Compressed Sensing Methods for Memory Reduction in Monte Carlo Simulations

TL;DR

Monte Carlo neutron transport simulations are memory-intensive. The authors propose overlapping coarse-tally compressed sensing and a sparse reconstruction in the discrete cosine transform domain, solved via the basis pursuit denoising objective $\min_{x}\left(\tfrac{1}{2}\|Ax-b\|_2^2+\lambda\|x\|_1\right)$. Reconstruction yields substantial memory reductions, up to 96.25% in 3D, with errors within a few standard deviations of high-fidelity references in some simple geometries; performance depends on the sparsity parameter $\lambda$ and problem geometry. The approach offers a practical path to memory-efficient MC simulations and motivates further optimization, parallelization, and extensions to additional dimensions such as energy, angle, and time.

Abstract

Monte Carlo simulations of neutronic systems are computationally intensive and demand significant memory resources for high-fidelity modeling. Compressed sensing enables accurate reconstruction of signals from significantly fewer samples than traditional methods. The specific implementation of compressed sensing investigated here involves the use of overlapping cells to collect tallies. Increasing the number of samples improves the reconstruction accuracy, although the marginal gains diminish with more samples. Reconstruction quality is strongly influenced by the sparsity parameter used in basis pursuit denoising. Across the three test cases considered, memory reductions of up to 81.25% (96.25%) are demonstrated for 2D (3D) reconstructions, with select scenarios achieving reconstruction errors within 1 standard deviation of the corresponding high-fidelity reference results.

Figures (15)

• Figure 1: (Left) An example of the overlapping cells in 2D, covering the vast majority of the problem space. (Right) A single coarse bin overlaid on a Cartesian mesh, with redistribution values shown for each cell. The matrix of redistribution values would be flattened and would be one row of the sampling matrix $\mathbf{S}$.
• Figure 2: Geometry of the pure-fission sphere problem. The sphere has a radius of 1.5 cm and its center is at the center of the cube with a side length of 4 cm.
• Figure 3: The geometry in the x-y plane of the Kobayashi dog-leg benchmark.
• Figure 4: The geometry of the modified Cooper-Larsen problem. The barrier extends 1 cm high, in line with the edge of the source.
• Figure 5: Reconstructions of the sphere problem in 2D using basis pursuit denoising for different values of $\lambda$. 300 bins of size 3$\times$3 pixels on the left; 1000 bins of size 2$\times$2 pixels on the right. The reference solution is shown in the top left of each set, run with $10^4$ particles. Errors associated with each reconstruction are plotted in Figure \ref{['fig:2D sphere errors']}.