AI 中文总结
该论文针对含结构与概率歧义的随机逆问题，提出结合混合模型密度的贝叶斯反演算法，经数值实验验证其可解决此类逆问题的概率歧义。
AI 中文摘要
在这篇简短论文中，我们研究一类计算型随机逆问题，该类问题通过前向模型中的非线性随机变量参数以及加性观测不确定性纳入模型不确定性。具有非线性参数依赖的随机逆问题可能出现在工程、地球物理、图像处理或不确定性量化领域。我们引入关于结构歧义的新视角，结构歧义源于前向模型的非单射性，同时结合概率歧义，为参数分配具有独立分量的混合模型密度，这会产生可能复杂的前向模型、观测模型和后验分布。因此，前向模型中的混合模型参数可被解释为同时描述非线性歧义与逆问题不确定性的各个方面。基于贝叶斯反演，我们针对三种观测场景提出基础求解算法，这些场景可针对给定输出样本或观测到的输出密度，得到输入的后验密度。将推导的算法应用于一维和二维二次模型，我们在数值上证明了所提算法可在哪些场景下解决随机逆问题求解中的概率歧义。研究表明，让残差结构歧义在后验中显现并展示其与概率歧义的相互作用是一个相关视角，包括解的数量有限和无限的情况。
英文摘要
In this paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We study a perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to an in general complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be viewed as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The presented solution algorithms are based on Bayesian inversion and Monte Carlo computation for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. We apply the derived algorithms to analytic 1D and 2D quadratic models, to an epidemiological inverse parameter estimation and to a heat equation for inverse source localization. We numerically demonstrate in which observation scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution and in which they cannot and show the interaction patterns between these two types of ambiguities. The results suggest that the proposed perspective is useful in making residual structural ambiguities visible in highly ambiguous inverse problems, including cases with a finite and an infinite number of solutions.
