Memory-Efficient Ray Tracing Methodology

• Memory-efficient ray tracing is an approach that reduces memory usage by grouping non-overlapping rays to simulate 3D diffuse radiation.
• It leverages structured ray casting and scalable wavefront schemes to decrease computational scaling from Nₘ² to Nₘ^(5⁄3), enabling efficient performance on modern GPUs and CPUs.
• The methodology eliminates costly atomic operations, enhances cache locality, and supports robust multi-node parallelism, as validated by astrophysical benchmarks.

Memory-efficient ray tracing refers to the set of algorithms, data structures, and parallelization techniques designed to minimize memory usage, data traffic, and synchronization costs in the simulation of light propagation—specifically focusing on diffuse radiation transfer in three-dimensional domains. The methodology described in the cited work introduces a ray-tracing scheme for 3D diffuse radiation that reduces computational and memory overheads by leveraging grouped non-overlapping rays, highly parallel architectures, and scalable inter-node wavefront strategies. The approach is validated across astrophysical benchmarks and is designed for deployment on modern GPUs and multi-/many-core CPUs.

1. Computational Complexity and Scaling

Ray tracing for diffuse radiative transfer in 3D meshes has historically been memory-intensive, with computational and memory costs scaling as $N_m^2$, where $N_m$ is the number of mesh cells. This is due to the necessity to model all interactions between source and destination cells for diffuse emission. The described methodology reduces this to a cost proportional to $N_m^{5/3}$ by introducing a structured set of equally spaced, parallel rays for a number of directions $N_d \propto N_m^{2/3}$. Rays are cast such that:

• The number of rays per direction equals the two-dimensional cross-section, $N_m^{2/3}$.
• Each ray traverses a line of $N_m^{1/3}$ cells.
• The total cost for all directions is then $N_m^{2/3} \cdot N_m^{1/3} \cdot N_d \sim N_m^{5/3}$.

This approach yields considerable memory savings, as the intermediate data and working sets needed for computation and communication are drastically reduced compared to traditional methods.

2. Conflict-Free Ray Grouping for Parallel Execution

One primary memory and performance bottleneck in parallel ray tracing is the occurrence of simultaneous writes by multiple rays traversing the same cell, which necessitates atomic operations or locks—these quickly become performance and memory overheads on GPU and CPU hardware.

The key innovation is to group rays so that, within each group, no two rays intersect the same cell. Each ray is started from a distinct, carefully-chosen subset of boundary cells such that their trajectories do not overlap through the domain. This enables:

• No atomic operations or locks needed (no concurrent writes).
• Predictable, regular memory access patterns, which facilitate cache locality and avoid memory bandwidth contention.
• Exclusive update regions per thread or GPU core, allowing maximal occupancy without increasing per-thread memory requirements.
• Near-perfect scaling with available computational resources.

The ray grouping is implemented by splitting the rays for each direction into a fixed number of interleaved groups per direction (e.g., 4 in 3D), with their origins selected to guarantee disjoint paths through the mesh.

3. Efficient Multi-node Parallelization: The Multiple Wave Front (MWF) Scheme

Distributed-memory parallelism (multi-node) introduces the need to transfer boundary data as rays cross processor or node domains. The introduced "multiple wave front" scheme divides all ray directions into 8 groups, corresponding to all combinations of direction cosine signs in 3D.

• Each group forms a "wave front" that progresses through the domain in a designated order, ensuring that data dependencies (rays from upstream subdomains) are respected.
• For different directional groups, wave fronts are allowed to process concurrently, maximizing resource utilization across nodes.
• Inter-node communication is minimized, scaling as $N_m^{4/3}$ (less steep than computational scaling), preventing communication costs from dominating as domain size increases.

Memory per node is limited to only the cells contained in the local subdomain and a modest per-ray state, rather than any global data structures.

4. Validation and Performance

The scheme is validated through benchmark problems such as HII region expansion, shadowing, and ionization front trapping. Results show:

• Computed fields (ionization, temperature, etc.) agree closely with established 1D and 3D codes.
• Angular fidelity is preserved so long as the number of ray directions (sampled, e.g., by HEALPix) is chosen to ensure the photon mean free path is small compared to the angular sampling scale.
• Empirical performance data confirm the $N_m^{5/3}$ computational cost scaling and demonstrate more than double the performance when using grouped rays (no atomics) versus atomic-operation-based schemes.

5. Integration of Efficiency Techniques

The methodology relies on several interlocking efficiency-minded techniques:

• Conflict-free ray grouping: avoids memory contention, locks, and atomics.
• Regular access patterns: optimize cache usage, prefetching, and memory hardware performance.
• Hierarchical angular sampling: enables tunable fidelity versus cost, leveraging HEALPix direction grids.
• Hybrid parallelization: combines OpenMP, CUDA, and MPI for optimal CPU+GPU and multi-node scaling.
• MWF wavefront advance: minimizes memory footprint and communication overhead for large-scale problems.

These techniques together enable the full utilization of state-of-the-art parallel computing hardware, with linear or near-linear scaling in both runtime and memory usage with increasing problem size and hardware resources.

6. Representative Mathematical Formulations

The memory-efficient scheme relies on key operational formulae, such as:

• Computational cost scaling:

$\text{Cost} \propto N_m^{5/3}$

• Averaged intensity update for each mesh cell (for ray direction $\hat{\mathbf{n}}$):

$$\bar{I}_\nu^{\text{ave}}(\hat{\mathbf{n}}) = \frac{\sum_j \Delta \tau_{\nu, j} \bar{I}_{\nu, j}(\hat{\mathbf{n}})}{\sum_j \Delta \tau_{\nu, j}}$$

where the sum is restricted to rays within the current group, enabling independent, conflict-free computation and memory storage.

• Scaling of communication cost for the MWF scheme:

$\text{comm. cost} \propto N_m^{4/3}$

This ensures communication does not become a dominating memory or performance cost as simulations scale.

7. Impact and Applicability

The introduced approach transforms 3D diffuse ray tracing into a task suitable for memory-limited, massively-parallel hardware. By ensuring that memory per processor remains proportional to local domain size and per-ray state, and by eliminating wasteful synchronization and redundant data movement, the scheme enables:

• Large-scale, physically accurate simulations that were previously intractable due to memory bottlenecks.
• On-the-fly coupling with hydrodynamics and time-dependent radiation fields.
• Robust deployment on current and future high-performance computing systems including GPU clusters and exascale platforms.

This methodology has become foundational for efficient, scalable radiative transfer in astrophysical and computational physics simulations where high spatial and angular resolution and memory efficiency are paramount.