away ray artifacts and reduce fluctuation in macroscopic observables (Li et al., 6 Oct 2025).

The reported complexity comparison in 2D is that DOM requires $S_1,S_2,\ldots,S_n$3 computational complexity for target accuracy $S_1,S_2,\ldots,S_n$4, and ROM also requires $S_1,S_2,\ldots,S_n$5, while benefiting from parallelization and reduced ray effects (Li et al., 2024). The 2025 convergence analysis gives a more specific guideline in terms of a deterministic DOM discretization with $S_1,S_2,\ldots,S_n$6 directions: if DOM with $S_1,S_2,\ldots,S_n$7 directions achieves $S_1,S_2,\ldots,S_n$8 accuracy, then ROM can use only $S_1,S_2,\ldots,S_n$9 random directions per sample together with $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$0 independent samples to reach the same $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$1 accuracy, with comparable total work, fewer directions per run, and lower memory usage (Li et al., 6 Oct 2025). The same source states that $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$2 is typically $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$3, depending on solution regularity (Li et al., 6 Oct 2025).

The practical strategy summarized in both studies is therefore not to run one very large DOM solve, but to run many smaller DOM-like solves and average them. This suggests that ROM is particularly compatible with distributed computing environments in which memory per solve and wall-clock scalability are critical constraints (Li et al., 2024).

5. Numerical behavior and mitigation of the ray effect

The numerical evidence reported for ROM covers slab geometry, X-Y geometry, and a lattice problem (Li et al., 2024). In slab geometry, three cases are described: smooth, non-differentiable, and discontinuous inflow boundaries. For these tests, ROM bias is reported to converge at order approximately $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$4–$$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$5, while the error converges at order approximately $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$6–$$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$7, often better than DOM when low regularity limits deterministic quadrature (Li et al., 2024).

In X-Y geometry, ROM is reported to achieve order approximately $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$8 for the error and order approximately $$\int_S f(u)\, du \approx \sum_{\ell=1}^n \omega_\ell f(u_\ell),$$9 for the bias in estimating the macroscopic density, matching the theoretical predictions summarized for that setting (Li et al., 2024). By contrast, DOM convergence is described as slow in the same problems because of non-smoothness and pronounced ray effects (Li et al., 2024).

The most direct demonstrations concern localized-source and lattice-source problems. There, DOM is reported to produce clear rays in density plots regardless of spatial grid refinement or quadrature nodes, while ROM “blurs” these artificial rays because each realization contains a different set of rays and their average cancels the directional artifact (Li et al., 2024). The numerical summary states that even with very few ordinates per run, such as $\omega_\ell = |S_\ell|/|S|.$0, averaging over many runs makes the ray effect vanish and improves accuracy (Li et al., 2024).

A recurrent misconception is that randomization alone should be sufficient. The reported results distinguish sharply between a single random realization and the ensemble average. A single realization still carries fluctuation, whereas the averaged solution is the object for which faster bias convergence and strong ray-effect mitigation are observed (Li et al., 6 Oct 2025). Another misconception is that ray effects are primarily a spatial discretization issue; the lattice and localized-source demonstrations indicate instead that refining the spatial grid does not remove the angular artifact generated by fixed ordinates (Li et al., 2024).

6. Relation to other randomized and quasi-random angular discretizations

ROM belongs to a broader family of approaches that alter angular sampling to reduce beam-like artifacts. Within that family, the quasi-random discrete ordinates method (QRDOM) replaces pseudo-random sampling by low-discrepancy quasi-random sequences such as the reverse Halton sequence or $\omega_\ell = |S_\ell|/|S|.$1, integrated into source iteration (Konzen et al., 2023). QRDOM preserves the main characteristics of DOM, mitigates the ray effect, and is described as using a deterministic structure in which each iteration is equivalent to a DOM solve with its own direction set (Konzen et al., 2023).

The relationship between the two methods is stated explicitly: QRDOM can be regarded as the deterministic, quasi-random variant of ROM (Konzen et al., 2023). The distinction is consequential. Monte Carlo-style random ordinates reduce ray effects because they do not privilege fixed directions, but they are described as suffering from slow $\omega_\ell = |S_\ell|/|S|.$2 convergence and high variance (Konzen et al., 2023). ROM retains a DOM-like discrete system and leverages ensemble averaging to obtain the bias and error orders reported in the convergence analysis, particularly in low-regularity regimes (Li et al., 6 Oct 2025). QRDOM, by contrast, seeks more uniform angular coverage through low-discrepancy sequences and is reported to converge as $\omega_\ell = |S_\ell|/|S|.$3 in smooth problems under quasi-Monte Carlo assumptions (Konzen et al., 2023).

The 2025 convergence analysis situates ROM relative to earlier ray-effect mitigation strategies by stating that it provides a first rigorous proof for a randomized ray effect mitigation strategy, complementing previous works that only offered heuristic or empirical justification, including rotated quadratures and adaptive methods (Li et al., 6 Oct 2025). This suggests a broader methodological significance: ROM is not only a computational recipe but also a framework in which bias, variance, and cost can be analyzed quantitatively for transport discretizations affected by low angular regularity.