This paper focuses on a challenging class of inverse problems that is often encountered in applications. The forward model is a complex non-linear black-box, potentially non-injective, whose outputs cover multiple decades in amplitude. Observations are supposed to be simultaneously damaged by additive and multiplicative noises and censorship. As needed in many applications, the aim of this work is to provide uncertainty quantification on top of parameter estimates. The resulting log-likelihood is intractable and potentially non-log-concave. An adapted Bayesian approach is proposed to provide credibility intervals along with point estimates. An MCMC algorithm is proposed to deal with the multimodal posterior distribution, even in a situation where there is no global Lipschitz constant (or it is very large). It combines two kernels, namely an improved version of (Preconditioned Metropolis Adjusted Langevin) PMALA and a Multiple Try Metropolis (MTM) kernel. Whenever smooth, its gradient admits a Lipschitz constant too large to be exploited in the inference process. This sampler addresses all the challenges induced by the complex form of the likelihood. The proposed method is illustrated on classical test multimodal distributions as well as on a challenging and realistic inverse problem in astronomy.

翻译：本文聚焦于应用中常见的一类具有挑战性的反问题。前向模型为复杂的非线性黑箱，可能非单射，其输出幅度跨越多个数量级。观测数据同时受到加性噪声、乘性噪声和删失的影响。如众多应用所需，本研究旨在参数估计之上提供不确定性量化。由此产生的对数似然函数难以处理且潜在非对数凹。我们提出了一种适应性贝叶斯方法，以提供置信区间和点估计。针对多模态后验分布，即便在没有全局Lipschitz常数（或其值极大）的情况下，我们设计了一种MCMC算法。该算法结合了两个核函数：改进版的（预条件Metropolis调整Langevin算法）PMALA和多尝试Metropolis（MTM）核。当梯度光滑时，其Lipschitz常数过大而无法在推断过程中直接利用。该采样器有效应对了似然函数复杂形式带来的所有挑战。所提方法在经典多模态测试分布以及天文学中一项具有挑战性的真实反问题上得到了验证。