In search of rogue waves: a novel proposal distribution for parallelized rejection sampling of the truncated KdV Gibbs measure

TL;DR

This work presents a novel proposal distribution for efficient, parallelizable rejection sampling of the truncated KdV Gibbs measure, leveraging a Sun–Moore–inspired anisotropic Gaussian that matches the linear TKdV limit. By carefully evaluating the acceptance ratio $f/g$ and precomputing a rejection constant, the method achieves 1–6 orders of magnitude higher acceptance than spectrally uniform proposals while maintaining uncorrelated samples, in contrast to MCMC's tuning needs. The approach enables generation of independent wave-field ensembles, including extreme, rogue-like events, across parameter regimes relevant to laboratory experiments. The ability to produce large, skewed, and localized wave fields rapidly has practical implications for predicting and characterizing rogue waves and for constructing databases to support downstream simulations and data-driven warning mechanisms.

Abstract

The Gibbs ensemble of the truncated KdV (TKdV) equation has been shown to accurately describe the anomalous wave statistics observed in laboratory experiments, in particular the emergence of extreme events. Here, we introduce a novel proposal distribution that facilitates efficient rejection sampling of the TKdV Gibbs measure. Within parameter regimes accessible to laboratory experiments and capable of producing extreme events, the proposal distribution generates 1-6 orders of magnitude more accepted samples than does a naive, uniform distribution. When equipped with the new proposal distribution, a simple rejection algorithm enjoys key advantages over a Markov chain Monte Carlo algorithm, include better parallelization properties and generation of uncorrelated samples.

Figures (9)

• Figure 1: The value $\alpha^*$ used in the approximating measure \ref{['dgSig2']} is selected as the root of \ref{['F_alpha']}. (a) $F(\alpha)$ from \ref{['F_alpha']} with $K = 32$ and $\beta' = 20$. (b) Root $\alpha^*$ for increasing values of $K$ with $\beta' = 20$.
• Figure 2: Visualization of the Dirichlet kernel for $K =$ 16 and 32. This function is used as the initial guess to numerically optimize the ratio $f/g$ on the unit hypersphere.
• Figure 3: Visualizing wavefield statistics in the case of linear TKdV ($C_3/C_2 = 0$) with $\beta' = 40$. (a) A representative wave field sampled by the rejection algorithm. (b) Histogram of surface displacements over 5,000 samples. The distribution is symmetric and nearly Gaussian. (c) Power spectrum averaged over all 5,000 samples. The spectrum decays gradually with wavenumber.
• Figure 4: The effects of strong nonlinearity, $C_3/C_2 = 120$, with $\beta' = 40$ as before. (a) The representative wave field features a prominent peak occurring near $\xi = 1.5$. (b) The histogram exhibits positive skewness, indicating that large, positive displacements are consistently favored in the ensemble of 5,000 accepted samples. (c) The spectrum decays gradually, similar to the previous case.
• Figure 5: The parallel speedup of the rejection algorithm versus the number of processors. Test conducted on an Apple M2 Ultra chip with 16 performance cores. The cutoff wavenumber is $K = 128$ and the number of nominal samples per thread is $10^4$.