Geometric optics approximation sampling: near-field case

TL;DR

The paper introduces Geometric Optics Approximation Sampling (GOAS), a gradient-free, dimension-independent sampler based on designing a near-field reflecting surface that pushes a source measure to a prescribed target measure via a reflecting map $T$. GOAS defines the geometric optics approximation measure as the push-forward of the source measure, proves its well-posedness and stability under domain perturbations, and derives explicit Hellinger-distance error bounds between the numerical and target measures. The algorithm combines an enhanced supporting ellipsoid method to construct a continuous reflector, softmin smoothing to enable stable sampling, and ray-tracing to generate independent samples, achieving dimensional independence and competitive performance relative to MCMC, especially for complex distributions and PDE-constrained Bayesian problems. Numerical experiments—including spherical reflectors, strongly non-Gaussian targets, and Bayesian inverse problems like acoustic source localization and nonlinear ADR initial-field reconstruction—demonstrate GOAS’s robustness, accuracy, and practical utility, highlighting its potential as a scalable alternative to traditional sampling methods in high-dimensional settings with costly density evaluations.

Abstract

In this paper, we propose a novel gradient-free and dimensionality-independent sampler, the Geometric Optics Approximation Sampling (GOAS), based on a near-field reflector system. The key idea involves constructing a reflecting surface that redirects rays from a source with a prescribed simple distribution toward a target domain, achieving the desired target measure. Once this surface is constructed, an arbitrary number of independent, uncorrelated samples can be drawn by re-simulating (ray-tracing) the reflector system, i.e., push-forward samples from the source distribution under a reflecting map. To compute the reflecting surface, we employ an enhanced supporting ellipsoid method for the near-field reflector problem. This approach does not require gradient information of the target density and discretizes the target measure using either a low-discrepancy or random sequence, ensuring dimensionality independence. Since the resulting surface is non-smooth (being a union of ellipsoidal sheets) but continuous, we apply a softmin smoothing technique to enable sampling. Theoretically, we define the geometric optics approximation measure as the push-forward of the source measure through the reflecting map. We prove that this measure is well-defined and stable with respect to perturbations of the target domain, ensuring robustness in sampling. Additionally, we derive error bounds between the numerical geometric optics approximation measure and the target measure under the Hellinger metric. Our numerical experiments validate the theoretical claims of GOAS, demonstrate its superior performance compared to MCMC for complex distributions, and confirm its practical effectiveness and broad applicability in solving Bayesian inverse problems.

Figures (6)

• Figure 1.1: Given rays $m$, drawing sample $z=T(m)$ from the non-Gaussian target distribution with Banana geometry $\mu_t$ using geometric optics approximation sampling method.
• Figure 4.1: Different sequences for discretizing target distribution.
• Figure 5.1: The spherical reflecting surface (a) obtained via Algorithm \ref{['AlgSE']} and softmin smoothing with $\lambda=10^{-4}$ and $K=561$, true density and the kernel density estimations (b-e) from the GOAS ($\lambda=10^{-4}$) with increasing $K$, and Hellinger distance between true measure and the push-forward of source measure $\tilde{T}_{\sharp}\mu_s$, where $\tilde{T}$ is computed by GOAS using uniform sequence, Hammersley sequence and random sequence, plotted against $K$ (f) and $\lambda$ (g).
• Figure 6.1: Comparison of GOAS and MCMCs for non-Gaussian distribution sampling. True densities and the kernel density estimations from the GOAS and MCMCs (first five columns), and computational time (seconds) and number of model (i.e., density) evaluations versus ESS (sixth column) for the MoG distribution sampling.
• Figure 6.2: Sample points and histograms from GOAS with $\lambda=10^{-4},K=602$ and MCMC.

Theorems & Definitions (33)

• Theorem 2.1
• Remark 2.1
• Definition 3.1: supporting ellipsoid
• Definition 3.2: convex reflecting surface
• Remark 3.1
• Proposition 3.1
• proof
• Definition 3.3: weak solution
• Theorem 3.2: Existence and uniqueness of weak solution
• Theorem 3.3: domain stability