Markov chain Monte Carlo (MCMC) simulations have been widely used to generate samples from the complex posterior distribution in Bayesian inferences. However, these simulations often require multiple computations of the forward model in the likelihood function for each drawn sample. This computational burden renders MCMC sampling impractical when the forward model is computationally expensive, such as in the case of partial differential equation models. In this paper, we propose a novel sampling approach called the geometric optics approximation method (GOAM) for Bayesian inverse problems, which entirely circumvents the need for MCMC simulations. Our method is rooted in the problem of reflector shape design, which focuses on constructing a reflecting surface that redirects rays from a source, with a predetermined density, towards a target domain while achieving a desired density distribution. The key idea is to consider the unnormalized Bayesian posterior as the density on the target domain within the optical system and define a geometric optics approximation measure with respect to posterior by a reflecting surface. Consequently, once such a reflecting surface is obtained, we can utilize it to draw an arbitrary number of independent and uncorrelated samples from the posterior measure for Bayesian inverse problems. In theory, we have shown that the geometric optics approximation measure is well-posed. The efficiency and robustness of our proposed sampler, employing the geometric optics approximation method, are demonstrated through several numerical examples provided in this paper.

翻译：马尔可夫链蒙特卡洛（MCMC）模拟被广泛用于从贝叶斯推断中的复杂后验分布生成样本。然而，这些模拟通常需要对每个抽取样本的似然函数中的正演模型进行多次计算。当正演模型计算成本高昂（例如偏微分方程模型）时，这种计算负担使得MCMC采样变得不切实际。本文提出一种名为几何光学近似方法（GOAM）的新型采样方法，用于贝叶斯反问题，该方法完全避免了MCMC模拟的需求。我们的方法植根于反射面形状设计问题，该问题专注于构建一个反射面，将具有预定密度的光源发出的光线重定向到目标域，同时实现所需的密度分布。核心思想是将未归一化的贝叶斯后验视为光学系统中目标域上的密度，并借助反射面定义相对于后验的几何光学近似测度。因此，一旦获得这样的反射面，即可利用它从贝叶斯反问题的后验测度中抽取任意数量的独立且不相关的样本。在理论上，我们证明了几何光学近似测度是适定的。通过本文提供的多个数值算例，验证了采用几何光学近似方法的所提采样器的效率和鲁棒性。