In this article, we propose a new dimensionality-independent and gradient-free sampler, called Geometric Optics Approximation Sampling, which is based on the reflector antenna system. The core idea is to construct a reflecting surface that redirects rays from a source with a predetermined simpler measure towards a output domain while achieving a desired distribution defined by the projection of a complex target measure of interest. Given a such reflecting surface, one can generate arbitrarily many independent and uncorrelated samples from the target measure simply by dual re-simulating or rays tracing the reflector antenna system and then projecting the traced rays onto target domain. In order to obtain a desired reflecting surface, we use the method of supporting paraboloid to solve the reflector antenna problem that does not require a gradient information regarding the density of the target measure. Furthermore, within the supporting paraboloid method, we utilize a low-discrepancy sequence or a random sequence to discretize the target measure, which in turn yields a dimensionality-independent approach for constructing the reflecting surface. Meanwhile, we present a dual re-simulation or ray tracing method based on its dual reflecting surface, which enables drawing samples from the target measure using the reflector antenna system obtained through the dimensionality-independent method. Several examples and numerical experiments comparing with measure transport samplers as well as traditional Markov chain Monte Carlo simulations are provided in this paper to demonstrate the efficiency and applicability of our geometric optics approximation sampling, especially in the context of Bayesian inverse problems. Additionally, these numerical results confirm the theoretical findings.

翻译：本文提出一种新的与维度无关且无需梯度的采样器，称为几何光学近似采样，其基于反射面天线系统。核心思想是构造一个反射面，将来自具有预定简单测度的光源的射线重定向至输出域，同时实现由复杂目标测度的投影所定义的期望分布。给定这样一个反射面，仅需对反射面天线系统进行对偶重模拟或射线追踪，然后将追踪到的射线投影至目标域，即可从目标测度中生成任意多个独立且不相关的样本。为获得期望的反射面，我们采用支撑抛物面方法求解反射面天线问题，该方法无需目标测度密度的梯度信息。此外，在支撑抛物面方法中，我们利用低差异序列或随机序列对目标测度进行离散化，从而得到一种与维度无关的反射面构造方法。同时，我们提出基于其对偶反射面的对偶重模拟或射线追踪方法，使得能够使用通过维度无关方法获得的反射面天线系统从目标测度中抽取样本。本文提供了若干示例和数值实验，与测度传输采样器以及传统马尔可夫链蒙特卡洛模拟进行比较，以展示我们几何光学近似采样的效率和适用性，特别是在贝叶斯反问题背景下。此外，这些数值结果验证了理论发现。