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.

翻译：本文提出一种新型无梯度且维度无关的采样器——几何光学近似采样（GOAS），其基于近场反射器系统。核心思想在于构建一个反射表面，将来自具有指定简单分布光源的光线重新导向目标域，从而实现期望的目标测度。一旦该表面构建完成，即可通过重新模拟（光线追踪）反射器系统，从源分布中抽取任意数量的独立、不相关样本，即通过反射映射下的前推样本。为计算反射表面，我们采用增强支撑椭球方法求解近场反射器问题。该方法无需目标密度的梯度信息，并使用低差异序列或随机序列对目标测度进行离散化，确保维度无关性。由于所得表面为非光滑（由椭球面片并集构成）但连续，我们应用软最小平滑技术以实现采样。理论上，我们将几何光学近似测度定义为源测度通过反射映射的前推测度。我们证明该测度是良定义的，且对目标域的扰动具有稳定性，从而确保采样的鲁棒性。此外，我们推导了数值几何光学近似测度与目标测度在海灵格度量下的误差界。数值实验验证了GOAS的理论主张，展示了其相较于MCMC在处理复杂分布时的优越性能，并证实了其在求解贝叶斯反问题中的实际有效性和广泛适用性。